- Email: [email protected]
Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos
Journal Pre-proof
Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos Guoyuan Qi , Jianbing Hu , Ze Wang PII: DOI: Reference:
S0307-904X(19)30511-6 https://doi.org/10.1016/j.apm.2019.08.023 APM 12995
To appear in:
Applied Mathematical Modelling
Received date: Accepted date:
27 June 2019 20 August 2019
Please cite this article as: Guoyuan Qi , Jianbing Hu , Ze Wang , Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.08.023
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Highlights
A 4D Euler equation satisfying symplectic structure is proposed for the dynamics of 4D rigid body.
A Hamiltonian conservative chaotic system is modelled having strong pseudo-randomness in terms of various measures.
An analytic Casimir power form is given as the key factor identifying whether the system produces periodic or aperiodic orbits.
The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map.
A circuit is implemented to physically verify the existence of the conservative chaos.
* Guoyuan Qi The correspondence author Email: [email protected]
Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos Guoyuan Qia,*, Jianbing Hub, Ze Wanga a
Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300387, China b School of Mechanical Engineering, Tianjin Polytechnic University, Tianjin 300387, China
ABSTRACT In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos. Key words:4D Euler equation, Hamiltonian conservative chaos, Rigid body, Pseudo-random number, Casimir power
2
1. Introduction The Euler rotation equation describing the rotation of a rigid body using the moment of inertia of the body’s principal axes is well-known and valuable in classical mechanics [1]. 3D Euler equation plays an important role in the analysis of 3D dissipative chaotic systems. Many 3D dissipative chaotic systems can be transformed into Kolmogorov system describing the dynamics of force-dissipative rigid body or fluid system [2-7]. Therefore, a dissipative chaotic system, such as Qi system [3], Qi four-wing chaotic system [4], Lorenz system [5], brushless DC motor chaotic system [2], etc. was analogous to a force-dissipative rigid body system. The vector fields of these systems have been regarded as the force fields and were decomposed into inertial torque, internal torque, dissipative torque and external torque. In the mechanism uncovering of various dynamics these systems, the Euler equation and rigid body analog take important role. Based on the work, Qi, et al. proposed the force analysis and energy analysis to reveal the dynamical behaviors, such as sink, periodic orbit, chaos using Hamiltonian energy and Casimir energy [2-7]. However, all these systems are 3D chaotic systems, to make the analyses of force and energy for 4D chaotic systems is not possible if there is no 4D Euler equation. Euler equation has three first-order differential equations for 3D system [2]. Gluhovsky [8] constructed several models by coupling Volterra gyrostats with 5, 7, and 8 variables. Shamolin [9] described the dynamics of a 4D rigid-body using six first-order differential equations. In mathematics, rigid-body dynamics, and the structure of symplectic manifolds and fluid dynamics, building a 4D Euler equation is paramount. The model of 4D Euler equation serves for the modeling of the coupled body. Therefore, building a 4D Euler equation on four variables is necessary. Qi proposed four 4D generalized sub-bodies, and modeled a 4D Euler equation by coupling two sub-bodies of the four [10]. However, so much remaining work has not been completed in [10]. Actually, the combination of any two of those four generalized sub-bodies given in [10] can be integrated into a 4D rigid body, and there are total six combinations. Only one of the six 4D rigid bodies was provided in a concrete form, consequently only one 4D Euler equation was modeled in [10] and some simple analyses were conducted. In this paper, we propose an another 4D Euler equation for a 4D rigid body by coupling another two sub-bodies and the 3
mechanism routes to chaos using energy transition and theoretical proof are performed. Dynamical systems described by ordinary differential equation are classified into three categories: dissipative, conservative, and expanding in phase-volume depending on the sign of the divergence, f 0 , f 0 , and f 0 , respectively [11-13], where f is the vector field of the dynamical system. Most of the dissipative chaotic systems have been mathematically generated, and they are variants of either the Chua-like multi-scroll attractors or the Lorenz-like multi-wing attractors or jerk functions [14]. Because of the property of f 0 for the dissipative systems, their steady state solutions are attractors. They can be nodes, stars, foci, limit cycles, quasiperiodic and chaotic attractors [11]. The phase space approaches to zero, so dimensions of dissipative chaotic systems are fractional, which result in the poor ergodicity, because the trajectories reach to zero volume space and huge space initially the trajectories entered is not occupied. In contrast, the phase space volume is conserved. The conservative systems can exhibit concentric periodic, quasiperiodic and chaotic solutions but these are not attractors [11]. The dimension of the conservative chaotic systems is integer and equals the system dimension, which brings about a better ergodic property than the dissipative system. Consequently, the probability distribution associated with conservative chaos is relatively flat like the uniform distribution of white noise, whereas dissipative chaos has either a single peak or multiple peaks, from which the main frequency characteristics can be identified. By the identification, the information encrypted based on dissipative chaos can be attacked or decrypted. Therefore, with both having the same bandwidths, the conservative chaotic system is more suitable as a pseudo-random number generator than the dissipative chaotic system. Reports in literature concerning conservative chaotic systems are rare compared with dissipative chaotic systems, thereby deserving to bring a valuable research effort on it. Sprott [14] proposed a conservative chaotic system. Thomas [15] proposed a conservative chaotic system with sinusoidal nonlinearity. Cang [16] identified a 4D conservative chaotic system using the theory of perpetual points. These three conservative systems satisfy zero divergence. Some conservative systems do not satisfy the requirement of zero divergence, f 0 , but meets a zero sum of LEs. Analytically, it is not a conservative system in phase space. A typical system is the Nosé-Hoover oscillator [17], also called Sprott A, as Sprott also obtained the system 4
independently [18, 19] with divergence f x3 0 . However, the sum of Lyapunov exponents equals the divergence of a system in average, therefore, statistically the Nosé-Hoover oscillator is a conservative chaotic system. Cang has built two dissipative chaotic systems derived from the generalized Hamiltonian systems [20]. In special cases, both satisfy zero sum of LEs [21, 22]. Vaidyanathan and Volos [22] proposed a conservative system, for which divergence is f 2x3 and LE1 0.0395 . Mahmoud and Ahmed [23] proposed a 6-D hyper-chaotic conservative system with f 2x5 . Dong, et al. [24] proposed a class of conservative chaotic systems using the generalized Hamilton system, however, no mechanism routes to chaos have been proposed. All the conservative chaotic systems aforementioned are non-Hamiltonian chaotic systems. In the literature, Hamiltonian conservative chaotic systems are reported rarely. Hénon-Heiles system [25, 11] is a prototype Hamiltonian chaotic system with LE1 0.002 , in which two of the variables are horizontal coordinates, and the other two variables are their corresponding conjugate momenta. The system meets both volume conservation, i.e. f 0 , and Hamiltonian conservation, i.e. H 0 . Eckhardt and Hose [26] used a Hamiltonian function to analyze the dynamical system of quantum mechanics. The quantum chaotic system is also Hamiltonian conservative with LE1 0.3 . Qi proposed a Hamiltonian chaotic system with large maximum LE and large bandwidth by adjusting the parameters [10], however, the mechanism routes to chaos using Hamiltonian energy transition and theoretical proof have not been performed, and the circuit implementation, a physical proof for chaos existence, was not provided. There are four main problems associated with the existing conservative chaotic systems: (1) The maximum LE of most existing conservative chaotic systems is too small, and the bandwidth of the power spectral density (PSD) is too narrow. (2) The trajectories of non-Hamiltonian conservative chaotic systems are not steady, and it easily converges or diverges in long run, and consequently, chaos disappears. However, the trajectories of Hamiltonian conservative chaotic systems are steady, because the Hamiltonian is conserved. Therefore, modeling a Hamiltonian chaotic system is necessary but challengeable with large maximum LE. (3) Many dissipative chaotic systems have been implemented by circuit to demonstrate the physical existence and realizable. However, there has been no conservative chaotic systems have 5
been implemented by circuit because the conservation with a little parameter change will disappear, and conservative chaos will become periodic or degenerated. (4) A mechanism routes or transition using Casimir energy and Hamiltonian energy from periodic orbit to quasiperiodic orbit to chaos to higher degree chaos have not been revealed. Normally, most of the existing dissipative and conservative chaotic systems have been generated by changing some terms of typical chaotic systems such as the Lorenz system, the Chua circuit, the jerk function and the Hamiltonian function via trial and error and numerical simulation. there is almost no reasonable procedure or regime to build a new system to realize the given objectives, such as raising the value of the maximum LE or the chaotic degree or the bandwidths of the PSD. In this paper, a 4D Euler equation is proposed with four first-order equations governing the behavior of a 4D rigid body by integrating two coupled 3D rigid bodies with common axes. A symplectic structure is met and both the Hamiltonian and Casimir energies are preserved. It is proved that the 4D Euler equation only produce periodic orbit which are the interception of Hamiltonian and two Casimir functions. Based on the 4D Euler equation, a Hamiltonian conservative chaotic system is found, and the mechanism of generating chaos is uncovered in a Casimir energy analysis. The proposed Hamiltonian conservative chaos is characterized by its strong pseudo-randomness in terms of the LE values, bandwidth, probability distribution. The mechanism routes from periodic to quasiperiodic, chaos and higher degree chaos are revealed through the Hamiltonian energy transition and Poincaré map. The circuit implementation is conducted. The paper is organized as follows: Section 2 reviews the playing role of Euler equation in 3D dissipative chaotic systems. Section 3 proposes a 4D Euler equation, and its periodic solution is proved. Section 4 presents a Hamiltonian conservative chaotic system using the Casimir energy. Section 5 gives the characteristics of equilibria of the Hamiltonian conservative chaotic system. Section 6 gives the dynamical analysis of 4D Hamiltonian chaotic system. The mechanism routes from periodic to quasiperiodic and chaos are investigated in Section 7. Circuit implementation is provided in Section 8. Section 9 summarizes the paper.
2. Role of Euler equation in dissipative chaotic systems 6
2.1 Description of rotational Euler equation The Newtonian rotational equation in a space frame is expressed as
dx Γ, dt space
(1)
The equation seen in both space frame and rotating body frame is described as (Ref. [12])
dx dx ω x Γ . dt space dt body
(2)
When it is only seen in rotating body space, Eq. (2) is written as
x x ω Γ.
(3)
Eq. (3) is forced rotational Euler equation. The Euler equation for a rigid body with free force is
x xω x x,
(4)
where x x1 , x2 , x3 , ω 1 , 2 , 3 , i xi Ii i xi , i I i1 , I i is the principle moment T
T
of inertia for the group SO(3), and xi is the angular momentum, diag (1 , 2 , 3 ) . The components form of the Euler equation is
x1 ( 3 2 ) x2 x3 , x2 (1 3 ) x1 x3 ,
(5)
x3 ( 2 1 ) x1 x2 . It can also be written using a Hamiltonian vector field x J (x)H (x)
(6)
where J (x) is a symplectic matrix with
0 x3 x2 J (x) J (x) x3 0 x1 , x2 x1 0 T
(7)
and H (x) is the Hamiltonian written in the form H ( x)
1 1 x12 2 x22 3 x32 . 2
(8)
The Euler equation for the symplectic structure in classical mechanics, rigid body dynamics 7
and fluid dynamics. Recently we found that Euler equation has important role in dissipative chaotic systems. 2.2 Role of Euler equation in the forced-dissipative chaotic systems The Euler equation is part of Kolmogorov system describing dissipative-forced dynamical systems or hydro-dynamic instability. The 3D form is written as [3, 4] x J (x)H (x) x f ,
(9)
where J (x)H (x) is the inertial torque and internal torque (when H (x) includes the kinetic energy and potential energy), x is the linear dissipative torque and f is the external torque. Many dissipative chaotic systems can be transformed into the Kolmogorov type system. For instance, the Qi chaotic system is presented in the form [3] x1 a ( x2 x1 ) x2 x3 , x2 cx1 x2 x1 x3 ,
(10)
x3 x1 x2 bx3 .
There are various dynamical behaviors generated from Eq. (10). The system is modelled mathematically. What are the physical meanings of Eq. (10), like the system object, physical quantity of variables xi , the force or torque on the system? The system is a dissipative autonomous system with f (a 1 b) 0 , therefore, it should converge to a sink, because it has no external force to supply the energy lost by the dissipation. Why does it run in a chaotic manner? All the questions can be answered by the transformation of the Kolmogorov model. After a transformation [3], Eq. (10) can be written as
x J (x)H (x) x f J ( x ) K ( x ) U ( x ) x f e1 x2 x3 e2 x1 x3 e3 x1 x2 Inertial torque
(11)
c1 x2 c1 x1 0
ax1 0 x 0 , 2 bx3 b Internal Dissipative External torque torque torque
Here, term J (x)K (x) is the inertial torque being the same as Euler equation (5) (referring to the 8
[3] for the detail) released by the kinetic energy, term J (x)U (x) is the internal torque released by the potential energy. Now the mathematical model, Qi chaotic system, is analogous to a forced-dissipative rigid body system with four torques. Prior to the transformation, it is an autonomous system, however, it is a forced system because of the term f which supplies the energy losing from the dissipative terms, which explained why it does not converge to a sink but run in a chaotic behavior. In addition, the variable in the Qi system was endowed the physical quantity, i.e., xi is the angular momentum, and each parameter was provided physical point. Similarly, the Lorenz system [5] the four-wing chaotic system [4], the brushless DC motor chaotic system [2], the permanent magnet synchronous motor chaotic system [6], plasma chaotic system [7] have been transformed into the Kolmogorov system to uncover the force decomposition and energy cycling, and further reveal the mechanism of various dynamics. The Euler equation not only appears in the Kolmogorov form, but in generalized Hamiltonian form as x M (x)H (x) g (x)u(x, t ) .
(12) for generalized Hamiltonian systems, M (x) is no longer skew symmetric and can be decomposed into the sum of a skew symmetric J (x) and a symmetric matrices R(x) (the dissipation of the system). Remark 1: Both the Kolmogorov system and the generalized Hamiltonian form are the generalized Euler rotational equation with dissipative term and external force term. Euler equation takes important role in rotational forced-dissipative chaotic systems. However, Euler equation has three first-order differential equations for 3D system. To make a 4D forced-dissipative chaotic system or a Hamiltonian conservative chaotic system, it is necessary to generate a 4D Euler equation.
3. Generation of 4D Euler equation and its property Qi [10] proposed six 4D generalized sub-bodies, and modeled a 4D Euler equation by coupling two sub-bodies of those six. The two generalized sub-bodies are sub-body123 with axes 1, 2, 3 and 9
sub-body134 with axes 1, 3, 4. The coupled 4D rigid body called body and 3. And a complete 4D Euler equation for body
13
13
has common axes 1
was proposed in [10].
In this paper, we select another two 4D rigid generalized sub-bodies: Sub-body123 of axes 1, 2 and 3 and Sub-body234 of axes 2, 3 and 4 given. The generalized 4D sub-Euler equation of Sub-body123 in the group SO(4) is expressed as
Sub-body123
x3 0 x1 0
x1 0 x 2 x3 x3 x2 x4 0
x2 x1 0 0
0 1 x1 0 2 x2 , 0 3 x3 0 4 x4
(13)
0 1 x1 x3 2 x2 . x2 3 x3 0 4 x4
(14)
and the generalized Euler equation of Sub-body234
Sub-body 234
x1 0 0 x 2 0 0 x3 0 x4 x4 0 x3
0 x4 0 x2
where i I i1 , I i is the moments of inertial, and xi Iii , i 1, 2, 3, 4 the angular momenta. We integrate the two sub-bodies to generate the whole 4D rigid body: Sub-body123 coupled with
Sub-body234 with common axes 2 and 3, labelled of Body
23
23
.Therefore, a 4D Euler rigid body equation
is proposed having the Hamiltonian vector field form
23
x J 23 (x)H (x) ,
(15)
where 0 x J 23 (x) 3 x2 0
x3 0 x1 x4 x3
x2 x1 x4 0 x2
0 x3 , x2 0
(16)
with Hamiltonian H ( x)
1 1 x12 2 x22 3 x32 4 x42 . 2
Theorem 1. The 4D Euler equation is conservative in the Hamiltonian. Proof. 10
(17)
Because J 23 (x) is a symplectic matrix, we have
H (x) H (x)T J 23 (x)H (x) 0 .
(18)
The proof is complete. The conservation means that given an initial condition x 0 , H 0 is fixed, therefore, the Hamiltonian forms an ellipsoid as
1 x12 2 x22 3 x32 4 x42 2H 0 .
(19)
Remark 2: Like role of the 3D Euler equation taking in 3D rigid body, the 4D Euler equation plays important role in constructing the symplectic structure of generalized Hamiltonian system or Kolmogorov for 4D systems. It forms of the inertial part of the 4D forced-dissipative chaotic systems. The Casimir function (energy), C , also called energy-momentum [2], is an important physical quantity in rigid body dynamics. It is also very useful in uncovering the mechanism of the expansion and contraction of dynamic behavior, and the global bound of a dynamical system [1, 3-6, 10]. C is defined by the kernel whose gradient is orthogonal to the vector field of the system, i.e.,
C C T (x) x 0 .
(20)
Casimir energy is also called the total angular momentum for the 3D rotational rigid body. When the external torque is absent, the Casimir energy is conservative [12, 3-7]. Differing from 3D system having single Casimir function, we obtain two Casimir functions for system
23
satisfying Eq. (20): C1 (x)
1 2 x1 x22 x32 x42 2
(21)
and C2 (x) x1 x4 .
(22)
From Eq. (20) and (16), we obtain
C1 C1T (x) x C1T (x) J 23H (x) 0.
(23)
This means that given an initial condition x 0 , C10 is fixed, therefore, a Casimir sphere is formed 11
as
x12 x22 x32 x42 2C0 .
(24)
For C2 (x) , we have
C2T (x) J 23 (x) [0, 0, 0, 0] ,
(25)
C2 C2T (x) x C2T (x) J 23H (x) 0.
(26)
so
Therefore, both C1 (x) and C2 (x) are Casimir functions. From Eq. (26), we conclude that given an initial condition x 0 , C2 (x) x10 x40 C20 is fixed, therefore, a Casimir plane is formed as
x1 x4 C20 .
(27)
Casimir function C1 (x) is an energy function, also called internal energy. The rate of change of the Casimir energy is called the Casimir power. For 3D systems, the Casimir energy form is similar with C1 (x) and the Casimir power, we concluded in Refs. [3, 4] as the following: (1) The Casimir energy represents the existing internal energy of a forced-dissipative system. The Casimir power is the exchange rate between the supplied power and the dissipative power of the system. (2) If the Casimir power is less than zero for the system at all times, the orbit converges to a sink; if it is equal to zero, i.e., a constant Casimir function at all time, signifies a conservative system; if it is greater than zero at all times, and the orbit diverges as a source; if it is oscillating irregularly around the zero line with bound, the system produces chaos. Remark 3: Casimir power can be taken as a criterion as to whether a system can produce chaos. Theorem 2: The Casimir energy is conservative, and the 4D Euler equation produces a periodic solution which is the interception of the Hamiltonian ellipsoid, the Casimir sphere and the Casimir plane. There is no chaotic behavior produced in the 4D Euler equation. Proof. From Eq. (23), we proved that the Casimir energy is conserved. 12
Given an initial condition, the solution of the 4D Euler equation
23
satisfies Hamiltonian
ellipsoid, Eq. (19), Casimir sphere (24) and Casimir plane (27). The interception of the Hamiltonian ellipsoid, Casimir sphere generates a 3D ellipsoid. And then by intercepting the 3D ellipsoid with the Casimir plane, a periodic orbit is shaped. There is no chaotic behavior produced in the 4D Euler equation. The proof is complete.
4. Modelling of a Hamiltonian conservative chaotic system To produce chaos, we have to modify the system to make the Casimir power oscillated irregularly. System
c 23
23
becomes
x J 23c (x)H (x) ,
(28)
where 0 x J 23c 3 x2 c
x3 0 x1 x4 x3
x2 x1 x4 0 x2
c x3 . x2 0
(29)
Theorem 3. The modified system, c23 , is conservative in the Hamiltonian. Proof. Because J 23c (x) is a skew-symmetric matrix, then
H (x) H (x)T J 23c (x)H (x) 0 ,
(30)
The proof is complete. Theorem 4. The modified system,
c 23
, is not conservative in the Casimir energy if c( 4 1 ) 0 .
Proof. For system
c 23
, we have 13
c C xT x xT J 23 (x)H (x) c( 4 1 ) x1 x4 .
(31)
Therefore, the modified systems are not conservative in the Casimir energy when c( 4 1 ) 0 . The proof is complete. Theorem 4 provides a procedure of producing quasiperiodic orbit and chaos. Because all diagonal elements of the matrix for both J 23 (x) and J 23c (x) of the two systems are zero, their divergences are 4
f i 1
f x
i
xi
0.
(32)
Remark 4: The Euler equation and the modified systems preserve volume in phase space.
5. Equilibria and their stability properties of 4D Hamiltonian chaotic system System
c 23
can be written as
x1 3 2 x2 x3 c 4 x4 ,
c 23
x2 1 3 x1 x3 4 3 x3 x4 ,
(33)
x3 2 1 x1 x2 2 4 x2 x4 , x4 3 2 x2 x3 c1 x1.
By letting xi 0 , we find that system
c 23
has point equilibrium E0 at origin, line equilibria
Ex2 0, x2 , 0, 0 and Ex3 0, 0, x3 , 0 . The Jacobi matrix of system T
0 ( ) x 3 3 D 1 ( 2 1 ) x2 c1
T
( 3 2 ) x3 0 ( 2 1 ) x1 ( 2 4 ) x4 ( 3 2 ) x3
( 3 2 ) x2 (1 3 ) x1 ( 4 3 ) x4 0 ( 3 2 ) x2
c 23
is
c 4 ( 4 3 ) x3 . (34) ( 2 4 ) x2 0
By substituting E0 into Eq. (34) and making I D0 0 , we find the eigenvalues of E0 as
1,2 0, 3,4 jc 1 4 .
(35)
Hence, the system is a center at E0 . For the equilibrium line Ex2 , substituting Ex2 into Eq. (34), the eigenvalues then are 14
analytically given by
1 0, 2 a1 , 3,4
a1 a3 2
(36)
a2 ,
where
a x2 a a6 j 3a5 x22 3 a1 5 2 a4 6 , a2 j a , a , a4 4 3 a4 3a4 2 3a4 2a4
3
a6 2 2 a5 x2 a7 a7 , 3 (37) 2 ( 4 2 2 )( 2 3 ) ( 4 )( 2 3 )c 2 x2 a5 1 , a6 c 21 4 , a7 1 , 3 2 3
where j is the imaginary unit. If a1 0 , we have 1,2 0 . Through a series derivation, we have either 3,4 jc 1 4 , or 3,4 j 3a4 . For 3,4 jc 1 4 , equilibrium line Ex2 are natural elliptic. For
3,4 j 3a4 , when a4 is real number, Ex2 are also natural elliptic, and when a4 is an imaginary number, Ex2 are saddles. If a1 0 , when a1 0 , and then 2 is positive real number, and the real parts of 3,4 must be negative; if a1 0 , we have 2 is negative real number, and the real parts of 3,4 must be positive. Thus, equilibrium line Ex2 are saddles. Similarly, we obtained equilibrium line Ex3 have same properties as Ex2 .
In sum, both
Ex2 and Ex3 are either natural elliptic or saddles. They must neither be focuses nor nodes, which means there are no attracting solution for system
c 23
. This is identical to the characteristics of
conservative systems: nodes and spirals are not possible in conservative systems, however, elliptic (center) and saddle equilibrium points can and do occur [11].
6. Dynamical analysis of 4D Hamiltonian chaotic system The Matlab ODE5 (Dormand-Prince) solver has been used in the simulation, and the 15
symplectic integration algorithm has been compared. We found that the two integrators’ results are practically identical. 6.1 Dynamics of 4D Euler equation When c 0 , system
23
reduces to Euler equation 23 . From Theorem 2, it must
c 23
produce periodic orbit. Take parameters
x10 , x20 , x30 , x40
T
c 23
becomes
1 , 2 , 3 , 4
T
2, 3, 4, 5 , initial conditions T
4, 5, 5, 4 , c 0 , and sampling time T 0.005 s. For this case, system
T
23
. As concluded in Theorems 1 and 2, system
23
is conservative in both
the Hamiltonian and Casimir energy, and from Theorem 2, it only can produce a periodic orbit which is interception of the Hamiltonian ellipsoid formed by H 0 , the sphere formed by Casimir
C10 and the plane constructed by C20 as shown in Fig. 1(a). The total Hamiltonian keeps constant with H 0 143.5 , and the Casimir energy also remains the same with C10 41 [Fig. 1(b)].
(a) Periodic orbit with c 0 .
(b) Time response of Hamiltonian and Casimir energy Fig. 1 Dynamics of 4D Euler system
23
.
The system has five adjustable parameters: i , i 1, 2, 3, 4 and c , and four adjustable initial values, xi 0 , to influence the orbital modes of the system, and each is physically meaningful: the initial values, xi 0 , determine the total Hamiltonian given i from Eq. (17) because the system 16
preserves the Hamiltonian; i are proportional to the frequency of the rigid body provided the initial Hamiltonian is fixed. If c( 4 1 ) 0 , the system either produces a quasi-periodic orbit or a chaotic orbit depending on the initial energy. The value of c( 4 1 ) quantitatively influences the orbital pattern and the degree of chaos. The system is conservative in phase volume from Remark 4; therefore, the system’s orbits cannot reach an equilibrium point asymptotically. This implies nodes and spirals are not possible in the conservative system. However, elliptic (center) and hyperbolic (saddle) equilibrium points do occur. We analyze what are the possible route and mechanisms in the transition from pseudo-periodic orbits to chaos for the system under condition c( 4 1 ) 0 . 6.2 Ordinary Hamiltonian conservative chaos All the parameters and initials remain the same as Section 6.1 except for c 3 , choosing sampling time, the orbits of system
c 23
is chaotic [Fig. 2(a), (b) and (c)], which can be verified
by the LEs LE1,2,3,4 [2.4241, 0, 0, 2.4241]T . The maximum of LEs is quite large compared with those of the conservative chaotic systems in aforementioned references. From Fig. 2(b) and (c), the orbits are seen not to approach a chaotic attractor asymptotically but visits the whole space that it initially enters that being identical to the integer dimension,
LD 4 . However, dissipative chaos asymptotically reaches a chaotic attractor regardless of the initial values set. From Theorem 3, the system is conservative in the Hamiltonian with H (t ) 143.999 for the given parameter and initial values; therefore, chaos cannot be characterized using the Hamiltonian. The positive LE does indicate chaos when the system is bounded. A numerical and statistical index of chaos; however, positive LE is a numerical and statistical index instead of the analytical form. From Theorem 4, the Casimir energy gives an analytical form. If the Casimir power oscillates irregularly around the zero line, it produces chaos. Fig. 2(d) clearly demonstrates the irregular oscillation.
17
(b) Phase portrait of x1 x2
(a) Time series of x2 and x3
(c) Phase portrait of x1 x3
(d) Casimir power of x2
Fig. 2. Numerical characteristics of system
c 23
.
6.3 Hamiltonian chaos with strong pseudo-randomness The system
c 23
not only produces chaos, but produces chaos of high quality and strong
pseudo-randomness. This can be realized by adjusting the five parameters and four initial values according to their functions in frequency and energy. To improve the frequency, i must be increased. To speed up the stretching and folding of orbits, and enlarge the spatial magnitude of orbits, the absolute of initial values, xi 0 , must be enhanced, which is equivalent to increasing of the impulse torque pushing the rigid body for rotation. Importantly, c must be improved for increasing the Casimir power of oscillation. Set 1 , 2 , 3 , 4 100, 150, 200, 50 , c 500 , T
18
T
and initial values
x10 , x20 , x30 , x40
T
800, 1000, 1000, 800 , and the sampling time T
T 3 107 s.
The times series for x2 is shown in Fig. 3(a). Since the trajectory is too dense and noisy to display, we have to limit the time interval to t [0, 0.01]s . The magnitudes of the angular momenta are much larger than in Figs. 2(a), (b) and (c) compared with those in Sub-Section 6.2. The difference in magnitudes arises from the initial Hamiltonian. The maximum of the LEs LE1 11398.1088 [Fig. 3(c)]. It is not imaginable for its largeness compared with those of existing dissipative and conservative chaotic systems. Note that the value of LE is the good measure of the degree of chaos or pseudo-randomness [11, 27, 28]. We can use the width of Power Spectral Density (PSD) to measure the pseudo-randomness of the chaotic system. The width PSD of system in Sub-Section 6.2 is 10Hz. Note that the width is even less for other existing dissipative chaotic systems. The Qi hyperchaotic system, which is dissipative has quite a wide bandwidth of 100Hz compared with other existing hyperchaotic systems [27]. However, the bandwidth of the PSD for the Hamiltonian conservative chaotic system in this parameter settings is 57600Hz [Fig. 3(d)], which is 5760 times that in Sub-Section 6.2, and 576 times than that of the Qi hyperchaotic system. Here, Welch’s overlapped segment averaging estimator with Hamming window for the PSD in Matlab is used; the magnitude of the PSD is normalized. The bandwidth is truncated when the magnitude of the PSD is less than 0.1 [Fig. 3(d)]. In chaos-based cryptography, the bandwidth of the PSD is an important measure of a pseudo-random number in regard to the security of encrypted messages. When the bandwidth is too narrow, the encryption can be broken even using simple filtering methods. Note that since all the parameters are physically meaningful, the bandwidth of a pseudo-random number of the system can be continually increased further to MHz and even GHz by adjusting the parameters as we have confirmed. Generally, the bandwidth of a chaotic system can be widened through the time-scale transformation; however, the scaled system is no longer the original one, and the system loses physical meanings. Nevertheless, the proposed system realizes the bandwidth objective through the adjustment of parameters and initial values, which do not lose their original physical meanings. For some chaotic systems, when parameters or initials are 19
adjusted to some regions, no chaotic orbit is generated. For instance, the initial values of the Hénon-Heiles system can only be set in a very small region; When outside this region, system is unstable, but inside the region, the system orbit is periodic. To further understand the statistical properties, we calculated the histogram reflecting the distribution of the random data. Normally, the main difference of the conservative chaos from dissipative chaos is that the former probability distribution is flatter like uniform distribution of white noise than that of the latter, which has a peak or multiple peaks form as for the normal distribution of Gaussian white noise. The reason is because of its integer dimension, the former has a better ergodicity than the latter. Therefore, the pseudo-random number generated from conservative chaotic system is more suitable than that generated from a dissipative chaotic system provided that both have the same bandwidth. Fig. 3(e) gives the probability density of the Hamiltonian conservative chaotic system using histogram, for which the distribution shape is very close to that for uniform white noise. The Poincaré map also displays the ergodicity and fractal dimension of a system. The Lorenz system shows poor ergodicity in only displaying ―several slim twigs‖ on the Poincaré map. The Poincaré map of x1 x3 of the proposed Hamiltonian chaos taking x4 0 as the section of Poincaré [Fig. 3(f)] shows excellent ergodicity, because the map almost fully occupies a large region of x1 x3 plane. We further investigate the proposed new method, the Casimir power method, to characterize and find the pathway to generating chaos. As given in Theorem 4, C c( 4 1 ) x1 x4 ; see Fig. 3(g). Both the frequency and magnitude of the irregular oscillation of the Casimir power are quite high demonstrating its chaotic degree. When c 0 , the system is conservative for both the Hamiltonian and Casimir energy, theoretically the system is periodic, as verified in Fig. 3(h). A further
test
is
taken.
Keeping
c 500 ,
but
taking
1 4 100 ,
we
have
C c( 4 1 ) x1 x4 0 , so the system is periodic also, as verified in Fig. 3(i). Note that because Fig. 3(h) and (i) has different parameters in terms of 4 and c , the shapes are different even though both are periodic. Therefore, Theorem 4 is further verified, and the Casimir power is a 20
proper evaluation index that characterizes whether a system is periodic.
(b) Phase portrait of x1 x3
(a) Time series of x2
(c) Lyapunov exponents of
c 23
(d) Power spectral density of x2
(f) Poincaré map x1 x3 .
(e) Probability distribution of x2
21
(h) Periodic orbit produced with c 0
(g) Casimir power of x2
(i)
Periodic
orbit
produced
(j) Hamiltonian and Casimir energies, where
with
(j1) and (j2) have 1 4 100, c 500 , and
1 4 100, c 500
(j3) and (j4) have 1 100, 4 50, c 500 .
Fig. 3 Strong quality of pseudo-randomness of
c 23
.
The KAM perturbation theorem is well-known in analyzing Hamiltonian chaotic systems [11]. It provides a starting point for an explanation of the transition from regular to chaotic motion. From the perturbation theory, when a Hamiltonian system with function H1 is perturbed by another Hamiltonian system with H 2 , the coupled Hamiltonian system will probably generate conservative chaos with H H1 H 2 because there is energy exchange between the two Hamiltonian functions. However, when the system Hamiltonian is not easy to decompose into two 22
functions, the KAM theory cannot explain or analyze the transition from regular to chaos. We analyze the transition for this situation from a new perspective, i.e., whether its Casimir power is oscillated around the zero line. When H is constant, and the energy momentum or total angular momentum or Casimir energy is changed, so the Casimir power can explain how the energy is exchanged. System (33) is conservative in the Hamiltonian whether c(1 4 ) is zero or not [see Figs. 3(j1) and (j3)], where (j1) has c(1 4 ) 0 , and (j3) does not. However, when c(1 4 ) 0 , there is Casimir energy exchange internally because terms c 4 x4 and c1 x1 trigger the changes in xi , i 1, 2, 3, 4 , the rates of change of angular momentums and the changes of in Casimir power,
C c(1 4 ) x1 x4 . Therefore, both terms equivalently take the mix role of dissipative and supplied energy in the total Casimir energy, because it is not clear about which one is dissipative or supplied, which is the main cause of generation of chaos as concluded in Refs. [3, 4]. Fig. 3(j2) shows that there is no Casimir energy exchange when c(1 4 ) 0 , whereas Fig. 3(j4) shows that terms c 4 x4 and c1 x1 strongly impact on the Casimir energy when c(1 4 ) 0 .
7. Role of Hamiltonian energy transition in dynamics change There are three predominant routes to chaos, namely, period doubling, quasiperiodic and intermittency bifurcations. The conservative chaotic systems preserve the phase space, hence, the chaos route is quasiperiodic to chaos. Although, Casimir power qualitatively determines whether the system produces periodic orbit or not, the Hamiltonian value quantitatively determines the degree of disorder of system
c 23
under condition that Casimir power is not equal to zero. Fixing parameters as
1 , 2 , 3 , 4
T
2, 3, 4, 5 , c 3 , T
x0 = 1.4, 1.4, x30 , 1.4
T
,
and
x30 [5, 5]
which
corresponds to the Hamiltonian H 0 [9.8, 59.8] , where H0
1 2 2 2 1 x102 2 x20 3 x30 4 x40 . 2 23
(38)
The bifurcation diagram of initial conditions along with their energy level is plotted in Fig. 4. The stripe colors represent Hamiltonian values labeled on energy bar. Normally, the low Hamiltonian energy (the dark blue and the blue) represents quasiperiodic orbits, and the high Hamiltonian energy (the light blue and the yellow) represents chaos. It hence can be inferred that system
c 23
with higher energy has a wider dynamic field or frequency domain. We may clearly find that the Hamiltonian energy level (color stripe) leap essentially labels the transition of the dynamics of the system.
Fig. 4 Bifurcation diagram of initial conditions corresponding to various energy levels.
There are complicated coexisting dynamics for system
c 23
. Under same Hamiltonian,
different initial conditions may produce different quasiperiodic orbits, even coexist both chaos and quasiperiodic orbits. To visualize the geometry of complicated orbits and investigate the coexistence and transition from quasiperiodic motion to chaotic motion in system
c 23
, the
Poincaré maps on surface of sections (SOS) for trajectories in terms of various Hamiltonian energy levels are plotted in Fig. 5. The SOS is taken as the extremals of Casimir energy, i.e.,
C1 (x) c( 4 1 ) x1 x4 0 . Take x0 = 1.4, 1.4, x30 , x40 , and H 0 13.18,17.02, 23.32, 34.3 , T
respectively. For each H 0 , as x30 varies, x40 must be correspondingly varied to keep the given
H 0 through Eq. (38). For each given H 0 , many pairs of x30 and x40 as initials are taken for simulations. Case 1 for H 0 13.18 . In Poincaré map, there are total eight primary islands labeled by 24
Ci , i 1,
, 8 [Fig. 5(a)], the center of each island is the elliptic equilibrim point of Poincaré map.
Each island has many concentric coexisting closed curves, which are the maps of the quasiperiodic solutions generated by system
c 23
in different initials with same Hamiltonian. Some islands are
separated by the separatrices, and some are intersected. Two intersected islands probably do not intersect in 2D plane. They may have different phase spaces because the SOS is not a plane but the extremals of Casimir energy, which allows the SOS has 3D geometric space. Case 2 for H 0 17.02 . As the Hamiltonian increases, each primary island gives rise to the secondary islands [Fig. 5(b)]. This is caused by the multi-frequency and resonance of the orbits generated by system
c 23
. The separatrices are either occupied by secondary islands or little
chaos. The coexistence become more complicated. Case 3 for H 0 23.32 . As more and more secondary islands enlarge and the third or fourth islands appears. To some extent, when H 0 23.32 , some primary islands disappeared, and the peripheral portion corresponding to the previous invariant curves has dissolved into the chaotic region [Fig. 5(c)], which is called chaotic sea. Case 4 for H 0 34.30 . As the energy is further increased to higher level, the chaotic region occupied by scattered points is gradually widening, and all the invariant curves is dissolved [Fig. 5(d)]. When the energy level is high enough, the quasiperiodic motion disappears and be replaced by the chaotic motion completely.
(a) Primary islands corresponding to the quasiperiodic orbit under relatively low Hamiltonian
25
(b) Mixed primary islands and secondary islands corresponding to multi-frequency quasiperiodic orbits with higher Hamiltonian
(c) coexistence of islands and chaotic sea corresponding to coexistence of quasiperiodic orbits and chaos with higher Hamiltonian
(d) No islands corresponding to chaos with high Hamiltonian
Fig. 5. The evolution of dynamics from quasiperiodic orbit to chaos of system
c 23
as the leap of Hamiltonian
energy level.
8. Circuit implementation For the nonlinear system,
c 23
, all the numerical simulation integration methods are
calculated through digitization and iteration, such as the ODE45 method, the symplectic algorithm and the Field-Programmable Gate Array (FPGA) digital implementation. Even if they have high accuracy, they have some errors aroused from algorithm, sampling time and computer accuracy. For a conservative system, even for a little error, and the accumulation of iteration will make the conservation of energy and phase space loose, and then conservative chaos will lead to either diverging or shrinking. However, because analog electronic circuit operate physically to implement the system, it is able to verify whether the system really is conservatively chaotic and confirm the suitability of numerical method selected in the paper. Especially for conservative systems, the general integral formula may be difficult to converge the solution or energy error, moreover, the diversity of conservation makes it extremely complicated to theoretically prove and design the applicable numerical integration methods such as preserving symplectic geometry or 26
invariants. As shown in Fig. 6, system
c 23
with fixed chaotic parameters is implemented by an analog
electronic circuit. Six quadratic terms in the system are accomplished by six analog multipliers, besides, several basic arithmetic units consists of eight operational amplifiers and some linear resistors and capacitors for performing addition, subtraction, and integral operations. For the parameter choices 1 , 2 , 3 , 4 2, 3, 4, 5 , c 3 , the experimental observations from the T
T
analog oscilloscope are shown in Fig. 7(a) and (b). The figures showing in different phase planes are identical to the figures showing numerically [Fig. 2(b) and (c)]. The implementation of analog circuit for the system verified that the system is a really conservative chaos and confirmed that the ODE5 selected for numerical method is suitable.
Fig. 6. Circuit implementation of system
c 23
, with electronic parameters:
R1,R2,R3,R4,R5,R6,R7,R8,R10,R12,R13,R16 10K;R9 16.66K; R11,R14 5K;R15 6.66K ; C1,C2,C3,C4 10nF . 27
(a)
(b)
(c)
(d) Fig. 7. Implementation of analog circuit of system
c 23
.
9. Conclusion This paper proposed a 4D Euler equation by integrating two generalized 3D Euler equations. The equation will be the inertial part of a rotational forced-dissipative system. Hence, it will be significant in generating and analyzing the generalized Hamiltonian systems and Kolmogorov systems encountered in rigid-body dynamics and fluid dynamics and chaos generation. The Hamiltonian, Casimir energy and phase volume of the 4D Euler equation are preserved. The periodic solution was proved for the 4D Euler equation. Based on the 4D Euler equation, using a Casimir power method, a Hamiltonian conservative chaotic system has been presented by breaking 28
the conservation of Casimir energy. An analytic form of Casimir power has been proposed, which provides the way of producing nonperiodic orbits: quasiperiodic orbit and chaos. The routes and mechanism from quasiperiodic orbit to chaos have been provided through energy bifurcation and Poincaré map. The Hamiltonian conservative chaos with strong pseudo-randomness was produced, which is assessed in the evaluations of Lyapunov exponent, the bandwidth of the power spectral density, the probability distribution, the Poincaré map. The circuit implementation has been conducted to physically verify the existence of Hamiltonian conservative chaos. Much open topic has been left for researchers: how to construct a forced-dissipative chaotic systems based on the 4D Euler equation and what types of hidden chaos can be found in the Hamiltonian conservative chaotic systems, and so on.
Acknowledgements This work is supported by the National Natural Science Foundation of China (61873186) and the Tianjin Natural Science Foundation (17JCZDJC38300).
References [1] J. E. Marsden, T. S. Ratiu, Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems, 2nd Edition, Springer, 2002, Chapter 1. [2] G. Qi, Energy cycle of brushless DC motor chaotic system, Appl. Math. Model. 51 (2017) 686–697. [3] G. Qi, J. Zhang, Energy cycle and bound of Qi chaotic system. Chaos Soliton. Fract. 99 (2017) 7-15 [4] G. Qi, X. Liang, Mechanism and energy cycling of Qi four-wing chaotic system. Int. J. Bifurcat. Chaos., 27 (2017) 1750180-1-15. [5] V. Pelino, F. Maimone, A. Pasini. Energy cycle for the Lorenz attractor. Chaos, Solitons & Fractals 2014; 64: 67-77. [6] G. Qi, J. Hu. Force Analysis and Energy Operation of Chaotic System of Permanent-Magnet Synchronous Motor. International Journal of Bifurcation and Chaos, 2017,27, 1750216 [7] Y. Yang, G. Qi. Comparing mechanical analysis with generalized-competitive-mode analysis for the plasma chaotic system. Physics Letters A, 383 (2019) 318–327. [8] A. Gluhovsky, Energy-conserving and Hamiltonian low-order models in geophysical fluid dynamics. Nonlin. Processes Geophys. 13(2006) 125–133. [9] M. V. Shamolin, Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field. J Math S. 165(2010) 743-754. [10] G. Qi. Modelings and mechanism analysis underlying both the 4D Euler equations and Hamiltonian 29
conservative chaotic systems. Nonlinear Dynamics, 2019, 95:2063–2077 [11] M. Lakshmanan, S. Rajasekar, Nonlinear Dynamics—Integrability, Chaos, and Patterns, Springer-Verlag, Berlin, 2012. [12] J. R. Taylor, Classical mechanics, University Science Books, Sausalito, 2005, [13] J. Gao, Y. Cao, W. Tung, J. Hu. Multi scale analysis of complex time series Integration of Chaos and Random Fractal Theory, and Beyond. John Wiley & Sons. Inc. Hoboken. New Jersey. 2007, page 248. [14] J. C. Sprott, Some simple chaotic jerk functions. A. J. Phys. 65(1997) 537-543. [15] R. Thomas, Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, ―labyrinth chaos‖, Int. J. Bifurcat. Chaos 9 (1999) 1889-1905. [16] S. Cang, A. Wu, Z. Wang, Z. Chen, Four-dimensional autonomous dynamical systems with conservative flows: two-case study, Nonlinear Dyn. 89(2017)2495–2508. [17] W. G. Hoover, Remark on ―some simple chaotic flows‖. Phys. Rev. E, 51(1), (1995) 759–760. [18] J. C. Sprott, Some simple chaotic flows, Phys. Rev. E, 50(2) (1994) 647–50. [19] J. C. Sprott, Elegant chaos—Algebraically simple chaotic flows, World Scientific Publishing, 2010, Chapter 4. [20] S. Cang, A. Wu, Z. Wang, Z. Chen, On a 3-D generalized Hamiltonian model with conservative and dissipative chaotic flows. Chaos, Chaos Soliton. Fract. 99 (2017) 45–51. [21] S. Cang, A. Wu, R. Zhang, Z. Wang, Z. Chen. Conservative chaos in a class of nonconservative systems: theoretical analysis and numerical demonstrations. Int. J. Bifurcat. Chaos., 28 (2018) 1850087-1-19. [22] S. Vaidyanathan, C. Volos, Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system, Archives of Control Sciences, 25(3)(2015) 333–353. [23] G. M. Mahmoud, M. E. Ahmed, Analysis of chaotic and hyperchaotic conservative complex nonlinear systems, Miskolc Math. Notes, 18(1) (2017) 315–326. [24] E. Dong, Yuan M., et al.: A new class of Hamiltonian conservative chaotic systems with multistability and design of pseudo-random number generator. Appl. Math. Model. 73(9), 40-71(2019) [25] M. Hénon, C. Heiles, The applicability of the third integral of motion: some numerical experiments. Astrophys. J, 69 (1964) 73–79. [26] B. Eckhardt, G. Hose, Quantum mechanics of a classically chaotic system: Observations on scars, periodic orbits, and vibrational adiabaticity. Phys. Rev. E 39(1989) 3776-3793. [27] G. Qi, M. Avan Wyk, B. J. van Wyk, G. Chen, On a new hyperchaotic system. Phys. Lett. A. 372/2(2008) 124-136. [28] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos. Second Edition, Springer, New York, 2003.
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Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos Guoyuan Qi , Jianbing Hu , Ze Wang PII: DOI: Reference:
S0307-904X(19)30511-6 https://doi.org/10.1016/j.apm.2019.08.023 APM 12995
To appear in:
Applied Mathematical Modelling
Received date: Accepted date:
27 June 2019 20 August 2019
Please cite this article as: Guoyuan Qi , Jianbing Hu , Ze Wang , Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.08.023
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Highlights
A 4D Euler equation satisfying symplectic structure is proposed for the dynamics of 4D rigid body.
A Hamiltonian conservative chaotic system is modelled having strong pseudo-randomness in terms of various measures.
An analytic Casimir power form is given as the key factor identifying whether the system produces periodic or aperiodic orbits.
The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map.
A circuit is implemented to physically verify the existence of the conservative chaos.
* Guoyuan Qi The correspondence author Email: [email protected]
Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos Guoyuan Qia,*, Jianbing Hub, Ze Wanga a
Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300387, China b School of Mechanical Engineering, Tianjin Polytechnic University, Tianjin 300387, China
ABSTRACT In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos. Key words:4D Euler equation, Hamiltonian conservative chaos, Rigid body, Pseudo-random number, Casimir power
2
1. Introduction The Euler rotation equation describing the rotation of a rigid body using the moment of inertia of the body’s principal axes is well-known and valuable in classical mechanics [1]. 3D Euler equation plays an important role in the analysis of 3D dissipative chaotic systems. Many 3D dissipative chaotic systems can be transformed into Kolmogorov system describing the dynamics of force-dissipative rigid body or fluid system [2-7]. Therefore, a dissipative chaotic system, such as Qi system [3], Qi four-wing chaotic system [4], Lorenz system [5], brushless DC motor chaotic system [2], etc. was analogous to a force-dissipative rigid body system. The vector fields of these systems have been regarded as the force fields and were decomposed into inertial torque, internal torque, dissipative torque and external torque. In the mechanism uncovering of various dynamics these systems, the Euler equation and rigid body analog take important role. Based on the work, Qi, et al. proposed the force analysis and energy analysis to reveal the dynamical behaviors, such as sink, periodic orbit, chaos using Hamiltonian energy and Casimir energy [2-7]. However, all these systems are 3D chaotic systems, to make the analyses of force and energy for 4D chaotic systems is not possible if there is no 4D Euler equation. Euler equation has three first-order differential equations for 3D system [2]. Gluhovsky [8] constructed several models by coupling Volterra gyrostats with 5, 7, and 8 variables. Shamolin [9] described the dynamics of a 4D rigid-body using six first-order differential equations. In mathematics, rigid-body dynamics, and the structure of symplectic manifolds and fluid dynamics, building a 4D Euler equation is paramount. The model of 4D Euler equation serves for the modeling of the coupled body. Therefore, building a 4D Euler equation on four variables is necessary. Qi proposed four 4D generalized sub-bodies, and modeled a 4D Euler equation by coupling two sub-bodies of the four [10]. However, so much remaining work has not been completed in [10]. Actually, the combination of any two of those four generalized sub-bodies given in [10] can be integrated into a 4D rigid body, and there are total six combinations. Only one of the six 4D rigid bodies was provided in a concrete form, consequently only one 4D Euler equation was modeled in [10] and some simple analyses were conducted. In this paper, we propose an another 4D Euler equation for a 4D rigid body by coupling another two sub-bodies and the 3
mechanism routes to chaos using energy transition and theoretical proof are performed. Dynamical systems described by ordinary differential equation are classified into three categories: dissipative, conservative, and expanding in phase-volume depending on the sign of the divergence, f 0 , f 0 , and f 0 , respectively [11-13], where f is the vector field of the dynamical system. Most of the dissipative chaotic systems have been mathematically generated, and they are variants of either the Chua-like multi-scroll attractors or the Lorenz-like multi-wing attractors or jerk functions [14]. Because of the property of f 0 for the dissipative systems, their steady state solutions are attractors. They can be nodes, stars, foci, limit cycles, quasiperiodic and chaotic attractors [11]. The phase space approaches to zero, so dimensions of dissipative chaotic systems are fractional, which result in the poor ergodicity, because the trajectories reach to zero volume space and huge space initially the trajectories entered is not occupied. In contrast, the phase space volume is conserved. The conservative systems can exhibit concentric periodic, quasiperiodic and chaotic solutions but these are not attractors [11]. The dimension of the conservative chaotic systems is integer and equals the system dimension, which brings about a better ergodic property than the dissipative system. Consequently, the probability distribution associated with conservative chaos is relatively flat like the uniform distribution of white noise, whereas dissipative chaos has either a single peak or multiple peaks, from which the main frequency characteristics can be identified. By the identification, the information encrypted based on dissipative chaos can be attacked or decrypted. Therefore, with both having the same bandwidths, the conservative chaotic system is more suitable as a pseudo-random number generator than the dissipative chaotic system. Reports in literature concerning conservative chaotic systems are rare compared with dissipative chaotic systems, thereby deserving to bring a valuable research effort on it. Sprott [14] proposed a conservative chaotic system. Thomas [15] proposed a conservative chaotic system with sinusoidal nonlinearity. Cang [16] identified a 4D conservative chaotic system using the theory of perpetual points. These three conservative systems satisfy zero divergence. Some conservative systems do not satisfy the requirement of zero divergence, f 0 , but meets a zero sum of LEs. Analytically, it is not a conservative system in phase space. A typical system is the Nosé-Hoover oscillator [17], also called Sprott A, as Sprott also obtained the system 4
independently [18, 19] with divergence f x3 0 . However, the sum of Lyapunov exponents equals the divergence of a system in average, therefore, statistically the Nosé-Hoover oscillator is a conservative chaotic system. Cang has built two dissipative chaotic systems derived from the generalized Hamiltonian systems [20]. In special cases, both satisfy zero sum of LEs [21, 22]. Vaidyanathan and Volos [22] proposed a conservative system, for which divergence is f 2x3 and LE1 0.0395 . Mahmoud and Ahmed [23] proposed a 6-D hyper-chaotic conservative system with f 2x5 . Dong, et al. [24] proposed a class of conservative chaotic systems using the generalized Hamilton system, however, no mechanism routes to chaos have been proposed. All the conservative chaotic systems aforementioned are non-Hamiltonian chaotic systems. In the literature, Hamiltonian conservative chaotic systems are reported rarely. Hénon-Heiles system [25, 11] is a prototype Hamiltonian chaotic system with LE1 0.002 , in which two of the variables are horizontal coordinates, and the other two variables are their corresponding conjugate momenta. The system meets both volume conservation, i.e. f 0 , and Hamiltonian conservation, i.e. H 0 . Eckhardt and Hose [26] used a Hamiltonian function to analyze the dynamical system of quantum mechanics. The quantum chaotic system is also Hamiltonian conservative with LE1 0.3 . Qi proposed a Hamiltonian chaotic system with large maximum LE and large bandwidth by adjusting the parameters [10], however, the mechanism routes to chaos using Hamiltonian energy transition and theoretical proof have not been performed, and the circuit implementation, a physical proof for chaos existence, was not provided. There are four main problems associated with the existing conservative chaotic systems: (1) The maximum LE of most existing conservative chaotic systems is too small, and the bandwidth of the power spectral density (PSD) is too narrow. (2) The trajectories of non-Hamiltonian conservative chaotic systems are not steady, and it easily converges or diverges in long run, and consequently, chaos disappears. However, the trajectories of Hamiltonian conservative chaotic systems are steady, because the Hamiltonian is conserved. Therefore, modeling a Hamiltonian chaotic system is necessary but challengeable with large maximum LE. (3) Many dissipative chaotic systems have been implemented by circuit to demonstrate the physical existence and realizable. However, there has been no conservative chaotic systems have 5
been implemented by circuit because the conservation with a little parameter change will disappear, and conservative chaos will become periodic or degenerated. (4) A mechanism routes or transition using Casimir energy and Hamiltonian energy from periodic orbit to quasiperiodic orbit to chaos to higher degree chaos have not been revealed. Normally, most of the existing dissipative and conservative chaotic systems have been generated by changing some terms of typical chaotic systems such as the Lorenz system, the Chua circuit, the jerk function and the Hamiltonian function via trial and error and numerical simulation. there is almost no reasonable procedure or regime to build a new system to realize the given objectives, such as raising the value of the maximum LE or the chaotic degree or the bandwidths of the PSD. In this paper, a 4D Euler equation is proposed with four first-order equations governing the behavior of a 4D rigid body by integrating two coupled 3D rigid bodies with common axes. A symplectic structure is met and both the Hamiltonian and Casimir energies are preserved. It is proved that the 4D Euler equation only produce periodic orbit which are the interception of Hamiltonian and two Casimir functions. Based on the 4D Euler equation, a Hamiltonian conservative chaotic system is found, and the mechanism of generating chaos is uncovered in a Casimir energy analysis. The proposed Hamiltonian conservative chaos is characterized by its strong pseudo-randomness in terms of the LE values, bandwidth, probability distribution. The mechanism routes from periodic to quasiperiodic, chaos and higher degree chaos are revealed through the Hamiltonian energy transition and Poincaré map. The circuit implementation is conducted. The paper is organized as follows: Section 2 reviews the playing role of Euler equation in 3D dissipative chaotic systems. Section 3 proposes a 4D Euler equation, and its periodic solution is proved. Section 4 presents a Hamiltonian conservative chaotic system using the Casimir energy. Section 5 gives the characteristics of equilibria of the Hamiltonian conservative chaotic system. Section 6 gives the dynamical analysis of 4D Hamiltonian chaotic system. The mechanism routes from periodic to quasiperiodic and chaos are investigated in Section 7. Circuit implementation is provided in Section 8. Section 9 summarizes the paper.
2. Role of Euler equation in dissipative chaotic systems 6
2.1 Description of rotational Euler equation The Newtonian rotational equation in a space frame is expressed as
dx Γ, dt space
(1)
The equation seen in both space frame and rotating body frame is described as (Ref. [12])
dx dx ω x Γ . dt space dt body
(2)
When it is only seen in rotating body space, Eq. (2) is written as
x x ω Γ.
(3)
Eq. (3) is forced rotational Euler equation. The Euler equation for a rigid body with free force is
x xω x x,
(4)
where x x1 , x2 , x3 , ω 1 , 2 , 3 , i xi Ii i xi , i I i1 , I i is the principle moment T
T
of inertia for the group SO(3), and xi is the angular momentum, diag (1 , 2 , 3 ) . The components form of the Euler equation is
x1 ( 3 2 ) x2 x3 , x2 (1 3 ) x1 x3 ,
(5)
x3 ( 2 1 ) x1 x2 . It can also be written using a Hamiltonian vector field x J (x)H (x)
(6)
where J (x) is a symplectic matrix with
0 x3 x2 J (x) J (x) x3 0 x1 , x2 x1 0 T
(7)
and H (x) is the Hamiltonian written in the form H ( x)
1 1 x12 2 x22 3 x32 . 2
(8)
The Euler equation for the symplectic structure in classical mechanics, rigid body dynamics 7
and fluid dynamics. Recently we found that Euler equation has important role in dissipative chaotic systems. 2.2 Role of Euler equation in the forced-dissipative chaotic systems The Euler equation is part of Kolmogorov system describing dissipative-forced dynamical systems or hydro-dynamic instability. The 3D form is written as [3, 4] x J (x)H (x) x f ,
(9)
where J (x)H (x) is the inertial torque and internal torque (when H (x) includes the kinetic energy and potential energy), x is the linear dissipative torque and f is the external torque. Many dissipative chaotic systems can be transformed into the Kolmogorov type system. For instance, the Qi chaotic system is presented in the form [3] x1 a ( x2 x1 ) x2 x3 , x2 cx1 x2 x1 x3 ,
(10)
x3 x1 x2 bx3 .
There are various dynamical behaviors generated from Eq. (10). The system is modelled mathematically. What are the physical meanings of Eq. (10), like the system object, physical quantity of variables xi , the force or torque on the system? The system is a dissipative autonomous system with f (a 1 b) 0 , therefore, it should converge to a sink, because it has no external force to supply the energy lost by the dissipation. Why does it run in a chaotic manner? All the questions can be answered by the transformation of the Kolmogorov model. After a transformation [3], Eq. (10) can be written as
x J (x)H (x) x f J ( x ) K ( x ) U ( x ) x f e1 x2 x3 e2 x1 x3 e3 x1 x2 Inertial torque
(11)
c1 x2 c1 x1 0
ax1 0 x 0 , 2 bx3 b Internal Dissipative External torque torque torque
Here, term J (x)K (x) is the inertial torque being the same as Euler equation (5) (referring to the 8
[3] for the detail) released by the kinetic energy, term J (x)U (x) is the internal torque released by the potential energy. Now the mathematical model, Qi chaotic system, is analogous to a forced-dissipative rigid body system with four torques. Prior to the transformation, it is an autonomous system, however, it is a forced system because of the term f which supplies the energy losing from the dissipative terms, which explained why it does not converge to a sink but run in a chaotic behavior. In addition, the variable in the Qi system was endowed the physical quantity, i.e., xi is the angular momentum, and each parameter was provided physical point. Similarly, the Lorenz system [5] the four-wing chaotic system [4], the brushless DC motor chaotic system [2], the permanent magnet synchronous motor chaotic system [6], plasma chaotic system [7] have been transformed into the Kolmogorov system to uncover the force decomposition and energy cycling, and further reveal the mechanism of various dynamics. The Euler equation not only appears in the Kolmogorov form, but in generalized Hamiltonian form as x M (x)H (x) g (x)u(x, t ) .
(12) for generalized Hamiltonian systems, M (x) is no longer skew symmetric and can be decomposed into the sum of a skew symmetric J (x) and a symmetric matrices R(x) (the dissipation of the system). Remark 1: Both the Kolmogorov system and the generalized Hamiltonian form are the generalized Euler rotational equation with dissipative term and external force term. Euler equation takes important role in rotational forced-dissipative chaotic systems. However, Euler equation has three first-order differential equations for 3D system. To make a 4D forced-dissipative chaotic system or a Hamiltonian conservative chaotic system, it is necessary to generate a 4D Euler equation.
3. Generation of 4D Euler equation and its property Qi [10] proposed six 4D generalized sub-bodies, and modeled a 4D Euler equation by coupling two sub-bodies of those six. The two generalized sub-bodies are sub-body123 with axes 1, 2, 3 and 9
sub-body134 with axes 1, 3, 4. The coupled 4D rigid body called body and 3. And a complete 4D Euler equation for body
13
13
has common axes 1
was proposed in [10].
In this paper, we select another two 4D rigid generalized sub-bodies: Sub-body123 of axes 1, 2 and 3 and Sub-body234 of axes 2, 3 and 4 given. The generalized 4D sub-Euler equation of Sub-body123 in the group SO(4) is expressed as
Sub-body123
x3 0 x1 0
x1 0 x 2 x3 x3 x2 x4 0
x2 x1 0 0
0 1 x1 0 2 x2 , 0 3 x3 0 4 x4
(13)
0 1 x1 x3 2 x2 . x2 3 x3 0 4 x4
(14)
and the generalized Euler equation of Sub-body234
Sub-body 234
x1 0 0 x 2 0 0 x3 0 x4 x4 0 x3
0 x4 0 x2
where i I i1 , I i is the moments of inertial, and xi Iii , i 1, 2, 3, 4 the angular momenta. We integrate the two sub-bodies to generate the whole 4D rigid body: Sub-body123 coupled with
Sub-body234 with common axes 2 and 3, labelled of Body
23
23
.Therefore, a 4D Euler rigid body equation
is proposed having the Hamiltonian vector field form
23
x J 23 (x)H (x) ,
(15)
where 0 x J 23 (x) 3 x2 0
x3 0 x1 x4 x3
x2 x1 x4 0 x2
0 x3 , x2 0
(16)
with Hamiltonian H ( x)
1 1 x12 2 x22 3 x32 4 x42 . 2
Theorem 1. The 4D Euler equation is conservative in the Hamiltonian. Proof. 10
(17)
Because J 23 (x) is a symplectic matrix, we have
H (x) H (x)T J 23 (x)H (x) 0 .
(18)
The proof is complete. The conservation means that given an initial condition x 0 , H 0 is fixed, therefore, the Hamiltonian forms an ellipsoid as
1 x12 2 x22 3 x32 4 x42 2H 0 .
(19)
Remark 2: Like role of the 3D Euler equation taking in 3D rigid body, the 4D Euler equation plays important role in constructing the symplectic structure of generalized Hamiltonian system or Kolmogorov for 4D systems. It forms of the inertial part of the 4D forced-dissipative chaotic systems. The Casimir function (energy), C , also called energy-momentum [2], is an important physical quantity in rigid body dynamics. It is also very useful in uncovering the mechanism of the expansion and contraction of dynamic behavior, and the global bound of a dynamical system [1, 3-6, 10]. C is defined by the kernel whose gradient is orthogonal to the vector field of the system, i.e.,
C C T (x) x 0 .
(20)
Casimir energy is also called the total angular momentum for the 3D rotational rigid body. When the external torque is absent, the Casimir energy is conservative [12, 3-7]. Differing from 3D system having single Casimir function, we obtain two Casimir functions for system
23
satisfying Eq. (20): C1 (x)
1 2 x1 x22 x32 x42 2
(21)
and C2 (x) x1 x4 .
(22)
From Eq. (20) and (16), we obtain
C1 C1T (x) x C1T (x) J 23H (x) 0.
(23)
This means that given an initial condition x 0 , C10 is fixed, therefore, a Casimir sphere is formed 11
as
x12 x22 x32 x42 2C0 .
(24)
For C2 (x) , we have
C2T (x) J 23 (x) [0, 0, 0, 0] ,
(25)
C2 C2T (x) x C2T (x) J 23H (x) 0.
(26)
so
Therefore, both C1 (x) and C2 (x) are Casimir functions. From Eq. (26), we conclude that given an initial condition x 0 , C2 (x) x10 x40 C20 is fixed, therefore, a Casimir plane is formed as
x1 x4 C20 .
(27)
Casimir function C1 (x) is an energy function, also called internal energy. The rate of change of the Casimir energy is called the Casimir power. For 3D systems, the Casimir energy form is similar with C1 (x) and the Casimir power, we concluded in Refs. [3, 4] as the following: (1) The Casimir energy represents the existing internal energy of a forced-dissipative system. The Casimir power is the exchange rate between the supplied power and the dissipative power of the system. (2) If the Casimir power is less than zero for the system at all times, the orbit converges to a sink; if it is equal to zero, i.e., a constant Casimir function at all time, signifies a conservative system; if it is greater than zero at all times, and the orbit diverges as a source; if it is oscillating irregularly around the zero line with bound, the system produces chaos. Remark 3: Casimir power can be taken as a criterion as to whether a system can produce chaos. Theorem 2: The Casimir energy is conservative, and the 4D Euler equation produces a periodic solution which is the interception of the Hamiltonian ellipsoid, the Casimir sphere and the Casimir plane. There is no chaotic behavior produced in the 4D Euler equation. Proof. From Eq. (23), we proved that the Casimir energy is conserved. 12
Given an initial condition, the solution of the 4D Euler equation
23
satisfies Hamiltonian
ellipsoid, Eq. (19), Casimir sphere (24) and Casimir plane (27). The interception of the Hamiltonian ellipsoid, Casimir sphere generates a 3D ellipsoid. And then by intercepting the 3D ellipsoid with the Casimir plane, a periodic orbit is shaped. There is no chaotic behavior produced in the 4D Euler equation. The proof is complete.
4. Modelling of a Hamiltonian conservative chaotic system To produce chaos, we have to modify the system to make the Casimir power oscillated irregularly. System
c 23
23
becomes
x J 23c (x)H (x) ,
(28)
where 0 x J 23c 3 x2 c
x3 0 x1 x4 x3
x2 x1 x4 0 x2
c x3 . x2 0
(29)
Theorem 3. The modified system, c23 , is conservative in the Hamiltonian. Proof. Because J 23c (x) is a skew-symmetric matrix, then
H (x) H (x)T J 23c (x)H (x) 0 ,
(30)
The proof is complete. Theorem 4. The modified system,
c 23
, is not conservative in the Casimir energy if c( 4 1 ) 0 .
Proof. For system
c 23
, we have 13
c C xT x xT J 23 (x)H (x) c( 4 1 ) x1 x4 .
(31)
Therefore, the modified systems are not conservative in the Casimir energy when c( 4 1 ) 0 . The proof is complete. Theorem 4 provides a procedure of producing quasiperiodic orbit and chaos. Because all diagonal elements of the matrix for both J 23 (x) and J 23c (x) of the two systems are zero, their divergences are 4
f i 1
f x
i
xi
0.
(32)
Remark 4: The Euler equation and the modified systems preserve volume in phase space.
5. Equilibria and their stability properties of 4D Hamiltonian chaotic system System
c 23
can be written as
x1 3 2 x2 x3 c 4 x4 ,
c 23
x2 1 3 x1 x3 4 3 x3 x4 ,
(33)
x3 2 1 x1 x2 2 4 x2 x4 , x4 3 2 x2 x3 c1 x1.
By letting xi 0 , we find that system
c 23
has point equilibrium E0 at origin, line equilibria
Ex2 0, x2 , 0, 0 and Ex3 0, 0, x3 , 0 . The Jacobi matrix of system T
0 ( ) x 3 3 D 1 ( 2 1 ) x2 c1
T
( 3 2 ) x3 0 ( 2 1 ) x1 ( 2 4 ) x4 ( 3 2 ) x3
( 3 2 ) x2 (1 3 ) x1 ( 4 3 ) x4 0 ( 3 2 ) x2
c 23
is
c 4 ( 4 3 ) x3 . (34) ( 2 4 ) x2 0
By substituting E0 into Eq. (34) and making I D0 0 , we find the eigenvalues of E0 as
1,2 0, 3,4 jc 1 4 .
(35)
Hence, the system is a center at E0 . For the equilibrium line Ex2 , substituting Ex2 into Eq. (34), the eigenvalues then are 14
analytically given by
1 0, 2 a1 , 3,4
a1 a3 2
(36)
a2 ,
where
a x2 a a6 j 3a5 x22 3 a1 5 2 a4 6 , a2 j a , a , a4 4 3 a4 3a4 2 3a4 2a4
3
a6 2 2 a5 x2 a7 a7 , 3 (37) 2 ( 4 2 2 )( 2 3 ) ( 4 )( 2 3 )c 2 x2 a5 1 , a6 c 21 4 , a7 1 , 3 2 3
where j is the imaginary unit. If a1 0 , we have 1,2 0 . Through a series derivation, we have either 3,4 jc 1 4 , or 3,4 j 3a4 . For 3,4 jc 1 4 , equilibrium line Ex2 are natural elliptic. For
3,4 j 3a4 , when a4 is real number, Ex2 are also natural elliptic, and when a4 is an imaginary number, Ex2 are saddles. If a1 0 , when a1 0 , and then 2 is positive real number, and the real parts of 3,4 must be negative; if a1 0 , we have 2 is negative real number, and the real parts of 3,4 must be positive. Thus, equilibrium line Ex2 are saddles. Similarly, we obtained equilibrium line Ex3 have same properties as Ex2 .
In sum, both
Ex2 and Ex3 are either natural elliptic or saddles. They must neither be focuses nor nodes, which means there are no attracting solution for system
c 23
. This is identical to the characteristics of
conservative systems: nodes and spirals are not possible in conservative systems, however, elliptic (center) and saddle equilibrium points can and do occur [11].
6. Dynamical analysis of 4D Hamiltonian chaotic system The Matlab ODE5 (Dormand-Prince) solver has been used in the simulation, and the 15
symplectic integration algorithm has been compared. We found that the two integrators’ results are practically identical. 6.1 Dynamics of 4D Euler equation When c 0 , system
23
reduces to Euler equation 23 . From Theorem 2, it must
c 23
produce periodic orbit. Take parameters
x10 , x20 , x30 , x40
T
c 23
becomes
1 , 2 , 3 , 4
T
2, 3, 4, 5 , initial conditions T
4, 5, 5, 4 , c 0 , and sampling time T 0.005 s. For this case, system
T
23
. As concluded in Theorems 1 and 2, system
23
is conservative in both
the Hamiltonian and Casimir energy, and from Theorem 2, it only can produce a periodic orbit which is interception of the Hamiltonian ellipsoid formed by H 0 , the sphere formed by Casimir
C10 and the plane constructed by C20 as shown in Fig. 1(a). The total Hamiltonian keeps constant with H 0 143.5 , and the Casimir energy also remains the same with C10 41 [Fig. 1(b)].
(a) Periodic orbit with c 0 .
(b) Time response of Hamiltonian and Casimir energy Fig. 1 Dynamics of 4D Euler system
23
.
The system has five adjustable parameters: i , i 1, 2, 3, 4 and c , and four adjustable initial values, xi 0 , to influence the orbital modes of the system, and each is physically meaningful: the initial values, xi 0 , determine the total Hamiltonian given i from Eq. (17) because the system 16
preserves the Hamiltonian; i are proportional to the frequency of the rigid body provided the initial Hamiltonian is fixed. If c( 4 1 ) 0 , the system either produces a quasi-periodic orbit or a chaotic orbit depending on the initial energy. The value of c( 4 1 ) quantitatively influences the orbital pattern and the degree of chaos. The system is conservative in phase volume from Remark 4; therefore, the system’s orbits cannot reach an equilibrium point asymptotically. This implies nodes and spirals are not possible in the conservative system. However, elliptic (center) and hyperbolic (saddle) equilibrium points do occur. We analyze what are the possible route and mechanisms in the transition from pseudo-periodic orbits to chaos for the system under condition c( 4 1 ) 0 . 6.2 Ordinary Hamiltonian conservative chaos All the parameters and initials remain the same as Section 6.1 except for c 3 , choosing sampling time, the orbits of system
c 23
is chaotic [Fig. 2(a), (b) and (c)], which can be verified
by the LEs LE1,2,3,4 [2.4241, 0, 0, 2.4241]T . The maximum of LEs is quite large compared with those of the conservative chaotic systems in aforementioned references. From Fig. 2(b) and (c), the orbits are seen not to approach a chaotic attractor asymptotically but visits the whole space that it initially enters that being identical to the integer dimension,
LD 4 . However, dissipative chaos asymptotically reaches a chaotic attractor regardless of the initial values set. From Theorem 3, the system is conservative in the Hamiltonian with H (t ) 143.999 for the given parameter and initial values; therefore, chaos cannot be characterized using the Hamiltonian. The positive LE does indicate chaos when the system is bounded. A numerical and statistical index of chaos; however, positive LE is a numerical and statistical index instead of the analytical form. From Theorem 4, the Casimir energy gives an analytical form. If the Casimir power oscillates irregularly around the zero line, it produces chaos. Fig. 2(d) clearly demonstrates the irregular oscillation.
17
(b) Phase portrait of x1 x2
(a) Time series of x2 and x3
(c) Phase portrait of x1 x3
(d) Casimir power of x2
Fig. 2. Numerical characteristics of system
c 23
.
6.3 Hamiltonian chaos with strong pseudo-randomness The system
c 23
not only produces chaos, but produces chaos of high quality and strong
pseudo-randomness. This can be realized by adjusting the five parameters and four initial values according to their functions in frequency and energy. To improve the frequency, i must be increased. To speed up the stretching and folding of orbits, and enlarge the spatial magnitude of orbits, the absolute of initial values, xi 0 , must be enhanced, which is equivalent to increasing of the impulse torque pushing the rigid body for rotation. Importantly, c must be improved for increasing the Casimir power of oscillation. Set 1 , 2 , 3 , 4 100, 150, 200, 50 , c 500 , T
18
T
and initial values
x10 , x20 , x30 , x40
T
800, 1000, 1000, 800 , and the sampling time T
T 3 107 s.
The times series for x2 is shown in Fig. 3(a). Since the trajectory is too dense and noisy to display, we have to limit the time interval to t [0, 0.01]s . The magnitudes of the angular momenta are much larger than in Figs. 2(a), (b) and (c) compared with those in Sub-Section 6.2. The difference in magnitudes arises from the initial Hamiltonian. The maximum of the LEs LE1 11398.1088 [Fig. 3(c)]. It is not imaginable for its largeness compared with those of existing dissipative and conservative chaotic systems. Note that the value of LE is the good measure of the degree of chaos or pseudo-randomness [11, 27, 28]. We can use the width of Power Spectral Density (PSD) to measure the pseudo-randomness of the chaotic system. The width PSD of system in Sub-Section 6.2 is 10Hz. Note that the width is even less for other existing dissipative chaotic systems. The Qi hyperchaotic system, which is dissipative has quite a wide bandwidth of 100Hz compared with other existing hyperchaotic systems [27]. However, the bandwidth of the PSD for the Hamiltonian conservative chaotic system in this parameter settings is 57600Hz [Fig. 3(d)], which is 5760 times that in Sub-Section 6.2, and 576 times than that of the Qi hyperchaotic system. Here, Welch’s overlapped segment averaging estimator with Hamming window for the PSD in Matlab is used; the magnitude of the PSD is normalized. The bandwidth is truncated when the magnitude of the PSD is less than 0.1 [Fig. 3(d)]. In chaos-based cryptography, the bandwidth of the PSD is an important measure of a pseudo-random number in regard to the security of encrypted messages. When the bandwidth is too narrow, the encryption can be broken even using simple filtering methods. Note that since all the parameters are physically meaningful, the bandwidth of a pseudo-random number of the system can be continually increased further to MHz and even GHz by adjusting the parameters as we have confirmed. Generally, the bandwidth of a chaotic system can be widened through the time-scale transformation; however, the scaled system is no longer the original one, and the system loses physical meanings. Nevertheless, the proposed system realizes the bandwidth objective through the adjustment of parameters and initial values, which do not lose their original physical meanings. For some chaotic systems, when parameters or initials are 19
adjusted to some regions, no chaotic orbit is generated. For instance, the initial values of the Hénon-Heiles system can only be set in a very small region; When outside this region, system is unstable, but inside the region, the system orbit is periodic. To further understand the statistical properties, we calculated the histogram reflecting the distribution of the random data. Normally, the main difference of the conservative chaos from dissipative chaos is that the former probability distribution is flatter like uniform distribution of white noise than that of the latter, which has a peak or multiple peaks form as for the normal distribution of Gaussian white noise. The reason is because of its integer dimension, the former has a better ergodicity than the latter. Therefore, the pseudo-random number generated from conservative chaotic system is more suitable than that generated from a dissipative chaotic system provided that both have the same bandwidth. Fig. 3(e) gives the probability density of the Hamiltonian conservative chaotic system using histogram, for which the distribution shape is very close to that for uniform white noise. The Poincaré map also displays the ergodicity and fractal dimension of a system. The Lorenz system shows poor ergodicity in only displaying ―several slim twigs‖ on the Poincaré map. The Poincaré map of x1 x3 of the proposed Hamiltonian chaos taking x4 0 as the section of Poincaré [Fig. 3(f)] shows excellent ergodicity, because the map almost fully occupies a large region of x1 x3 plane. We further investigate the proposed new method, the Casimir power method, to characterize and find the pathway to generating chaos. As given in Theorem 4, C c( 4 1 ) x1 x4 ; see Fig. 3(g). Both the frequency and magnitude of the irregular oscillation of the Casimir power are quite high demonstrating its chaotic degree. When c 0 , the system is conservative for both the Hamiltonian and Casimir energy, theoretically the system is periodic, as verified in Fig. 3(h). A further
test
is
taken.
Keeping
c 500 ,
but
taking
1 4 100 ,
we
have
C c( 4 1 ) x1 x4 0 , so the system is periodic also, as verified in Fig. 3(i). Note that because Fig. 3(h) and (i) has different parameters in terms of 4 and c , the shapes are different even though both are periodic. Therefore, Theorem 4 is further verified, and the Casimir power is a 20
proper evaluation index that characterizes whether a system is periodic.
(b) Phase portrait of x1 x3
(a) Time series of x2
(c) Lyapunov exponents of
c 23
(d) Power spectral density of x2
(f) Poincaré map x1 x3 .
(e) Probability distribution of x2
21
(h) Periodic orbit produced with c 0
(g) Casimir power of x2
(i)
Periodic
orbit
produced
(j) Hamiltonian and Casimir energies, where
with
(j1) and (j2) have 1 4 100, c 500 , and
1 4 100, c 500
(j3) and (j4) have 1 100, 4 50, c 500 .
Fig. 3 Strong quality of pseudo-randomness of
c 23
.
The KAM perturbation theorem is well-known in analyzing Hamiltonian chaotic systems [11]. It provides a starting point for an explanation of the transition from regular to chaotic motion. From the perturbation theory, when a Hamiltonian system with function H1 is perturbed by another Hamiltonian system with H 2 , the coupled Hamiltonian system will probably generate conservative chaos with H H1 H 2 because there is energy exchange between the two Hamiltonian functions. However, when the system Hamiltonian is not easy to decompose into two 22
functions, the KAM theory cannot explain or analyze the transition from regular to chaos. We analyze the transition for this situation from a new perspective, i.e., whether its Casimir power is oscillated around the zero line. When H is constant, and the energy momentum or total angular momentum or Casimir energy is changed, so the Casimir power can explain how the energy is exchanged. System (33) is conservative in the Hamiltonian whether c(1 4 ) is zero or not [see Figs. 3(j1) and (j3)], where (j1) has c(1 4 ) 0 , and (j3) does not. However, when c(1 4 ) 0 , there is Casimir energy exchange internally because terms c 4 x4 and c1 x1 trigger the changes in xi , i 1, 2, 3, 4 , the rates of change of angular momentums and the changes of in Casimir power,
C c(1 4 ) x1 x4 . Therefore, both terms equivalently take the mix role of dissipative and supplied energy in the total Casimir energy, because it is not clear about which one is dissipative or supplied, which is the main cause of generation of chaos as concluded in Refs. [3, 4]. Fig. 3(j2) shows that there is no Casimir energy exchange when c(1 4 ) 0 , whereas Fig. 3(j4) shows that terms c 4 x4 and c1 x1 strongly impact on the Casimir energy when c(1 4 ) 0 .
7. Role of Hamiltonian energy transition in dynamics change There are three predominant routes to chaos, namely, period doubling, quasiperiodic and intermittency bifurcations. The conservative chaotic systems preserve the phase space, hence, the chaos route is quasiperiodic to chaos. Although, Casimir power qualitatively determines whether the system produces periodic orbit or not, the Hamiltonian value quantitatively determines the degree of disorder of system
c 23
under condition that Casimir power is not equal to zero. Fixing parameters as
1 , 2 , 3 , 4
T
2, 3, 4, 5 , c 3 , T
x0 = 1.4, 1.4, x30 , 1.4
T
,
and
x30 [5, 5]
which
corresponds to the Hamiltonian H 0 [9.8, 59.8] , where H0
1 2 2 2 1 x102 2 x20 3 x30 4 x40 . 2 23
(38)
The bifurcation diagram of initial conditions along with their energy level is plotted in Fig. 4. The stripe colors represent Hamiltonian values labeled on energy bar. Normally, the low Hamiltonian energy (the dark blue and the blue) represents quasiperiodic orbits, and the high Hamiltonian energy (the light blue and the yellow) represents chaos. It hence can be inferred that system
c 23
with higher energy has a wider dynamic field or frequency domain. We may clearly find that the Hamiltonian energy level (color stripe) leap essentially labels the transition of the dynamics of the system.
Fig. 4 Bifurcation diagram of initial conditions corresponding to various energy levels.
There are complicated coexisting dynamics for system
c 23
. Under same Hamiltonian,
different initial conditions may produce different quasiperiodic orbits, even coexist both chaos and quasiperiodic orbits. To visualize the geometry of complicated orbits and investigate the coexistence and transition from quasiperiodic motion to chaotic motion in system
c 23
, the
Poincaré maps on surface of sections (SOS) for trajectories in terms of various Hamiltonian energy levels are plotted in Fig. 5. The SOS is taken as the extremals of Casimir energy, i.e.,
C1 (x) c( 4 1 ) x1 x4 0 . Take x0 = 1.4, 1.4, x30 , x40 , and H 0 13.18,17.02, 23.32, 34.3 , T
respectively. For each H 0 , as x30 varies, x40 must be correspondingly varied to keep the given
H 0 through Eq. (38). For each given H 0 , many pairs of x30 and x40 as initials are taken for simulations. Case 1 for H 0 13.18 . In Poincaré map, there are total eight primary islands labeled by 24
Ci , i 1,
, 8 [Fig. 5(a)], the center of each island is the elliptic equilibrim point of Poincaré map.
Each island has many concentric coexisting closed curves, which are the maps of the quasiperiodic solutions generated by system
c 23
in different initials with same Hamiltonian. Some islands are
separated by the separatrices, and some are intersected. Two intersected islands probably do not intersect in 2D plane. They may have different phase spaces because the SOS is not a plane but the extremals of Casimir energy, which allows the SOS has 3D geometric space. Case 2 for H 0 17.02 . As the Hamiltonian increases, each primary island gives rise to the secondary islands [Fig. 5(b)]. This is caused by the multi-frequency and resonance of the orbits generated by system
c 23
. The separatrices are either occupied by secondary islands or little
chaos. The coexistence become more complicated. Case 3 for H 0 23.32 . As more and more secondary islands enlarge and the third or fourth islands appears. To some extent, when H 0 23.32 , some primary islands disappeared, and the peripheral portion corresponding to the previous invariant curves has dissolved into the chaotic region [Fig. 5(c)], which is called chaotic sea. Case 4 for H 0 34.30 . As the energy is further increased to higher level, the chaotic region occupied by scattered points is gradually widening, and all the invariant curves is dissolved [Fig. 5(d)]. When the energy level is high enough, the quasiperiodic motion disappears and be replaced by the chaotic motion completely.
(a) Primary islands corresponding to the quasiperiodic orbit under relatively low Hamiltonian
25
(b) Mixed primary islands and secondary islands corresponding to multi-frequency quasiperiodic orbits with higher Hamiltonian
(c) coexistence of islands and chaotic sea corresponding to coexistence of quasiperiodic orbits and chaos with higher Hamiltonian
(d) No islands corresponding to chaos with high Hamiltonian
Fig. 5. The evolution of dynamics from quasiperiodic orbit to chaos of system
c 23
as the leap of Hamiltonian
energy level.
8. Circuit implementation For the nonlinear system,
c 23
, all the numerical simulation integration methods are
calculated through digitization and iteration, such as the ODE45 method, the symplectic algorithm and the Field-Programmable Gate Array (FPGA) digital implementation. Even if they have high accuracy, they have some errors aroused from algorithm, sampling time and computer accuracy. For a conservative system, even for a little error, and the accumulation of iteration will make the conservation of energy and phase space loose, and then conservative chaos will lead to either diverging or shrinking. However, because analog electronic circuit operate physically to implement the system, it is able to verify whether the system really is conservatively chaotic and confirm the suitability of numerical method selected in the paper. Especially for conservative systems, the general integral formula may be difficult to converge the solution or energy error, moreover, the diversity of conservation makes it extremely complicated to theoretically prove and design the applicable numerical integration methods such as preserving symplectic geometry or 26
invariants. As shown in Fig. 6, system
c 23
with fixed chaotic parameters is implemented by an analog
electronic circuit. Six quadratic terms in the system are accomplished by six analog multipliers, besides, several basic arithmetic units consists of eight operational amplifiers and some linear resistors and capacitors for performing addition, subtraction, and integral operations. For the parameter choices 1 , 2 , 3 , 4 2, 3, 4, 5 , c 3 , the experimental observations from the T
T
analog oscilloscope are shown in Fig. 7(a) and (b). The figures showing in different phase planes are identical to the figures showing numerically [Fig. 2(b) and (c)]. The implementation of analog circuit for the system verified that the system is a really conservative chaos and confirmed that the ODE5 selected for numerical method is suitable.
Fig. 6. Circuit implementation of system
c 23
, with electronic parameters:
R1,R2,R3,R4,R5,R6,R7,R8,R10,R12,R13,R16 10K;R9 16.66K; R11,R14 5K;R15 6.66K ; C1,C2,C3,C4 10nF . 27
(a)
(b)
(c)
(d) Fig. 7. Implementation of analog circuit of system
c 23
.
9. Conclusion This paper proposed a 4D Euler equation by integrating two generalized 3D Euler equations. The equation will be the inertial part of a rotational forced-dissipative system. Hence, it will be significant in generating and analyzing the generalized Hamiltonian systems and Kolmogorov systems encountered in rigid-body dynamics and fluid dynamics and chaos generation. The Hamiltonian, Casimir energy and phase volume of the 4D Euler equation are preserved. The periodic solution was proved for the 4D Euler equation. Based on the 4D Euler equation, using a Casimir power method, a Hamiltonian conservative chaotic system has been presented by breaking 28
the conservation of Casimir energy. An analytic form of Casimir power has been proposed, which provides the way of producing nonperiodic orbits: quasiperiodic orbit and chaos. The routes and mechanism from quasiperiodic orbit to chaos have been provided through energy bifurcation and Poincaré map. The Hamiltonian conservative chaos with strong pseudo-randomness was produced, which is assessed in the evaluations of Lyapunov exponent, the bandwidth of the power spectral density, the probability distribution, the Poincaré map. The circuit implementation has been conducted to physically verify the existence of Hamiltonian conservative chaos. Much open topic has been left for researchers: how to construct a forced-dissipative chaotic systems based on the 4D Euler equation and what types of hidden chaos can be found in the Hamiltonian conservative chaotic systems, and so on.
Acknowledgements This work is supported by the National Natural Science Foundation of China (61873186) and the Tianjin Natural Science Foundation (17JCZDJC38300).
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