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Energy analysis of Sprott-A system and generation of a new Hamiltonian conservative chaotic system with coexisting hidden attractors
Chaos, Solitons and Fractals 133 (2020) 109635
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Energy analysis of Sprott-A system and generation of a new Hamiltonian conservative chaotic system with coexisting hidden attractorsR Hongyan Jia a,∗, Wenxin Shi a, Lei Wang a, Guoyuan Qi b a b
Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, PR China School of Electrical Engineering and Automation, Tiangong University, Tianjin 300387, PR China
a r t i c l e
i n f o
Article history: Received 15 October 2019 Revised 19 December 2019 Accepted 15 January 2020
Keywords: Energy analysis Hamiltonian energy Casimir energy Conservative systems Coexisting hidden attractors
a b s t r a c t The paper firstly investigates energy cycle of the Sprott-A system by transforming the Sprott-A system into the Kolmogorov-type system. We found the dynamics of the Sprott-A system are influenced by the change along the energy exchange between the conservative energy and the external supplied energy. And the action of the external supplied torque is the main reason that the Sprott-A system generates chaos. Secondly, based on energy analysis of the Sprott-A system, a new four-dimension (4-D) chaotic system is obtained. The new 4-D chaotic system is a conservative system with a constant Hamiltonian energy. Besides, it is also a no-equilibrium system, this means that the new 4-D chaotic system can exhibit hidden characteristics. Further, the coexisting hidden attractors are found when selecting different initial points. Finally, the new 4-D chaotic system is implemented by FPGA, and the coexisting attractors observed are consistent with those found in numerical analysis, which in experiment verifies the existence of coexisting hidden attractors of the new 4-D chaotic system from physical point of view. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Since a Kolmogorov System describing a dissipative dynamical system or a hydrodynamical instability with a Hamiltonian energy function was reported in 1991 [1], energy analysis for nonlinear chaotic systems has attracted more and more attention [2–11]. The above research is mainly based on the Kolmogorov System, and focuses on investigating the physical meaning of variables in the chaotic systems, finding the reasons for chaos generation, analyzing the variation of dynamics characteristics caused by energy cycle, and further providing feasible chaotic models for application. That is, energy analysis is a new tool to investigate, analyze and utilize chaos in nonlinear systems from viewpoint of energy cycle. In addition, conservative chaotic systems (CCSs) are also increasingly emerged as a new research interest [12–18]. Generally speaking, CCSs are referred to those systems with conservative quantities, such as a constant Hamiltonian energy, a zero divergence or a zero sum of Lyapunov exponents. Compared with dissi-
R This work was supported in part by the National Natural Science Foundation of China (Grant nos. 61573199 and 61873186)) and the Tianjin Natural Science Foundation (Grant nos. 17JCZDJC38300 and 18JCQNJC74000). ∗ Corresponding author. E-mail address: [email protected] (H. Jia).
https://doi.org/10.1016/j.chaos.2020.109635 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
pative chaotic systems (DCSs) whose asymptotic motion settle onto a set of a strange attractor, CCSs are more suitable to be used in information security. The reasons are mainly as follows [12,16]: (a) CCSs do not produce attractor, (b) ergodicity of CCSs is richer than that of DCSs, (c) for CCSs, different initial conditions lead to distinct different dynamics. Therefore, CCSs begin to attract more and more attention. According to Refs. [12–17], CCSs are generally classified as two categories: the Hamiltonian conservative chaotic systems (HCCSs) and the non-Hamiltonian conservative chaotic systems (non-HCCSs). And compared with non-HCCSs which is only conservative in volume or sum of Lyapunov exponents, HCCS is a category of stricter conservative chaotic systems [12]. Therefore, the generation and analysis of HCCSs have been attracting more and more research interest. Sprott-A system is one of some simple chaotic flows reported in 1994, and it is also a special case of equations describing Nosé -Hoover dynamics systems [19,20]. It is generally considered as the oldest example of rare with no equilibria and the simplest mechanical system exhibiting chaos. Besides, because of a zero sum of Lyapunov exponents and no equilibria, it is classified as a conservative system with hidden coexisting attractors [21–24]. It is worth mentioning that hidden coexisting attractors are also a recent new research hotspot [25–28]. Therefore, the research based on the Sprott-A system begin to attract increasing attention [29–
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32]. Besides, with the development of chaos application, the FPGA implementation of chaotic systems has attracted more and more attention [10,18]. However, the research on FPGA implementation of HCCSs is seldomly reported, which can limit their application in engineering fields. FPGAs can be reprogrammed indefinitely, loading a new design solution takes only a few hundred milliseconds, and reconfiguration can reduce hardware overhead. The operating frequency of an FPGA is determined by the FPGA chip and the design. Some harsh requirements can be met by modifying the design or replacing a faster chip. Using the FPGA’s parallel processing algorithms and rich logic resources, video chaos-based cryptography needing fast process can be implemented. In this paper, the Sprott-A system is first transformed into the Kolmogorov-type system. Then, energy cycle of the Sprott-A system, the affection of the variation of energy on its dynamics and the key reason why it generates chaos are respectively discussed. Subsequently, according to energy analysis of the Sprott-A system, the paper obtains a new 4-D conservative chaotic system without equilibrium. Further, it is a Hamiltonian conservative chaotic system, i.e., its corresponding Hamiltonian is constant. In addition, according to numerical analysis, some coexisting hidden attractors are found to exist in the new 4-D chaotic system when selecting different initial points. Finally, the new 4-D chaotic system is also implemented by FPGA hardware, which further shows the existence of coexisting attractors of the new 4-D conservative chaotic system from physical point of view. The rest of this paper is organized as follows: in Section 2, we give the transformation of the Sprott-A system to the Kolmogorovtype system, which only includes two parts: the conservative part and the external supply part. In Section 3, we discuss and analyze energy cycle of the Sprott-A system. In Section 4, based on the analysis of the Hamiltonian energy, we obtain a 4-D Hamiltonian conservative chaotic system without equilibrium. And the corresponding numerical analysis and FPGA implementation are also given to show the characteristics of the 4-D Hamiltonian conservative chaotic system. Finally, the paper is concluded in Section 5.
Generally, Kolmogorov system is a very useful bridge which can help analyze the mechanical properties of dynamics systems, find the reason why chaos can generate and investigate the effect of variation of energy on dynamics of systems. Kolmogorov system can be written as follows [1,2]:
x˙ = {x, H (x )} − x + u = x × ∇ H (x ) − x + u (1)
where x ∈ Rn represents the state vector, H (x ) : Rn → R refers to Hamiltonian energy, J (x ) ∈ Rn×n represents a skew-symmetric matrix. {x, H (x )}, x and u represents the conservative part, the dissipative part and the external supply of Eq. (1), respectively. According to Refs. [3–12], some conclusions can be concluded as follows: Remark 1. (1) The term {x, H (x )} = x × ∇ H (x ) = J (x )∇ H (x ) represents Hamiltonian energy conservation, i.e., when x and u in (1) are absent, Hamiltonian energy is a nonzero constant, and the rate of change of Hamiltonian is H˙ (x ) = 0. (2) Hamiltonian energy can be defined as H (x ) = K (x ) + U (x ) n where K (x ) = 12 i x2i , i = 1, 2, ..., n and U (x ) refer to the kinetic energy and the potential energy, respectively.
Furthermore, according to Refs. [12–18], some more comments on Hamiltonian conservative chaotic systems can be concluded as follows: Remark 2. (1) There generally exists two classes of Hamiltonian conservative chaotic systems (HCCSs): traditional HCCSs and generalized HCCSs. The former can be described by a Hamiltonian function H (x )(q, p, t ), and they need to satisfy even-dimensional Hamiltonian equation. The latter can be expressed by a state equation x˙ = {x, H (x )}, and they can be found in either evendimensional systems or odd-dimensional systems. (2) Generalized HCCSs refer to a system whose Hamiltonian energy is a nonzero constant, and the rate of change of Hamiltonian is H˙ (x ) = 0. (3) Some feature of HCCSs can be valuable for security communication. 2.2. Sprott-A system In 1994, Sprott reported some algebraically simple threedimensional chaotic systems, one of those is subsequently called as the Sprott-A system, as follows [18]:
x˙ = y y˙ = −x + yz, z˙ = 1 − y2
2.1. Preliminaries
i=1
i=1
ergy is called as the Casimir power C˙ (x ), which represents the rate of change of energy exchange between the dissipative part and the external supply of the system. (4) When i = 1, i = 1, 2, ..., n are constant,K (x ) is consistent with Casimir energy C (x ), and Hamiltonian energy can be written as H ( x ) = C ( x ) + U ( x ). (5) Chaos can be found to exist in a Hamiltonian energy conservative system when C˙ (x ) oscillates irregularly near zero with bound.
2. Transformation of Sprott-A system into Kolmogorov-type system
= J (x )∇ H (x ) − x + u,
(3) For Eq. (1), Casimir energy is generally defined as the C (x ) = n 1 2 xi , i = 1, 2, ..., n, and the rate of change of the Casimir en2
(2)
It is also called the Nosé-Hoover oscillator, because it is a special case of equations describing the Nosé-Hoover dynamics systems [19–22]. According to Refs. [19–23], some characteristics of the Sprott A system are summarized as follows: Remark 3. (1) It is the simplest mechanical system exhibiting chaos. (2) It is a conservative system, because the sum of Lyapunov exponents is zero, which reflects that divergence in phase volume is zero from statistic point of view. (3) It is the oldest example of rare, ows with no equilibria, and can show a hidden attractor coexisting with a set of nested tori. Next, in order to investigate the key reason why it can generate chaos, and discuss how the variation of energy affects its dynamics. The Sprott-A system will be firstly transformed to the Kolmogorovtype system. 2.3. Kolmogorov-type transformation We rewrite the Sprott-A system as
x˙ y˙ z˙
= {x, H (x )} + u = Js (x )∇ H (x ) + u,
(3)
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3
Fig. 1. Hamiltonian energy and its derivative of Eq. (12).
where
Js (x ) =
0 −1 0
1 0 −y
0 y ,u = 0
0 1 2 0 , H (x ) = x + y2 + z 2 , 2 1
then we can obtain the rate of change of Hamiltonian energy as follows:
H˙ (x ) =
∂ H (x ) ∂ H (x ) x˙ = ( {x, H ( x )} ) = 0. ∂x ∂x
(6)
which implies the energy is a constant, i.e.
furthermore, the following conclusion can be obtained: Conclusion 1
H (x ) =
(1) The Sprott-A system is transformed into the Kolmogorov-type system, e.g., Eq. (3). (2) There are two parts in (3): the conservative part and the external supply part. The first term represents the conservative torque described by H (x ) = 12 x2 + y2 + z2 , and the second term represents the external supplied torque, respectively. (3) Eq. (3) can display chaotic behavior, and it is not Hamiltonian conservative because of the existence of the external supplied torque u. (4) For Eq. (3), C (x ) = H (x ) = 12 x2 + y2 + z2 .
1 2 x + y2 + z2 = c, 2
where c represents constant. This indicates that Eq. (5) is a Hamiltonian energy conservative system. In addition, for Eq. (5), the Casimir energy C(x) is same as the Hamiltonian Energy H(x), i.e., the Sprott-A system is also a Casimir energy conservative system when the external supplied torque is absent. According to Refs. [5,8,12], chaos generally can’t happen. Furthermore, we √ also find that H(x) implies an invariant sphere whose radius is 2C , and shows that the energy exchange does not happen in Eq. (5). Therefore, the orbit starting from an initial point (x0 , y0 , z0 ) keeps on r the surface of the corresponding sphere Sr0 described by
3. Energy of Sprott-A System
(7)
Srr0 : x2 + y2 + z2 = r02 ,
(8)
3.1. Energy cycle
where r0 =
Generally, linear or nonlinear vectors in equations describing dynamic systems are often coupled, which means that it is difficult to directly study the torques. However, every energy is related to its corresponding torque or force. Therefore, investigating energy cycling is a useful method to study the reason why some behavior can generate. In this study, Casimir energy and Hamiltonian energy adopted in some research on mechanics and energy analysis of chaotic systems [2–13] are mainly used to investigate the affection of the variation of the energy on the dynamics of Sprott-A system. By analyzing Eq. (3), it can be found that two forms of energies exist in the Sprott-A system: the conservative energy and the external supplied energy, which relates to the conservative torque described by Hamiltonian energy and the external supplied torque described by the external supplied energy, respectively. Next, Eq. (3) is rewritten as follows:
Case 2. Energy circle under the conservative torque and the external supplied torque, i.e., Eq. (4), then we have the rate of change of Hamiltonian as follows:
x˙ = {x, H (x )} + x, S(x ),
(4)
= ∂ S∂(xx ) , S(x ) represents the external supplied en-
where x, S(x ) ergy. Next, energy cycle of the Sprott-A system will be discussed as follows: Case 1. Energy circle under the conservative torque is
x˙ = {x, H (x )},
(5)
x20 + y20 + z02 .
∂ H (x ) x˙ ∂x ∂ H (x ) = ( {x , H ( x )} + x , S ( x ) ) ∂x = H ( x ), S ( x ).
H˙ (x ) =
(9)
which reflects the rate of change of the conservative torque is only related to the external supplied torque. At the same time, Eq. (9) also shows that the energy exchange in Eq. (4) happens between the conservative energy and the external supplied energy. According to Refs. [3–12], the reason why the Sprott-A system can generate chaos is found to be the action of the external supplied torque which makes the Hamiltonian energy change. Furthermore, because energy exchange exists in the Sprott-A system, the orbit starting from an initial point (x0 , y0 , z0 ) departs from the surr face of the corresponding sphere Sr0 , and then shows chaos. 3.2. Energy analysis In this section, two types of torques of the Sprott-A system will be numerically discussed to investigate the key factors that chaos generates.
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Fig. 2. Orbits of Eq. (12).
Fig. 3. Hamiltonian energy and its derivative of Eq. (2).
Fig. 4. Orbits of Eq. (2).
Firstly, we define two the invariant spheres:
2
Sr1 : x2 + y2 + z
= 1,
(10)
and
Sr10 : x2 + y2 + z2 = 100.
(11)
Then two groups of initial points from the two spheres are selected to numerically investigate dynamics of the Sprott-A system. One √ √ group of initial points are Ir1 (1, 0, 0 ), Ir1 (0, 1, 0 ) and Ir1 ( 22 , 22 , 0 ) on the surface of the invariant sphere Sr1 . The other of initial √ group √ points are Ir10 (10, 0, 0 ), Ir10 (0, 10, 0 ) and Ir10 (−5 2, 5 2, 0 ) on the surface of the invariant sphere Sr10 . In subsequent study, the orbits
H. Jia, W. Shi and L. Wang et al. / Chaos, Solitons and Fractals 133 (2020) 109635
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Fig. 5. Lyapunov exponents, bifurcation diagram and chaotic attractor of Eq. (18).
√
√
starting from Ir1 (0, 1, 0 ), Ir1 (1, 0, 0 ) and Ir1 ( 22 , 22 , 0 ) are marked in red, green and blue, respectively. Similarly, √ √ the orbits starting from Ir10 (0, 10, 0 ), Ir10 (10, 0, 0 ) and Ir10 (−5 2, 5 2, 0 ) are also marked in red, green and blue, respectively. Next, analysis of Hamiltonian energy will be only given to investigate energy cycle, because we define C (x ) = H (x ) = 12 x2 +
y2 + z2 for the Sprott-A system. Analysis for Case 1: Analysis under the conservative torque, i.e.,
x˙ y˙ z˙
= Js (x )∇ H (x ) =
0 −1 0
1 0 −y
0 y 0
x y . z
(12)
Its biggest Lyapunov exponent is 0, and the corresponding divergence is
∇ · f = z,
(13)
which indicates it is difficult to determine whether Eq. (12) is conservative or dissipative in volume. But according to Remark 1, we found Eq. (12) is Hamiltonian conservative, and its corresponding Hamiltonian energy is a constant. Energy exchange does not take place in Eq. (12). For the two groups of initial points on the surface of the spheres Sr1 and Sr10 , the corresponding Hamiltonian energy and its derivative are shown in Fig. 1(a) and (b), respectively. And
the corresponding orbits are observed to keep on the surface of the corresponding sphere Sr1 and Sr10 , as shown in Fig. 2(a) and (b), respectively. All these numerical results are consistent with Eqs. (6)– (8). Furthermore, we found the orbits in Fig. 2(a) and (b) are actually is on heteroclinic orbits described by Ref. [31]. Based on the above analysis, the Sprott-A system without the external torque is furtherly verified to be Hamiltonian conservative. But the chaotic dynamic is not found. Analysis for Case 2: Eq. (2) is the object of study, in other word, analysis for Eq. (2) is actually to investigate how dynamics of a Hamiltonian conservative system change via adding the external excitation. Similarly, for the two groups of initial points on the surface of the spheres Sr1 and Sr10 ,the corresponding Hamiltonians and their derivative are shown in Fig. 3(a) and (b), respectively, which are consistent with Eq. (9). We found the corresponding Hamiltonian energy of Eq. (2) is the variable with H˙ (x ) = 0, which means energy exchange happens. And the corresponding orbits are observed to depart from the surface of the corresponding sphere Sr1 and Sr10 , and show some attractors of (2), as shown in Fig. 4(a) and (b), respectively, which are consistent with dynamics of the Sprott-A system. Furthermore, from viewpoint of numerical analysis, the action of the external supplied torque is found to be the main reason why the Sprott-A system can generate chaos. According to the above analysis, we found there are only two parts in the Sprott-A system, i.e., the conservative toque and the external supplied torque, and the external supplied torque described by the external supplied energy is the main reason why it can generate chaos. Now, one question occurs: whether is the external supplied
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where
⎡ ⎢
J4D_s (x ) = ⎣
0 −1 0 0
1 0 −y 0
0 y 0 −1
⎤
0 0⎥ , 1⎦ 0
and then a conclusion can be obtained that Eq. (16) is a Hamiltonian conservative system described by the Hamiltonian energy H4D (x ) = 12 x2 + y2 + z2 + w. Proof: because J4D_s is a skew-symmetric matrix, according to Remark 1, then
H˙ 4D (x ) = ∇ H4TD (x )x˙ = ∇ H4TD (x )J4D_s (x )∇ H4D (x ) = 0.
(17)
Therefore, Eq. (16) is a Hamiltonian conservative system. Now, according to Eq. (16), we can obtain a general Hamiltonian conservative system as follows:
Fig. 6. Energy of Eq. (18).
energy included in Hamiltonian energy, and can a Hamiltonian energy conservative system be obtained?
⎧ ⎪ ⎨x˙ = y
y˙ = −x + yz
2 ⎪ ⎩z˙ = −y + 1
,
(18)
w˙ = −z
4. Generation and analysis of a new Hamiltonian energy conservative system 4.1. A new Hamiltonian energy conservative chaotic system Now, according to Eq. (9), we rewrite the Casimir power of the Sprott-A system as follows:
C˙ (x ) =
∂ C (x ) ({x, C (x )} + x, S(x ) ) = z, ∂x
(14)
which shows that C˙ (x ) is variable and is related to z. Next, according to Eq. (14), we can obtain a new Hamiltonian energy function with both kinetic energy and potential energy as follows:
H4D (x ) =
1 2 x + y2 + z2 + w, 2
(15)
where w = zdt represents the potential energy, and thus a new 4-D system can be obtained,
⎡ ⎤ x˙
⎢ y˙ ⎥ ⎣ z˙ ⎦ = J4D_s (x )∇ H4D (x ), w˙
(16)
Therefore, by constructing a new Hamiltonian energy, the above question is solved. Obviously, Eq. (18) is conservative in the Hamiltonian, and it is a system without equilibrium. Generally, an attractor is hidden when its basin of attraction does not intersect with a small neighborhood of any equilibrium point. Therefore, for Eq. (18) without equilibrium, all attractors are hidden and conservative in Hamiltonian energy. Furthermore, Eq. (18) can be written as follows:
x˙ = {x, H4D (x )}.
(19)
Which shows there is a conservative torque in Eq. (19), and the corresponding conservative torque consists of kinetic energy K (x ) = 12 x2 + y2 + z2 and potential energy U (x ) = w. 4.2. Energy circle of the new Hamiltonian conservative system Here, Hamiltonian energy circle, kinetic energy circle, and potential energy circle will be respectively discussed. Hamiltonian energy circle: according to Refs. [3–12], we can obtain the rate of energy exchange in Hamiltonian energy as follows:
H˙ 4D (x ) =
∂ H4D (x ) ∂ H4D (x ) x˙ = ({x, H4D (x )} ) = 0, ∂x ∂x
Fig. 7. Coexisting attractors of Eq. (18).
(20)
H. Jia, W. Shi and L. Wang et al. / Chaos, Solitons and Fractals 133 (2020) 109635
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Fig. 8. Coexisting attractors of Eq. (18).
which implies that the Hamiltonian energy is a constant and the energy exchange does not happen in Eq. (19), and the corresponding Hamiltonian energy is only related to the initial points (x0 , y0 , z0 , w0 ), i.e. H4D (x ) = 12 x20 + y20 + z02 + w0 . Potential energy circle: similarly, we can obtain the rate of energy exchange in potential energy as follows:
∂ U (x ) x˙ ∂x ∂ U (x ) = ({x, H4D (x )} ) = {U (x ), H4D (x )} ∂x = {U (x ), K (x )} = −K˙ (x ).
U˙ (x ) =
(21)
which reflects the rate of energy exchange in potential energy is only related to its instantaneous exchange with kinetic energy. Kinetic energy circle: similarly, we can obtain the rate of energy exchange in kinetic energy as follows:
∂ K (x ) x˙ ∂x ∂ K (x ) = ({x, H4D (x )} ) = {K (x ), H4D (x )} ∂x = {K (x ), U (x )} = −U˙ (x ),
K˙ (x ) =
(22)
which reflects the rate of energy exchange in kinetic energy is only related to its instantaneous exchange with potential energy.
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Based on the above analysis, for Eq. (18), energy exchange only happens between kinetic energy and potential energy, and therefore its Hamiltonian energy is conservative.
in this paper can provide a new physical model for chaos application, such as secure communication.
4.3. Numerical analysis and FPGA implementation
Declaration of Competing Interest
Now, some numerical analysis will be done to discuss the dynamics of the new Hamiltonian conservative system. Firstly, a numerical investigation will be done to study whether Eq. (18) is chaotic or not. When selecting the initial point (2, y0 , 0, 0), where y0 vary from −5 to 5, the corresponding Lyapunov exponents diagram and bifurcation diagram are provided, as shown in Fig. 5(a) and (b), respectively.This verifies system Eq. (18) is chaotic from numerical point of view. Furthermore, when selecting the initial point (2, −3.5, 0, 0 ), the corresponding phase portraits and Poincaré section are also given in Fig. 5(c) and (b), respectively. The chaotic dynamics of Eq. (18) is further demonstrated. Besides, the attractor in Fig. 5 is also hidden because equilibrium points of Eq. (18) do not exist. Furthermore, when selecting the initial point (2, 0.5, 0, 0), the corresponding kinetic energy K, potential energy U, and Hamiltonian energy H are obtained and marked in blue, red and green, respectively, as shown in Fig. 6. Where we can find U is symmetric to K, while H is constant and related to initial points, i.e., H = 2.215 for the initial point (2, 0.5, 0, 0). Which further shows that Eq. (18) is a Hamiltonian conservative system and energy exchange only happens between kinetic energy and potential energy from numerical analysis point of view. In addition, when selecting different initial points to investigate the dynamics of Eq. (18), some coexisting attractors are found. For example, the orbits starting from I p−0 (0.1, −0.1, 0.5, 1 ), I p−1 (−0.1, 0.1, 0.5, 1 ) and I p−2 (0.8, −1, 0.5, 1 ) are observed to show two chaotic attractors and one periodic attractor, marked in red, green and blue, respectively, as shown in Fig. 7(a). and the orbits starting from Ic−0 (−5, 1, 0.5, 1 ), Ic−1 (5, 1, 0.5, 1 ) and Ic−2 (0.8, −1, 0.5, 1 ) are observed to show some other chaotic attractors, and marked in red, green and blue, respectively, as shown in Fig. 7(b). Next, in order to investigate the coexisting hidden attractors from physical point of view, Eq. (18) is converted into hardware description language (HDL) which can be easily compiled, downloaded and recognized by FPGA hardware. And then the new Hamiltonian conservative system (18) is further implemented by FPGA, and the coexisting attractors found in numerical analysis are all in experiments observed by the oscilloscope, as shown in Fig. 8, these attractors are consistent with those in Fig. 7. All these results show the coexisting hidden attractors observed by FPGA hardware implementation are consistent with those observed by numerical analysis.
5. Conclusion In this paper, mechanics and energy of the Sprott-A System is first studied. It is found that the dynamics of the Sprott-A system change along the energy exchange between the conservative energy and the external supplied energy, and the action of the external supplied torque is the reason generating chaos. Subsequently, based on energy analysis of the Sprott-A System, the new Hamiltonian energy function is obtained. Furthermore, the new 4-D system is constructed, and the distinct characteristic is that the new 4-D system is a conservative system with a constant Hamiltonian energy. Besides, it is also a no-equilibrium system with some coexisting hidden attractors. Finally, both numerical analysis and FPGA implementation are given to show chaotic characteristic of the 4-D system and the existence of coexisting hidden attractors. The work
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement Hongyan Jia: Conceptualization, Methodology, Writing - original draft. Wenxin Shi: Investigation, Software, Validation. Lei Wang: Investigation, Software, Data curation. Guoyuan Qi: Methodology, Supervision, Funding acquisition. References [1] Arnold VI. Kolmogorovs hydrodynamic attractors. In: Vladimir I. Arnold-Collected Works. Springer; 1991. p. 429–32. [2] Pasini A, Pelino V. A unified view of Kolmogorov and Lorenz systems. Phys Lett A 20 0 0;275(5–6):435–46. [3] Pelino V, Maimone F, Pasini A. Energy cycle for the Lorenz attractor. Chaos Solitons Fractals 2014;64:67–77. [4] Qi G, Liang X. Force analysis of Qi chaotic system. Int J Bifurc Chaos 2016;26(14):1650237. [5] Qi G, Zhang J. Energy cycle and bound of Qi chaotic system. Chaos Solitons Fractals 2017;99:7–15. [6] Qi G. Energy cycle of brushless DC motor chaotic system. Appl Math Model 2017;51:686–97. [7] Yang Y, Qi G. Comparing mechanical analysis with generalized-competitive– mode analysis for the plasma chaotic system. Phys Lett A 2019;383(4):318–27. [8] Qi G, Liang X. Mechanism and energy cycling of the Qi four-wing chaotic system. Int J Bifurc Chaos 2017;27(12):1750180. [9] Liang X, Qi G. Mechanical analysis and energy cycle of Chen chaotic system. Braz J Phys 2017;47(3):288–94. [10] Jia H, Guo Z, Wang S, Chen Z. Mechanics analysis and hardware implementation of a new 3D chaotic system. Int J Bifurc Chaos 2018;28(13):1850161. [11] Qi G, Liang X. Mechanical analysis of Qi four-wing chaotic system. Nonlinear Dyn 2016;86(2):1095–106. [12] Qi G. Modelings and mechanism analysis underlying both the 4D euler equations and Hamiltonian conservative chaotic systems. Nonlinear Dyn 2019;95(3):2063–77. [13] Mahmoud GM, Ahmed M. Analysis of chaotic and hyperchaotic conservative complex nonlinear systems. Miskolc Math Notes 2017;18(1):315–26. [14] Cang S, Wu A, Wang Z, Chen Z. Four-dimensional autonomous dynamical systems with conservative flows: two-case study. Nonlinear Dyn 2017;89(4):2495–508. [15] Vaidyanathan S, Volos C. Analysis and adaptive control of a novel 3-Dconservative no-equilibrium chaotic system. Arch Control Sci 2015;25(3):333–53. [16] Cang S, Wu A, Zhang R, Wang Z, Chen Z. Conservative chaos in a class of nonconservative systems: theoretical analysis and numerical demonstrations. Int J Bifurc Chaos 2018;28(7):1850087. [17] Singh JP, Roy BK. Five new 4-D autonomous conservative chaotic systems with various type of non-hyperbolic and lines of equilibria. Chaos Solitons Fractals 2018;114:81–91. [18] Dong E, Yuan M, Du S, Chen Z. A new class of Hamiltonian conservative chaotic systems with multistability and design of pseudo-random number generator. Appl Math Model 2019;73:40–71. [19] Sprott JC. Some simple chaotic flows. Phys Rev E 1994;50(2):R647. [20] Sprott J. Some simple chaotic jerk functions. Am J Phys 1997;65(6):537–43. [21] Hoover WG. Remark on some simple chaotic flows. Phys Rev E 1995;51(1):759. [22] Posch HA, Hoover WG, Vesely FJ. Canonical dynamics of the nosé oscillator: Stability, order, and chaos. Phys Rev A 1986;33(6):4253. [23] Hoover WG. Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 1985;31(3):1695. [24] Jafari S, Sprott J, Nazarimehr F. Recent new examples of hidden attractors. Eur Phys J Spec Top 2015;224(8):1469–76. [25] Liu L, Du C, Zhang X, Li J, Shi S. Dynamics and entropy analysis for a new 4-Dhyperchaotic system with coexisting hidden attractors. Entropy 2019;21(3):287. [26] Lai Q, Nestor T, Kengne J, Zhao X-W. Coexisting attractors and circuit implementation of a new 4D chaotic system with two equilibria. Chaos Solitons Fractals 2018;107:92–102. [27] Kengne J, Njitacke Z, Fotsin H. Dynamical analysis of a simple autonomous jerk system with multiple attractors. Nonlinear Dyn 2016;83(1-2):751–65. [28] Bao B, Bao H, Wang N, Chen M, Xu Q. Hidden extreme multistability in memristive hyperchaotic system. Chaos Solitons Fractals 2017;94:102–11.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Energy analysis of Sprott-A system and generation of a new Hamiltonian conservative chaotic system with coexisting hidden attractorsR Hongyan Jia a,∗, Wenxin Shi a, Lei Wang a, Guoyuan Qi b a b
Department of Automation, Tianjin University of Science and Technology, Tianjin 300222, PR China School of Electrical Engineering and Automation, Tiangong University, Tianjin 300387, PR China
a r t i c l e
i n f o
Article history: Received 15 October 2019 Revised 19 December 2019 Accepted 15 January 2020
Keywords: Energy analysis Hamiltonian energy Casimir energy Conservative systems Coexisting hidden attractors
a b s t r a c t The paper firstly investigates energy cycle of the Sprott-A system by transforming the Sprott-A system into the Kolmogorov-type system. We found the dynamics of the Sprott-A system are influenced by the change along the energy exchange between the conservative energy and the external supplied energy. And the action of the external supplied torque is the main reason that the Sprott-A system generates chaos. Secondly, based on energy analysis of the Sprott-A system, a new four-dimension (4-D) chaotic system is obtained. The new 4-D chaotic system is a conservative system with a constant Hamiltonian energy. Besides, it is also a no-equilibrium system, this means that the new 4-D chaotic system can exhibit hidden characteristics. Further, the coexisting hidden attractors are found when selecting different initial points. Finally, the new 4-D chaotic system is implemented by FPGA, and the coexisting attractors observed are consistent with those found in numerical analysis, which in experiment verifies the existence of coexisting hidden attractors of the new 4-D chaotic system from physical point of view. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Since a Kolmogorov System describing a dissipative dynamical system or a hydrodynamical instability with a Hamiltonian energy function was reported in 1991 [1], energy analysis for nonlinear chaotic systems has attracted more and more attention [2–11]. The above research is mainly based on the Kolmogorov System, and focuses on investigating the physical meaning of variables in the chaotic systems, finding the reasons for chaos generation, analyzing the variation of dynamics characteristics caused by energy cycle, and further providing feasible chaotic models for application. That is, energy analysis is a new tool to investigate, analyze and utilize chaos in nonlinear systems from viewpoint of energy cycle. In addition, conservative chaotic systems (CCSs) are also increasingly emerged as a new research interest [12–18]. Generally speaking, CCSs are referred to those systems with conservative quantities, such as a constant Hamiltonian energy, a zero divergence or a zero sum of Lyapunov exponents. Compared with dissi-
R This work was supported in part by the National Natural Science Foundation of China (Grant nos. 61573199 and 61873186)) and the Tianjin Natural Science Foundation (Grant nos. 17JCZDJC38300 and 18JCQNJC74000). ∗ Corresponding author. E-mail address: [email protected] (H. Jia).
https://doi.org/10.1016/j.chaos.2020.109635 0960-0779/© 2020 Elsevier Ltd. All rights reserved.
pative chaotic systems (DCSs) whose asymptotic motion settle onto a set of a strange attractor, CCSs are more suitable to be used in information security. The reasons are mainly as follows [12,16]: (a) CCSs do not produce attractor, (b) ergodicity of CCSs is richer than that of DCSs, (c) for CCSs, different initial conditions lead to distinct different dynamics. Therefore, CCSs begin to attract more and more attention. According to Refs. [12–17], CCSs are generally classified as two categories: the Hamiltonian conservative chaotic systems (HCCSs) and the non-Hamiltonian conservative chaotic systems (non-HCCSs). And compared with non-HCCSs which is only conservative in volume or sum of Lyapunov exponents, HCCS is a category of stricter conservative chaotic systems [12]. Therefore, the generation and analysis of HCCSs have been attracting more and more research interest. Sprott-A system is one of some simple chaotic flows reported in 1994, and it is also a special case of equations describing Nosé -Hoover dynamics systems [19,20]. It is generally considered as the oldest example of rare with no equilibria and the simplest mechanical system exhibiting chaos. Besides, because of a zero sum of Lyapunov exponents and no equilibria, it is classified as a conservative system with hidden coexisting attractors [21–24]. It is worth mentioning that hidden coexisting attractors are also a recent new research hotspot [25–28]. Therefore, the research based on the Sprott-A system begin to attract increasing attention [29–
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32]. Besides, with the development of chaos application, the FPGA implementation of chaotic systems has attracted more and more attention [10,18]. However, the research on FPGA implementation of HCCSs is seldomly reported, which can limit their application in engineering fields. FPGAs can be reprogrammed indefinitely, loading a new design solution takes only a few hundred milliseconds, and reconfiguration can reduce hardware overhead. The operating frequency of an FPGA is determined by the FPGA chip and the design. Some harsh requirements can be met by modifying the design or replacing a faster chip. Using the FPGA’s parallel processing algorithms and rich logic resources, video chaos-based cryptography needing fast process can be implemented. In this paper, the Sprott-A system is first transformed into the Kolmogorov-type system. Then, energy cycle of the Sprott-A system, the affection of the variation of energy on its dynamics and the key reason why it generates chaos are respectively discussed. Subsequently, according to energy analysis of the Sprott-A system, the paper obtains a new 4-D conservative chaotic system without equilibrium. Further, it is a Hamiltonian conservative chaotic system, i.e., its corresponding Hamiltonian is constant. In addition, according to numerical analysis, some coexisting hidden attractors are found to exist in the new 4-D chaotic system when selecting different initial points. Finally, the new 4-D chaotic system is also implemented by FPGA hardware, which further shows the existence of coexisting attractors of the new 4-D conservative chaotic system from physical point of view. The rest of this paper is organized as follows: in Section 2, we give the transformation of the Sprott-A system to the Kolmogorovtype system, which only includes two parts: the conservative part and the external supply part. In Section 3, we discuss and analyze energy cycle of the Sprott-A system. In Section 4, based on the analysis of the Hamiltonian energy, we obtain a 4-D Hamiltonian conservative chaotic system without equilibrium. And the corresponding numerical analysis and FPGA implementation are also given to show the characteristics of the 4-D Hamiltonian conservative chaotic system. Finally, the paper is concluded in Section 5.
Generally, Kolmogorov system is a very useful bridge which can help analyze the mechanical properties of dynamics systems, find the reason why chaos can generate and investigate the effect of variation of energy on dynamics of systems. Kolmogorov system can be written as follows [1,2]:
x˙ = {x, H (x )} − x + u = x × ∇ H (x ) − x + u (1)
where x ∈ Rn represents the state vector, H (x ) : Rn → R refers to Hamiltonian energy, J (x ) ∈ Rn×n represents a skew-symmetric matrix. {x, H (x )}, x and u represents the conservative part, the dissipative part and the external supply of Eq. (1), respectively. According to Refs. [3–12], some conclusions can be concluded as follows: Remark 1. (1) The term {x, H (x )} = x × ∇ H (x ) = J (x )∇ H (x ) represents Hamiltonian energy conservation, i.e., when x and u in (1) are absent, Hamiltonian energy is a nonzero constant, and the rate of change of Hamiltonian is H˙ (x ) = 0. (2) Hamiltonian energy can be defined as H (x ) = K (x ) + U (x ) n where K (x ) = 12 i x2i , i = 1, 2, ..., n and U (x ) refer to the kinetic energy and the potential energy, respectively.
Furthermore, according to Refs. [12–18], some more comments on Hamiltonian conservative chaotic systems can be concluded as follows: Remark 2. (1) There generally exists two classes of Hamiltonian conservative chaotic systems (HCCSs): traditional HCCSs and generalized HCCSs. The former can be described by a Hamiltonian function H (x )(q, p, t ), and they need to satisfy even-dimensional Hamiltonian equation. The latter can be expressed by a state equation x˙ = {x, H (x )}, and they can be found in either evendimensional systems or odd-dimensional systems. (2) Generalized HCCSs refer to a system whose Hamiltonian energy is a nonzero constant, and the rate of change of Hamiltonian is H˙ (x ) = 0. (3) Some feature of HCCSs can be valuable for security communication. 2.2. Sprott-A system In 1994, Sprott reported some algebraically simple threedimensional chaotic systems, one of those is subsequently called as the Sprott-A system, as follows [18]:
x˙ = y y˙ = −x + yz, z˙ = 1 − y2
2.1. Preliminaries
i=1
i=1
ergy is called as the Casimir power C˙ (x ), which represents the rate of change of energy exchange between the dissipative part and the external supply of the system. (4) When i = 1, i = 1, 2, ..., n are constant,K (x ) is consistent with Casimir energy C (x ), and Hamiltonian energy can be written as H ( x ) = C ( x ) + U ( x ). (5) Chaos can be found to exist in a Hamiltonian energy conservative system when C˙ (x ) oscillates irregularly near zero with bound.
2. Transformation of Sprott-A system into Kolmogorov-type system
= J (x )∇ H (x ) − x + u,
(3) For Eq. (1), Casimir energy is generally defined as the C (x ) = n 1 2 xi , i = 1, 2, ..., n, and the rate of change of the Casimir en2
(2)
It is also called the Nosé-Hoover oscillator, because it is a special case of equations describing the Nosé-Hoover dynamics systems [19–22]. According to Refs. [19–23], some characteristics of the Sprott A system are summarized as follows: Remark 3. (1) It is the simplest mechanical system exhibiting chaos. (2) It is a conservative system, because the sum of Lyapunov exponents is zero, which reflects that divergence in phase volume is zero from statistic point of view. (3) It is the oldest example of rare, ows with no equilibria, and can show a hidden attractor coexisting with a set of nested tori. Next, in order to investigate the key reason why it can generate chaos, and discuss how the variation of energy affects its dynamics. The Sprott-A system will be firstly transformed to the Kolmogorovtype system. 2.3. Kolmogorov-type transformation We rewrite the Sprott-A system as
x˙ y˙ z˙
= {x, H (x )} + u = Js (x )∇ H (x ) + u,
(3)
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3
Fig. 1. Hamiltonian energy and its derivative of Eq. (12).
where
Js (x ) =
0 −1 0
1 0 −y
0 y ,u = 0
0 1 2 0 , H (x ) = x + y2 + z 2 , 2 1
then we can obtain the rate of change of Hamiltonian energy as follows:
H˙ (x ) =
∂ H (x ) ∂ H (x ) x˙ = ( {x, H ( x )} ) = 0. ∂x ∂x
(6)
which implies the energy is a constant, i.e.
furthermore, the following conclusion can be obtained: Conclusion 1
H (x ) =
(1) The Sprott-A system is transformed into the Kolmogorov-type system, e.g., Eq. (3). (2) There are two parts in (3): the conservative part and the external supply part. The first term represents the conservative torque described by H (x ) = 12 x2 + y2 + z2 , and the second term represents the external supplied torque, respectively. (3) Eq. (3) can display chaotic behavior, and it is not Hamiltonian conservative because of the existence of the external supplied torque u. (4) For Eq. (3), C (x ) = H (x ) = 12 x2 + y2 + z2 .
1 2 x + y2 + z2 = c, 2
where c represents constant. This indicates that Eq. (5) is a Hamiltonian energy conservative system. In addition, for Eq. (5), the Casimir energy C(x) is same as the Hamiltonian Energy H(x), i.e., the Sprott-A system is also a Casimir energy conservative system when the external supplied torque is absent. According to Refs. [5,8,12], chaos generally can’t happen. Furthermore, we √ also find that H(x) implies an invariant sphere whose radius is 2C , and shows that the energy exchange does not happen in Eq. (5). Therefore, the orbit starting from an initial point (x0 , y0 , z0 ) keeps on r the surface of the corresponding sphere Sr0 described by
3. Energy of Sprott-A System
(7)
Srr0 : x2 + y2 + z2 = r02 ,
(8)
3.1. Energy cycle
where r0 =
Generally, linear or nonlinear vectors in equations describing dynamic systems are often coupled, which means that it is difficult to directly study the torques. However, every energy is related to its corresponding torque or force. Therefore, investigating energy cycling is a useful method to study the reason why some behavior can generate. In this study, Casimir energy and Hamiltonian energy adopted in some research on mechanics and energy analysis of chaotic systems [2–13] are mainly used to investigate the affection of the variation of the energy on the dynamics of Sprott-A system. By analyzing Eq. (3), it can be found that two forms of energies exist in the Sprott-A system: the conservative energy and the external supplied energy, which relates to the conservative torque described by Hamiltonian energy and the external supplied torque described by the external supplied energy, respectively. Next, Eq. (3) is rewritten as follows:
Case 2. Energy circle under the conservative torque and the external supplied torque, i.e., Eq. (4), then we have the rate of change of Hamiltonian as follows:
x˙ = {x, H (x )} + x, S(x ),
(4)
= ∂ S∂(xx ) , S(x ) represents the external supplied en-
where x, S(x ) ergy. Next, energy cycle of the Sprott-A system will be discussed as follows: Case 1. Energy circle under the conservative torque is
x˙ = {x, H (x )},
(5)
x20 + y20 + z02 .
∂ H (x ) x˙ ∂x ∂ H (x ) = ( {x , H ( x )} + x , S ( x ) ) ∂x = H ( x ), S ( x ).
H˙ (x ) =
(9)
which reflects the rate of change of the conservative torque is only related to the external supplied torque. At the same time, Eq. (9) also shows that the energy exchange in Eq. (4) happens between the conservative energy and the external supplied energy. According to Refs. [3–12], the reason why the Sprott-A system can generate chaos is found to be the action of the external supplied torque which makes the Hamiltonian energy change. Furthermore, because energy exchange exists in the Sprott-A system, the orbit starting from an initial point (x0 , y0 , z0 ) departs from the surr face of the corresponding sphere Sr0 , and then shows chaos. 3.2. Energy analysis In this section, two types of torques of the Sprott-A system will be numerically discussed to investigate the key factors that chaos generates.
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Fig. 2. Orbits of Eq. (12).
Fig. 3. Hamiltonian energy and its derivative of Eq. (2).
Fig. 4. Orbits of Eq. (2).
Firstly, we define two the invariant spheres:
2
Sr1 : x2 + y2 + z
= 1,
(10)
and
Sr10 : x2 + y2 + z2 = 100.
(11)
Then two groups of initial points from the two spheres are selected to numerically investigate dynamics of the Sprott-A system. One √ √ group of initial points are Ir1 (1, 0, 0 ), Ir1 (0, 1, 0 ) and Ir1 ( 22 , 22 , 0 ) on the surface of the invariant sphere Sr1 . The other of initial √ group √ points are Ir10 (10, 0, 0 ), Ir10 (0, 10, 0 ) and Ir10 (−5 2, 5 2, 0 ) on the surface of the invariant sphere Sr10 . In subsequent study, the orbits
H. Jia, W. Shi and L. Wang et al. / Chaos, Solitons and Fractals 133 (2020) 109635
5
Fig. 5. Lyapunov exponents, bifurcation diagram and chaotic attractor of Eq. (18).
√
√
starting from Ir1 (0, 1, 0 ), Ir1 (1, 0, 0 ) and Ir1 ( 22 , 22 , 0 ) are marked in red, green and blue, respectively. Similarly, √ √ the orbits starting from Ir10 (0, 10, 0 ), Ir10 (10, 0, 0 ) and Ir10 (−5 2, 5 2, 0 ) are also marked in red, green and blue, respectively. Next, analysis of Hamiltonian energy will be only given to investigate energy cycle, because we define C (x ) = H (x ) = 12 x2 +
y2 + z2 for the Sprott-A system. Analysis for Case 1: Analysis under the conservative torque, i.e.,
x˙ y˙ z˙
= Js (x )∇ H (x ) =
0 −1 0
1 0 −y
0 y 0
x y . z
(12)
Its biggest Lyapunov exponent is 0, and the corresponding divergence is
∇ · f = z,
(13)
which indicates it is difficult to determine whether Eq. (12) is conservative or dissipative in volume. But according to Remark 1, we found Eq. (12) is Hamiltonian conservative, and its corresponding Hamiltonian energy is a constant. Energy exchange does not take place in Eq. (12). For the two groups of initial points on the surface of the spheres Sr1 and Sr10 , the corresponding Hamiltonian energy and its derivative are shown in Fig. 1(a) and (b), respectively. And
the corresponding orbits are observed to keep on the surface of the corresponding sphere Sr1 and Sr10 , as shown in Fig. 2(a) and (b), respectively. All these numerical results are consistent with Eqs. (6)– (8). Furthermore, we found the orbits in Fig. 2(a) and (b) are actually is on heteroclinic orbits described by Ref. [31]. Based on the above analysis, the Sprott-A system without the external torque is furtherly verified to be Hamiltonian conservative. But the chaotic dynamic is not found. Analysis for Case 2: Eq. (2) is the object of study, in other word, analysis for Eq. (2) is actually to investigate how dynamics of a Hamiltonian conservative system change via adding the external excitation. Similarly, for the two groups of initial points on the surface of the spheres Sr1 and Sr10 ,the corresponding Hamiltonians and their derivative are shown in Fig. 3(a) and (b), respectively, which are consistent with Eq. (9). We found the corresponding Hamiltonian energy of Eq. (2) is the variable with H˙ (x ) = 0, which means energy exchange happens. And the corresponding orbits are observed to depart from the surface of the corresponding sphere Sr1 and Sr10 , and show some attractors of (2), as shown in Fig. 4(a) and (b), respectively, which are consistent with dynamics of the Sprott-A system. Furthermore, from viewpoint of numerical analysis, the action of the external supplied torque is found to be the main reason why the Sprott-A system can generate chaos. According to the above analysis, we found there are only two parts in the Sprott-A system, i.e., the conservative toque and the external supplied torque, and the external supplied torque described by the external supplied energy is the main reason why it can generate chaos. Now, one question occurs: whether is the external supplied
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where
⎡ ⎢
J4D_s (x ) = ⎣
0 −1 0 0
1 0 −y 0
0 y 0 −1
⎤
0 0⎥ , 1⎦ 0
and then a conclusion can be obtained that Eq. (16) is a Hamiltonian conservative system described by the Hamiltonian energy H4D (x ) = 12 x2 + y2 + z2 + w. Proof: because J4D_s is a skew-symmetric matrix, according to Remark 1, then
H˙ 4D (x ) = ∇ H4TD (x )x˙ = ∇ H4TD (x )J4D_s (x )∇ H4D (x ) = 0.
(17)
Therefore, Eq. (16) is a Hamiltonian conservative system. Now, according to Eq. (16), we can obtain a general Hamiltonian conservative system as follows:
Fig. 6. Energy of Eq. (18).
energy included in Hamiltonian energy, and can a Hamiltonian energy conservative system be obtained?
⎧ ⎪ ⎨x˙ = y
y˙ = −x + yz
2 ⎪ ⎩z˙ = −y + 1
,
(18)
w˙ = −z
4. Generation and analysis of a new Hamiltonian energy conservative system 4.1. A new Hamiltonian energy conservative chaotic system Now, according to Eq. (9), we rewrite the Casimir power of the Sprott-A system as follows:
C˙ (x ) =
∂ C (x ) ({x, C (x )} + x, S(x ) ) = z, ∂x
(14)
which shows that C˙ (x ) is variable and is related to z. Next, according to Eq. (14), we can obtain a new Hamiltonian energy function with both kinetic energy and potential energy as follows:
H4D (x ) =
1 2 x + y2 + z2 + w, 2
(15)
where w = zdt represents the potential energy, and thus a new 4-D system can be obtained,
⎡ ⎤ x˙
⎢ y˙ ⎥ ⎣ z˙ ⎦ = J4D_s (x )∇ H4D (x ), w˙
(16)
Therefore, by constructing a new Hamiltonian energy, the above question is solved. Obviously, Eq. (18) is conservative in the Hamiltonian, and it is a system without equilibrium. Generally, an attractor is hidden when its basin of attraction does not intersect with a small neighborhood of any equilibrium point. Therefore, for Eq. (18) without equilibrium, all attractors are hidden and conservative in Hamiltonian energy. Furthermore, Eq. (18) can be written as follows:
x˙ = {x, H4D (x )}.
(19)
Which shows there is a conservative torque in Eq. (19), and the corresponding conservative torque consists of kinetic energy K (x ) = 12 x2 + y2 + z2 and potential energy U (x ) = w. 4.2. Energy circle of the new Hamiltonian conservative system Here, Hamiltonian energy circle, kinetic energy circle, and potential energy circle will be respectively discussed. Hamiltonian energy circle: according to Refs. [3–12], we can obtain the rate of energy exchange in Hamiltonian energy as follows:
H˙ 4D (x ) =
∂ H4D (x ) ∂ H4D (x ) x˙ = ({x, H4D (x )} ) = 0, ∂x ∂x
Fig. 7. Coexisting attractors of Eq. (18).
(20)
H. Jia, W. Shi and L. Wang et al. / Chaos, Solitons and Fractals 133 (2020) 109635
7
Fig. 8. Coexisting attractors of Eq. (18).
which implies that the Hamiltonian energy is a constant and the energy exchange does not happen in Eq. (19), and the corresponding Hamiltonian energy is only related to the initial points (x0 , y0 , z0 , w0 ), i.e. H4D (x ) = 12 x20 + y20 + z02 + w0 . Potential energy circle: similarly, we can obtain the rate of energy exchange in potential energy as follows:
∂ U (x ) x˙ ∂x ∂ U (x ) = ({x, H4D (x )} ) = {U (x ), H4D (x )} ∂x = {U (x ), K (x )} = −K˙ (x ).
U˙ (x ) =
(21)
which reflects the rate of energy exchange in potential energy is only related to its instantaneous exchange with kinetic energy. Kinetic energy circle: similarly, we can obtain the rate of energy exchange in kinetic energy as follows:
∂ K (x ) x˙ ∂x ∂ K (x ) = ({x, H4D (x )} ) = {K (x ), H4D (x )} ∂x = {K (x ), U (x )} = −U˙ (x ),
K˙ (x ) =
(22)
which reflects the rate of energy exchange in kinetic energy is only related to its instantaneous exchange with potential energy.
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Based on the above analysis, for Eq. (18), energy exchange only happens between kinetic energy and potential energy, and therefore its Hamiltonian energy is conservative.
in this paper can provide a new physical model for chaos application, such as secure communication.
4.3. Numerical analysis and FPGA implementation
Declaration of Competing Interest
Now, some numerical analysis will be done to discuss the dynamics of the new Hamiltonian conservative system. Firstly, a numerical investigation will be done to study whether Eq. (18) is chaotic or not. When selecting the initial point (2, y0 , 0, 0), where y0 vary from −5 to 5, the corresponding Lyapunov exponents diagram and bifurcation diagram are provided, as shown in Fig. 5(a) and (b), respectively.This verifies system Eq. (18) is chaotic from numerical point of view. Furthermore, when selecting the initial point (2, −3.5, 0, 0 ), the corresponding phase portraits and Poincaré section are also given in Fig. 5(c) and (b), respectively. The chaotic dynamics of Eq. (18) is further demonstrated. Besides, the attractor in Fig. 5 is also hidden because equilibrium points of Eq. (18) do not exist. Furthermore, when selecting the initial point (2, 0.5, 0, 0), the corresponding kinetic energy K, potential energy U, and Hamiltonian energy H are obtained and marked in blue, red and green, respectively, as shown in Fig. 6. Where we can find U is symmetric to K, while H is constant and related to initial points, i.e., H = 2.215 for the initial point (2, 0.5, 0, 0). Which further shows that Eq. (18) is a Hamiltonian conservative system and energy exchange only happens between kinetic energy and potential energy from numerical analysis point of view. In addition, when selecting different initial points to investigate the dynamics of Eq. (18), some coexisting attractors are found. For example, the orbits starting from I p−0 (0.1, −0.1, 0.5, 1 ), I p−1 (−0.1, 0.1, 0.5, 1 ) and I p−2 (0.8, −1, 0.5, 1 ) are observed to show two chaotic attractors and one periodic attractor, marked in red, green and blue, respectively, as shown in Fig. 7(a). and the orbits starting from Ic−0 (−5, 1, 0.5, 1 ), Ic−1 (5, 1, 0.5, 1 ) and Ic−2 (0.8, −1, 0.5, 1 ) are observed to show some other chaotic attractors, and marked in red, green and blue, respectively, as shown in Fig. 7(b). Next, in order to investigate the coexisting hidden attractors from physical point of view, Eq. (18) is converted into hardware description language (HDL) which can be easily compiled, downloaded and recognized by FPGA hardware. And then the new Hamiltonian conservative system (18) is further implemented by FPGA, and the coexisting attractors found in numerical analysis are all in experiments observed by the oscilloscope, as shown in Fig. 8, these attractors are consistent with those in Fig. 7. All these results show the coexisting hidden attractors observed by FPGA hardware implementation are consistent with those observed by numerical analysis.
5. Conclusion In this paper, mechanics and energy of the Sprott-A System is first studied. It is found that the dynamics of the Sprott-A system change along the energy exchange between the conservative energy and the external supplied energy, and the action of the external supplied torque is the reason generating chaos. Subsequently, based on energy analysis of the Sprott-A System, the new Hamiltonian energy function is obtained. Furthermore, the new 4-D system is constructed, and the distinct characteristic is that the new 4-D system is a conservative system with a constant Hamiltonian energy. Besides, it is also a no-equilibrium system with some coexisting hidden attractors. Finally, both numerical analysis and FPGA implementation are given to show chaotic characteristic of the 4-D system and the existence of coexisting hidden attractors. The work
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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