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Subharmonic bifurcations and chaotic motions for a class of inverted pendulum system
Chaos, Solitons and Fractals 99 (2017) 270–277
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Subharmonic bifurcations and chaotic motions for a class of inverted pendulum system Liangqiang Zhou∗, Shanshan Liu, Fangqi Chen∗ Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China
a r t i c l e
i n f o
Article history: Received 26 October 2016 Revised 9 January 2017 Accepted 3 April 2017
Keywords: Inverted pendulum Chaos Subharmonic bifurcation Melnikov method Heteroclinic orbit
a b s t r a c t Using both analytical and numerical methods, global dynamics including subharmonic bifurcations and chaotic motions for a class of inverted pendulum system are investigated in this paper. The expressions of the heteroclinic orbits and periodic orbits are obtained analytically. Chaos arising from heteroclinic intersections is studied with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are obtained. The conditions for subharmonic bifurcations are also obtained. It is proved that the system can be chaotically excited through finite subharmonic bifurcations. Some new dynamical phenomena are presented. Numerical simulations are given, which verify the analytical results.
1. Introduction The inverted pendulum system has wide applications in precision instruments, robot control, missile intercept control system, spacecraft attitude control and so on. Therefore, it is of great significance to study nonlinear dynamics of this system. A lot of results on this subject have been obtained in the past two decades. Bifurcations in the inverted pendulum system have been investigated by many researchers in the past years. Via the normal form theory, perturbation analysis and equivariant singularity theory, a reversible bifurcation analysis of the inverted pendulum was given by Broer et al. [1]. Using an approximating integrable normal form and equivariant singularity theory, Broer et al. [2] studied the qualitative dynamics of the Poincare´ map corresponding to the central periodic solution and bifurcations of the inverted pendulum system. Ponce et al. [17] investigated bifurcations of an inverted pendulum with saturated Hamiltonian control laws. Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation were investigated by Lenci and Rega [12]. It was found that the existence and the stability of the cycles depended on both classical (saddle-node and period-doubling) and non-classical bifurcations. With a quantitative theory together with numerical simulations, Butikov [4] studied the dynamic stabilization of an inverted pendulum. The dynamics of an inverted pendulum subjected to high-frequency excitation was studied by Yabuno et al. [22]. The stability of the stable equilibrium states under the effect ∗
Corresponding author. E-mail addresses: [email protected] (L. Zhou), [email protected] (F. Chen).
http://dx.doi.org/10.1016/j.chaos.2017.04.004 0960-0779/© 2017 Elsevier Ltd. All rights reserved.
© 2017 Elsevier Ltd. All rights reserved.
of the tilt was discussed non-locally. An analogy of the bifurcation of the inverted pendulum to that of the buckling phenomenon was also presented. By using the method of multiple scales and numerical simulations, Yang et al. [23] investigated stability and Hopf bifurcation in an inverted pendulum with delayed feedback control. Via multiple delayed proportional gains, Boussaada et al. [3] studied a codimension-three triple zero bifurcation of an inverted pendulum on a cart moving horizontally. Using the centre manifold, Sieber and Krauskopf [18] investigated a codimension-three triplezero eigenvalue bifurcation of an inverted pendulum with delayed feedback control. Using a center manifold reduction, Landry et al. [9] investigated local dynamics of an inverted pendulum with delayed feedback control. It was shown that the system undergoes a supercritical Hopf bifurcation at the critical delay. Employing a combination of analytical and numerical methods, the stability and bifurcations of two types of double impact periodic orbits for an inverted pendulum impacting between two rigid walls were studied by Shen and Du [19]. Especially, grazing bifurcations were presented there. Chaotic motions of the inverted pendulum system have been also investigated in the past years. With numerical methods, Kim and Hu [8] studied bifurcations and transitions to chaos in an inverted pendulum. It was found that an infinite sequence of perioddoubling bifurcations, leading to chaos, followed each destabilization of the inverted state. By using a Neyman–Pearson lemma like technique, Lenci [10] investigated the suppression of chaos by means of bounded excitations in an inverted pendulum. With experimental methods, Chen et al. [5] studied chaotic motions of a inverted pendulum system. It was found that the system may
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271
A = r0 ωd2 /g = R0 ω2 ≡ f ω2 , r0 is the excitation amplitude, R is the length of the pendulum, ωd is the angular frequency of motor. Denoting θ = x,then system (1) is written as follows : r
⎧ dx ⎪ ⎨ =y dτ ⎪ ⎩ dy = sin x − k˜ x − γ y + f ω2 cos x cos ωτ dτ
(2)
Assuming the elastic coefficient of the spring together with the damping coefficient and the parameter f are small, setting k˜ = ε k¯ , γ = ε γ¯ , f = ε f¯, system (2) can be written as
⎧ dx ⎪ ⎨ =y dτ ⎪ ⎩ dy = sin x − ε k¯ x − ε γ¯ y + ε f¯ω2 cos x cos ωτ dτ
When ε = 0, the unperturbed system of (3) is
Fig. 1. The physical model of the inverted pendulum.
undergo chaos through period doubling bifurcations. Via a systematic numerical investigation method, Lenci et al. [11] studied the nonlinear dynamics of an inverted pendulum between lateral rebounding harriers. Three different families of considerably variable attractors-periodic, chaotic, and rest positions with subsequent chattering were found. Employing the harmonic balance method and Melnikov theory, nonlinear dynamics of an inverted pendulum driven by airflow was investigated by Nbendjo [14]. Horseshoes chaos may exist in this system. With the Melnikov method, homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external quasi periodic excitations was analyzed by Gao and Du [7]. Smale horseshoe-type chaotic dynamics may occur in this system. Via Poincare´ maps, Gandhi and Meena [6] studied chaotic dynamics of an inverted flexible pendulum with tip mass. By means of the normal form theory, the Melnikov method and numerical methods, Perez-Polo et al. [16] studied the stability and chaotic behavior of a plus integral plus derivative (PID) controlled inverted pendulum subjected to harmonic base excitations. It was shown that when the pendulum was close to the unstable pointing-up position, the PID parameters were changed and the chaotic motion was destroyed. In this paper, global dynamics including subharmonic bifurcations and chaotic motions for a class of inverted pendulum system are investigated analytically with the subharmonic Melnikov method and Melnikov method, respectively. The mechanism and parameter conditions of chaotic motions are obtained rigorously. The critical curves separating the chaotic and non-chaotic regions are plotted. The chaotic feature on the system parameters is discussed in detail. The conditions for subharmonic bifurcations are also presented. It is proved that the system can be chaotically excited through finite subharmonic bifurcations. Numerical simulations verify the analytical results. 2. The dynamic model and analysis of the orbits for the inverted pendulum
where τ =
g
˜ R t, k =
k a2 mgR ,
ficient of the spring, γ =
(1)
g/R, k is the elastic coef-
ω = ωd / √γ˜ , γ˜ is the damping coefficient,
m
g/R
⎧ dx ⎪ ⎨ =y dτ ⎪ ⎩ dy = sin x dτ
(4)
which is a planar Hamiltonian system with the Hamiltonian
H (x, y ) =
y2 + cos(x ) − 1 ≡ h 2
(5)
System (4) has saddles (2l π , 0 )(l = 0, ±1, ±2, · · · ), and centers ((2l + 1 )π , 0 )(l = 0, ±1, ±2, · · · ). There exists a pair of heteroclinic orbits connecting (2lπ , 0)) to ((2l + 2 )π , 0 ) when h = 0. Due to the periodic symmetry, we just need to consider the dynamic behaviors of the system on the interval [0, 2π ] of the x-axis. In this case, system (4) has two saddles O(0, 0), A(2π , 0), and one center B(π , 0). To solve the expressions of the heteroclinic orbits, setting h = 0 in (5) one can obtain that
dx = y = ± 2(1 − cos x ) dτ
(6)
Separating variables for (6) and integrating, we can get
tan
x 4
= e ±τ
(7)
Therefore we can obtain the expressions of the homoclinic orbits as follows
+ :
x+ (τ ) = 4 arctan(eτ ) y+ (τ ) = 2 sech(τ )
(8)
− :
x− (τ ) = 4 arctan(e−τ ) y− (τ ) = −2 sech(τ )
(9)
There also exist a family of periodic orbits ± (k) around the center B inside ± for −2 < h < 0. To solve the expressions of the periodic orbits, we rewrite (5) as follows
cos x = (h + 1 ) −
Considering the model as in Fig. 1, assuming O is the center of the reciprocating motion for the system, choosing this point as the origin of the inertial coordinate system and the connection point of the inverted pendulum rod and the trolley as the origin of the non inertial coordinate system, then the dynamic equation of the ball in the non inertial coordinate system is [5]
d2 θ dθ = sin θ − k˜ θ − γ + A cos θ cos ωτ dτ dτ 2
(3)
y2 2
(10)
consequently, 2
1 − sin x =
y4 − ( h + 1 )y2 + ( h + 1 )2 4
(11)
i.e.,
dy 1 = (C 2 − y2 )(k2C 2 + k2 y2 ) dτ C
(12)
where k = 1 − k2 . Comparing the coefficients of like powers, we can obtain that
C = ±2k, h = 2(k2 − 1 )
(13)
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Thus, M+ (τ0 ) has simple zeros and chaotic motions occur if and only if
k¯ I0+ + γ¯ I1+ f¯I ω2 < 1
(17)
2+
i.e.,
πω π 2 k¯ + 4γ¯ < f¯|(π − 2 ) − 2ω2 |ω2 sech ≡ R+ ∞ (ω ) 2
(18)
Next, we compute the Melnikov function of system (3) along the heteroclinic orbit − as follows:
M − ( τ0 ) =
+∞ −∞
y− (τ )[−k¯ x− (τ ) − γ¯ y− (τ )
+ f¯ω2 cos x− (τ )y− (τ ) cos ω (τ + τ0 )]dτ ≡ −k¯ I0− − γ¯ I1− + f¯ω2 I2− cos ωτ0 where
I0− = I1− =
Fig. 2. The phase portrait of system (4).
Therefore we can obtain the expressions of the periodic orbits as follows
I2− =
± ( k ) :
(14)
where sn, and cn are Jacobi elliptic functions, and 0 < k < 1 is the modulus of the Jacobi elliptic functions. For each k, ± (k) has the period T± (k ) = 4K (k ), where K(k) is the complete elliptic integral of the first kind. The sketch of the phase portrait is shown as in Fig. 2. From Fig. 2 one can see that when the total energy including the kinetic energy and potential energy equal zero, there exists a pair of heteroclinic orbits connecting O and A. If the total energy is between −2 and 0, there exist a family of periodic orbits around B.
+∞
−∞
+∞ −∞
+∞ −∞
x− (τ )y− (τ )dτ = −2π 2 , [ y + ( τ )] 2 d τ = 8 , y+ (τ ) cos x+ (τ ) cos ωτ dτ
πω
= 2[(π − 2 ) − 2ω2 ] sech
x± (τ , k ) = π ∓ 2 arcsin(ksn(τ , k )), y± (τ , k ) = ±2kcn(τ , k )
(19)
(20)
2
Thus, M− (τ0 ) has simple zeros and chaotic motions occur if and only if
k¯ I0− + γ¯ I1− f¯I ω2 < 1
(21)
2−
i.e.,
πω | − π 2 k¯ + 4γ¯ | < f¯|(π − 2 ) − 2ω2 |ω2 sech ≡ R− ∞ (ω ) (22) 2
3.2. Chaotic feature on the system parameters 3. Chaotic motions of the system In this section, we use the Melnikov method [20] to investigate the chaotic motions of system (3).
Now we discuss the chaotic feature on the system parameters. First, supposing k¯ and γ¯ are fixed, then we can obtain the conditions of chaotic motions as follows: For + :
3.1. Melnikov functions and parameter conditions of chaos for the system
1 |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) f+ < ≡ R∞ (ω ) ¯f π 2 k¯ + 4γ¯
We compute the Melnikov function of system (3) along the heteroclinic orbit + as follows:
M + ( τ0 ) =
+∞ −∞
y+ (τ )[−k¯ x+ (τ ) − γ¯ y+ (τ )
+ f¯ω2 cos x+ (τ )y+ (τ ) cos ω (τ + τ0 )]dτ ≡ −k¯ I0+ − γ¯ I1+ + f¯ω2 I2+ cos ωτ0 where
I0+ = I1+ = I2+ =
+∞
−∞
+∞ −∞
+∞ −∞
x + ( τ ) y + ( τ )d τ = 2 π 2 , [ y + ( τ )] 2 d τ = 8 , y+ (τ ) cos x+ (τ ) cos ωτ dτ
πω
= 2[(π − 2 ) − 2ω2 ] sech
2
For − :
1 |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) f− < ≡ R∞ (ω ) f¯ | − π 2 k¯ + 4γ¯ | f+
(15)
(16)
(23)
f−
(24)
Obviously, for any ω > 0, R∞ (ω ) < R∞ (ω ), so the chaotic zone of − is wider than that of + . Therefore, as the increasing of the excitation amplitude, first the orbit − is chaotically excited, then both − and + are chaotically excited. For example, taking k¯ = 1, γ¯ = 1, the chaotic region in (1/ f¯, γ ) plane is shown as in Fig. 3. From Fig. 3 we find a new interesting phenomenon. The critical curve of chaos first decreases quickly to zero and then increases, at last it decreases to zero as the frequency ω increases from zero. f± There exists a zero point ω∗ = π 2−2 for R∞ (ω ). This means there exists a “controllable frequency” near ω∗ for both + and − (for example, taking ω = 0.7), excited at which chaotic motions do not take place no matter how large the excitation amplitude is. Next, fixing k¯ and f¯, then the conditions for chaos are as follows:
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
Fig. 3. The critical curves of f¯ for chaos.
273
Fig. 5. The critical curves of k¯ for chaos.
Lastly, assuming γ¯ and f¯ are fixed, then the conditions of chaotic oscillations are as follows: For + :
4γ¯ f¯ k¯ < 2 |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) − 2
π
(28)
π
For − :
max <
4γ¯
π2
f¯
π2
−
f¯
π2
|(π − 2 ) − 2ω |ω sech(π ω/2 ), 0 < k¯ 2
2
|(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) +
4γ¯
(29)
π2
Similarly, taking γ¯ = 1, f¯ = 1, since for any ω > 0,
f¯
π2
For + :
f¯ 4
π 2 k¯
max
π 2 k¯ 4
−
(25)
4
For − :
f¯ |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ), 0 4
π2
<0
(30)
4. Subharmonic bifurcations of the system In this section, subharmonic bifurcations of system (3) are investigated with the subharmonic Melnikov method [13].
< γ¯
π 2 k¯ f¯ < |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) + 4 4
(26)
4.1. Parameter conditions for subharmonic bifurcations
(27)
For any positive integers m and n, which are prime to each other, there exists a unique k ∈ (0, 1), such that T (k ) = 4K (k ) = m 2π m n T = nω . It can be computed that the subharmonic Melnikov function of system (3) along the periodic orbit + (k ) satisfying the resonance condition mT = nT (k ) is
For example, taking k¯ = 1, f¯ = 1, since for any ω > 0,
π 2 k¯ f¯ |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) − <0 4 4
4γ¯
On the other hand, the elastic coefficient k¯ > 0, so inequality (28) can not be satisfied. Therefore, the orbit + can not be excited in this case. The chaotic region of − in (k¯ , ω ) plane is shown as in Fig. 5. The critical curve is similar to the case of Fig. 4.
Fig. 4. The critical curves of γ¯ for chaos.
γ¯ < |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) −
|(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) −
On the other hand, the damping coefficient γ¯ > 0, so inequality (25) can not be satisfied, therefore the orbit + can not be excited in this case. The chaotic region of − in (γ¯ , ω ) plane is shown as in Fig. 4. From Fig. 4 we can see that there exist a chaotic band for − in this case. Especially, the chaotic band becomes a point with the excitation frequency ω = π 2−2 . This means, the system can not be chaotically excited at this frequency no matter what the value of the damping is.
m/n M+ (τ0 ) = −k¯
mT
0
+ f¯ω2
x+ (τ , k )y+ (τ , k )dτ − γ¯
mT 0
mT 0
[ y + ( τ , k )] 2 d τ
cos x+ (τ , k )y+ (τ , k ) cos ω (τ + τ0 )dτ
≡ −k¯ I0+ (m, n ) − γ¯ I1+ (m, n ) + f¯ω2 I2+ (m, n ) cos ωτ0 (31)
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L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
where
Consequently,
I0+ (m, n ) = I1+
(m, n ) =
mT
0
0
mT
x + ( τ , k ) y + ( τ , k )d τ = 0 , [y+ (τ , k )]2 dτ = 16n(E − k2 K ),
mT
cos x+ (τ , k )y+ (τ , k ) cos ωτ dτ ⎧0 0, n = 1 or m is even; ⎪ ⎪ ⎨ 2 ( 8 k − 4 ) π K − π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] = sech(K ω ), ⎪ K2 ⎪ ⎩
I2+ (m, n ) =
n = 1 and m is odd
(32) and E(k) is the complete elliptic integral of the second kind, k2 = 1 − k2 . Consequently, m/n M+ ( τ0 )
⎧−16n(E − k2 K )γ¯ , n = 1 or m is even; ⎪ ⎪ ⎪ ⎪ ⎨−16(E − k2 K )γ¯ − (8k2 − 4 )π K − π [m2 π 2 − 4K 2 (1 − 2k2 )] K2 = ⎪ ¯ 2 ⎪ f ω sech(K ω ) cos ωτ0 , ⎪ ⎪ ⎩ n = 1 and m is odd
(33) Therefore when
( 2 k 2 − 1 ) π K − π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] 2 ω sech(K ω ) < 4 ( E − k2 K )K 2 f¯ 2 2 2 π ( 2k − 1 ) πω π (1 − 2k ) 2 ω sech(K ω ) = − + 4 ( E − k2 K )K E − k2 K E − k2 K
γ¯
≡ Rm + (ω )
subharmonic bifurcations of m (odd) orders arising from + (k ) will occur. The subharmonic Melnikov function of system (3) along the periodic orbit − (k ) satisfying the resonance condition mT = nTkl is
mT 0
+ f¯ω
x− (τ , k )y− (τ , k )dτ − γ¯
2 0
mT
mT
0
[ y − ( τ , k )] 2 d τ
I0− (m, n ) = I1− (m, n ) = I2−
(m, n ) =
0
0
mT
mT
Therefore when
π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] − ( 2 k 2 − 1 ) π K 2 ω sech(K ω ) < 4 ( E − k2 K )K 2 f¯ π ω2 π (2k2 − 1 ) π (1 − 2k2 ) 2 ω sech(K ω ) = − − 2 E−k K 4 ( E − k2 K )K E − k2 K ≡ Rm (38) − (ω )
γ¯
subharmonic bifurcations of m (odd) orders arising from − (k ) will occur. m From (34) and (38) one can obtain that Rm + (ω ) = R− (ω ). Furthermore, according to the difinition of elliptic integral, for any positive odd number m, it is easy to see that
( 2 k 2 − 1 ) π K − π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] < 0 , 4 ( E − k2 K )K 2 > 0, k ∈ ( 0, 1 )
(39)
So when
γ¯
π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] − ( 2 k 2 − 1 ) π K 2 ω sech(K ω ) 4 ( E − k2 K )K 2 f¯ π ( 2k2 − 1 ) π ( 1 − 2k2 ) 2 π ω2 = − − ω sech(K ω ) E − k2 K 4 ( E − k2 K )K E − k2 K ≡ Rm ( ω ) (40) <
4.2. The relation between subharmonic bifurcations and chaos Now we discuss the relation between subharmonic bifurcations and chaos. We just discuss the case that the elastic coefficient k¯ and the damping coefficient γ¯ is fixed. The discussions of other cases are similar. In the following discussions, we consider the case of k¯ = γ¯ = 1. First we give some propositions which will be used in the following discussions.
lim Rm (ω ) = π (1 + ω2 )sech(π ω/2 ).
cos x− (τ , k )y− (τ , k ) cos ω (τ + τ0 )dτ
m→∞
where mT
(37)
Proposition 1. For any fixed ω,
≡ −k¯ I0− (m, n ) − γ¯ I1− (m, n ) + f¯ω2 I2− (m, n ) cos ωτ0 (35)
n = 1 and m is odd
subharmonic bifurcations of m (odd) orders will occur.
(34)
m/n M− (τ0 ) = −k¯
( τ0 ) ⎧−16n(E − k2 K )γ¯ , n = 1 or m is even; ⎪ ⎪ ⎪ ⎪ ⎨−16(E − k2 K )γ¯ + (8k2 − 4 )π K − π [m2 π 2 − 4K 2 (1 − 2k2 )] K2 = ⎪ 2 ⎪ ¯ f ω sech(K ω ) cos ωτ0 , ⎪ ⎪ ⎩ m/n M−
Proof. For any fixed ω, when m → ∞, we have k → 1, and K → ∞, K → π2 , E → 1, k2 K → 0, so
( 2k2 − 1 )π π = lim = 0, k→1 4 ( E − k2 K )K k→1 4K π ( 1 − 2k2 ) π ω2 lim = −π , lim = π ω2 . 2 k→1 E − k K k→1 ( E − k2 K ) lim
x − ( τ , k ) y − ( τ , k )d τ = 0 , [y− (τ , k )]2 dτ = 16n(E − k2 K ),
(41)
therefore
cos x− (τ , k )y− (τ , k ) cos ωτ dτ
⎧0 0, n = 1 or m is even; ⎪ ⎪ ⎨ π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] − ( 8 k 2 − 4 ) π K = sech(K ω ), ⎪ K2 ⎪ ⎩ n = 1 and m is odd
(36)
lim Rm (ω ) = π (1 + ω2 )sech(π ω/2 ) ≡ R∞ (ω ).
(42)
m→∞
Proposition 2. For any fixed ω f+
)−2ω2 | > 0, when π 2 k¯ + 4γ¯ < |(ππ−2 , (1+ω2 )
f−
we have R∞ (ω ) < R∞ (ω ) < R∞ (ω ).
Proof. The conclusion is obviously due to k¯ > 0, γ¯ > 0.
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
(a)
275
(b)
Fig. 6. The time history curves of system (2) for f = 0.1.
Fig. 7. The phase portraits of system (2) for f = 0.1.
Fig. 8. The Poincare´ sections of system (2) for f = 0.1.
According to the resonance condition mπ = 2K ω, we have K =
2 mπ
π ω . Since K ≥ 2 , so frequency condition for m order subhar2 monic bifurcation is ω ≥ π . 4m
It can be verified that while ω is small, R1 (ω) < R∞ (ω), for example taking ω = 3, we have R1 (ω ) − R∞ (ω ) = −0.3196, therefore when ω is small, 1 order subharmonic bifurcation implies chaos for this system. While ω is large, R1 (ω) > R∞ (ω), for example, taking ω = 5, we have R1 (ω ) − R∞ (ω ) = 0.2809; on the other hand for any ω > 0, Rm (ω) increases monotonically with m, together with Propositions 1 and 2, one can obtain that Theorem 1. When k¯ , γ¯ and ω satisfy
π 2 k¯ + 4γ¯ <
| ( π − 2 ) − 2ω 2 | , π (1 + ω 2 )
(43)
system (3) can be chaotically excited through finite subharmonic bifurcations. 5. Numerical simulations Using the fourth-order Runge-Kutta method, the phase portraits,time history curves and Poincare´ section of system (2) are obtained in this section. Choosing the system parameters k˜ = 0.1, γ = 0.1, f = 0.1, ω = 1.0, the initial value (x(0 ), y(0 )) = (3.1, 0.001 ), the time history
Fig. 9. The Lyapunov exponent spectrum of system (2) for f = 0.1.
curves and phase portraits are shown as in Figs. 6 and 7, respectively. The Poincare´ section is shown as in Fig. 8. Using the method of [15] and [21], the Lyapunov exponent spectrum is shown as in Fig. 9. The Lyapunov exponents are 0, −0.0056 and −0.0944, respectively. The maximum Lyapunov exponent is
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L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
Fig. 10. The time history curves of system (2) for f = 2.
Fig. 11. The phase portraits of system (2) for f = 2.
Fig. 12. The Poincare´ sections of system (2) for f = 2.
0. The Kolmogorov-Sinai entropies in this case is K1 = 0. From Figs. 6–8 we can see that the system undergo periodic motions. Next, choosing f = 2.0, other system parameters are chosen as before, the time history curves and phase portraits are shown as in Figs. 10 and 11, respectively. The Poincare´ section is shown as in Fig. 12. The Lyapunov exponent spectrum is shown as in Fig. 13. The Lyapunov exponents are 0.1499, −0.1210 and −0.1289, respectively. The maximum Lyapunov exponent is 0.1499 > 0. The Kolmogorov–Sinai entropies in this case is K2 = 0.1499 > 0. From Figs. 10–12, one can see that the system is chaotically excited. Notice that the first choices of system parameters in above the critical thresholds computed by the Melnikov method, while the second one in below the thresholds computed by the Melnikov method, so numerical simulations agree with the analytical results. 6. Conclusions Using both analytical and numerical methods, the subharmonic bifurcations and chaotic motions of an inverted pendulum are investigated in this paper. The critical curves separating the chaotic and non-chaotic regions are obtained. It is shown that there exist caotic bands for this system. Especially, there exists a “controllable frequency” for this system. It is also proved that the system can be chaotically excited through finite subharmonic bifurcations. Abun-
Fig. 13. The Lyapunov exponent spectrum of system (2) for f = 2.
dant dynamical behaviors are presented. The results provide some inspiration and guidance for the analysis and dynamic design of this class of systems. For example, we can chose suitable value of the excitation frequency which is near the “controllable frequency”, so that the system can not undergo chaotic motions, et al.
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
Acknowledgments The project was supported by the Fundamental Research Funds for the Central Universities (NO. NS2017046). References [1] Broer HW, Hoveijn I, Noort MV. A reversible bifurcation analysis of the inverted pendulum. Physica D 1998;112:50–63. [2] Broer HW, Hoveijn I, Noort MV, Vegter G. The inverted pendulum: a singularity theory approach. J Differ Equ 1999;157:120–49. [3] Boussaada I, Morarescu IC, Niculescu SI. Inverted pendulum stabilization: characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains. Syst Control Lett 2015;82:1–9. [4] Butikov EI. On the dynamic stabilization of an inverted pendulum. Am J Phys 2001;69:755–68. [5] Chen LH, Qiao HH, Peng JH, Zhang Y, Liu J. Chaotic demonstration system of the inverted pendulum driven by periodic external force (in chinese). Phys Exp 2005;25:34–7. [6] Gandhi P, Meena J. Chaos in inverted flexible pendulum with tip mass. ASME international mechanical engineering congress and exposition; 2014. 4B: V04BT04A030 [7] Gao JM, Du ZD. Homoclinic bifurcation in a quasi periodically excited impact inverted pendulum. Nonlinear Dyn 2015;79:1061–74. [8] Kim SY, Hu BB. Bifurcations and transitions to chaos in an inverted pendulum. Phys Rev E 1998;58:3028–35. [9] Landry M, Campbell SA, Morris K, Aguilar CO. Dynamics of an inverted pendulum with delayed feedback control. SIAM J Appl Dyn Syst 2005;4:333–51. [10] Lenci S. On the suppression of chaos by means of bounded excitations in an inverted pendulum. SIAM J Appl Math 1998;58:1116–27. [11] Lenci S, Demeio L, Petrini M. Response scenario and nonsmooth features in the nonlinear dynamics of an impacting inverted pendulum. J Comput Nonlinear Dyn 2006;1:56–64.
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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
Subharmonic bifurcations and chaotic motions for a class of inverted pendulum system Liangqiang Zhou∗, Shanshan Liu, Fangqi Chen∗ Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China
a r t i c l e
i n f o
Article history: Received 26 October 2016 Revised 9 January 2017 Accepted 3 April 2017
Keywords: Inverted pendulum Chaos Subharmonic bifurcation Melnikov method Heteroclinic orbit
a b s t r a c t Using both analytical and numerical methods, global dynamics including subharmonic bifurcations and chaotic motions for a class of inverted pendulum system are investigated in this paper. The expressions of the heteroclinic orbits and periodic orbits are obtained analytically. Chaos arising from heteroclinic intersections is studied with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are obtained. The conditions for subharmonic bifurcations are also obtained. It is proved that the system can be chaotically excited through finite subharmonic bifurcations. Some new dynamical phenomena are presented. Numerical simulations are given, which verify the analytical results.
1. Introduction The inverted pendulum system has wide applications in precision instruments, robot control, missile intercept control system, spacecraft attitude control and so on. Therefore, it is of great significance to study nonlinear dynamics of this system. A lot of results on this subject have been obtained in the past two decades. Bifurcations in the inverted pendulum system have been investigated by many researchers in the past years. Via the normal form theory, perturbation analysis and equivariant singularity theory, a reversible bifurcation analysis of the inverted pendulum was given by Broer et al. [1]. Using an approximating integrable normal form and equivariant singularity theory, Broer et al. [2] studied the qualitative dynamics of the Poincare´ map corresponding to the central periodic solution and bifurcations of the inverted pendulum system. Ponce et al. [17] investigated bifurcations of an inverted pendulum with saturated Hamiltonian control laws. Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation were investigated by Lenci and Rega [12]. It was found that the existence and the stability of the cycles depended on both classical (saddle-node and period-doubling) and non-classical bifurcations. With a quantitative theory together with numerical simulations, Butikov [4] studied the dynamic stabilization of an inverted pendulum. The dynamics of an inverted pendulum subjected to high-frequency excitation was studied by Yabuno et al. [22]. The stability of the stable equilibrium states under the effect ∗
Corresponding author. E-mail addresses: [email protected] (L. Zhou), [email protected] (F. Chen).
http://dx.doi.org/10.1016/j.chaos.2017.04.004 0960-0779/© 2017 Elsevier Ltd. All rights reserved.
© 2017 Elsevier Ltd. All rights reserved.
of the tilt was discussed non-locally. An analogy of the bifurcation of the inverted pendulum to that of the buckling phenomenon was also presented. By using the method of multiple scales and numerical simulations, Yang et al. [23] investigated stability and Hopf bifurcation in an inverted pendulum with delayed feedback control. Via multiple delayed proportional gains, Boussaada et al. [3] studied a codimension-three triple zero bifurcation of an inverted pendulum on a cart moving horizontally. Using the centre manifold, Sieber and Krauskopf [18] investigated a codimension-three triplezero eigenvalue bifurcation of an inverted pendulum with delayed feedback control. Using a center manifold reduction, Landry et al. [9] investigated local dynamics of an inverted pendulum with delayed feedback control. It was shown that the system undergoes a supercritical Hopf bifurcation at the critical delay. Employing a combination of analytical and numerical methods, the stability and bifurcations of two types of double impact periodic orbits for an inverted pendulum impacting between two rigid walls were studied by Shen and Du [19]. Especially, grazing bifurcations were presented there. Chaotic motions of the inverted pendulum system have been also investigated in the past years. With numerical methods, Kim and Hu [8] studied bifurcations and transitions to chaos in an inverted pendulum. It was found that an infinite sequence of perioddoubling bifurcations, leading to chaos, followed each destabilization of the inverted state. By using a Neyman–Pearson lemma like technique, Lenci [10] investigated the suppression of chaos by means of bounded excitations in an inverted pendulum. With experimental methods, Chen et al. [5] studied chaotic motions of a inverted pendulum system. It was found that the system may
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
271
A = r0 ωd2 /g = R0 ω2 ≡ f ω2 , r0 is the excitation amplitude, R is the length of the pendulum, ωd is the angular frequency of motor. Denoting θ = x,then system (1) is written as follows : r
⎧ dx ⎪ ⎨ =y dτ ⎪ ⎩ dy = sin x − k˜ x − γ y + f ω2 cos x cos ωτ dτ
(2)
Assuming the elastic coefficient of the spring together with the damping coefficient and the parameter f are small, setting k˜ = ε k¯ , γ = ε γ¯ , f = ε f¯, system (2) can be written as
⎧ dx ⎪ ⎨ =y dτ ⎪ ⎩ dy = sin x − ε k¯ x − ε γ¯ y + ε f¯ω2 cos x cos ωτ dτ
When ε = 0, the unperturbed system of (3) is
Fig. 1. The physical model of the inverted pendulum.
undergo chaos through period doubling bifurcations. Via a systematic numerical investigation method, Lenci et al. [11] studied the nonlinear dynamics of an inverted pendulum between lateral rebounding harriers. Three different families of considerably variable attractors-periodic, chaotic, and rest positions with subsequent chattering were found. Employing the harmonic balance method and Melnikov theory, nonlinear dynamics of an inverted pendulum driven by airflow was investigated by Nbendjo [14]. Horseshoes chaos may exist in this system. With the Melnikov method, homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external quasi periodic excitations was analyzed by Gao and Du [7]. Smale horseshoe-type chaotic dynamics may occur in this system. Via Poincare´ maps, Gandhi and Meena [6] studied chaotic dynamics of an inverted flexible pendulum with tip mass. By means of the normal form theory, the Melnikov method and numerical methods, Perez-Polo et al. [16] studied the stability and chaotic behavior of a plus integral plus derivative (PID) controlled inverted pendulum subjected to harmonic base excitations. It was shown that when the pendulum was close to the unstable pointing-up position, the PID parameters were changed and the chaotic motion was destroyed. In this paper, global dynamics including subharmonic bifurcations and chaotic motions for a class of inverted pendulum system are investigated analytically with the subharmonic Melnikov method and Melnikov method, respectively. The mechanism and parameter conditions of chaotic motions are obtained rigorously. The critical curves separating the chaotic and non-chaotic regions are plotted. The chaotic feature on the system parameters is discussed in detail. The conditions for subharmonic bifurcations are also presented. It is proved that the system can be chaotically excited through finite subharmonic bifurcations. Numerical simulations verify the analytical results. 2. The dynamic model and analysis of the orbits for the inverted pendulum
where τ =
g
˜ R t, k =
k a2 mgR ,
ficient of the spring, γ =
(1)
g/R, k is the elastic coef-
ω = ωd / √γ˜ , γ˜ is the damping coefficient,
m
g/R
⎧ dx ⎪ ⎨ =y dτ ⎪ ⎩ dy = sin x dτ
(4)
which is a planar Hamiltonian system with the Hamiltonian
H (x, y ) =
y2 + cos(x ) − 1 ≡ h 2
(5)
System (4) has saddles (2l π , 0 )(l = 0, ±1, ±2, · · · ), and centers ((2l + 1 )π , 0 )(l = 0, ±1, ±2, · · · ). There exists a pair of heteroclinic orbits connecting (2lπ , 0)) to ((2l + 2 )π , 0 ) when h = 0. Due to the periodic symmetry, we just need to consider the dynamic behaviors of the system on the interval [0, 2π ] of the x-axis. In this case, system (4) has two saddles O(0, 0), A(2π , 0), and one center B(π , 0). To solve the expressions of the heteroclinic orbits, setting h = 0 in (5) one can obtain that
dx = y = ± 2(1 − cos x ) dτ
(6)
Separating variables for (6) and integrating, we can get
tan
x 4
= e ±τ
(7)
Therefore we can obtain the expressions of the homoclinic orbits as follows
+ :
x+ (τ ) = 4 arctan(eτ ) y+ (τ ) = 2 sech(τ )
(8)
− :
x− (τ ) = 4 arctan(e−τ ) y− (τ ) = −2 sech(τ )
(9)
There also exist a family of periodic orbits ± (k) around the center B inside ± for −2 < h < 0. To solve the expressions of the periodic orbits, we rewrite (5) as follows
cos x = (h + 1 ) −
Considering the model as in Fig. 1, assuming O is the center of the reciprocating motion for the system, choosing this point as the origin of the inertial coordinate system and the connection point of the inverted pendulum rod and the trolley as the origin of the non inertial coordinate system, then the dynamic equation of the ball in the non inertial coordinate system is [5]
d2 θ dθ = sin θ − k˜ θ − γ + A cos θ cos ωτ dτ dτ 2
(3)
y2 2
(10)
consequently, 2
1 − sin x =
y4 − ( h + 1 )y2 + ( h + 1 )2 4
(11)
i.e.,
dy 1 = (C 2 − y2 )(k2C 2 + k2 y2 ) dτ C
(12)
where k = 1 − k2 . Comparing the coefficients of like powers, we can obtain that
C = ±2k, h = 2(k2 − 1 )
(13)
272
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
Thus, M+ (τ0 ) has simple zeros and chaotic motions occur if and only if
k¯ I0+ + γ¯ I1+ f¯I ω2 < 1
(17)
2+
i.e.,
πω π 2 k¯ + 4γ¯ < f¯|(π − 2 ) − 2ω2 |ω2 sech ≡ R+ ∞ (ω ) 2
(18)
Next, we compute the Melnikov function of system (3) along the heteroclinic orbit − as follows:
M − ( τ0 ) =
+∞ −∞
y− (τ )[−k¯ x− (τ ) − γ¯ y− (τ )
+ f¯ω2 cos x− (τ )y− (τ ) cos ω (τ + τ0 )]dτ ≡ −k¯ I0− − γ¯ I1− + f¯ω2 I2− cos ωτ0 where
I0− = I1− =
Fig. 2. The phase portrait of system (4).
Therefore we can obtain the expressions of the periodic orbits as follows
I2− =
± ( k ) :
(14)
where sn, and cn are Jacobi elliptic functions, and 0 < k < 1 is the modulus of the Jacobi elliptic functions. For each k, ± (k) has the period T± (k ) = 4K (k ), where K(k) is the complete elliptic integral of the first kind. The sketch of the phase portrait is shown as in Fig. 2. From Fig. 2 one can see that when the total energy including the kinetic energy and potential energy equal zero, there exists a pair of heteroclinic orbits connecting O and A. If the total energy is between −2 and 0, there exist a family of periodic orbits around B.
+∞
−∞
+∞ −∞
+∞ −∞
x− (τ )y− (τ )dτ = −2π 2 , [ y + ( τ )] 2 d τ = 8 , y+ (τ ) cos x+ (τ ) cos ωτ dτ
πω
= 2[(π − 2 ) − 2ω2 ] sech
x± (τ , k ) = π ∓ 2 arcsin(ksn(τ , k )), y± (τ , k ) = ±2kcn(τ , k )
(19)
(20)
2
Thus, M− (τ0 ) has simple zeros and chaotic motions occur if and only if
k¯ I0− + γ¯ I1− f¯I ω2 < 1
(21)
2−
i.e.,
πω | − π 2 k¯ + 4γ¯ | < f¯|(π − 2 ) − 2ω2 |ω2 sech ≡ R− ∞ (ω ) (22) 2
3.2. Chaotic feature on the system parameters 3. Chaotic motions of the system In this section, we use the Melnikov method [20] to investigate the chaotic motions of system (3).
Now we discuss the chaotic feature on the system parameters. First, supposing k¯ and γ¯ are fixed, then we can obtain the conditions of chaotic motions as follows: For + :
3.1. Melnikov functions and parameter conditions of chaos for the system
1 |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) f+ < ≡ R∞ (ω ) ¯f π 2 k¯ + 4γ¯
We compute the Melnikov function of system (3) along the heteroclinic orbit + as follows:
M + ( τ0 ) =
+∞ −∞
y+ (τ )[−k¯ x+ (τ ) − γ¯ y+ (τ )
+ f¯ω2 cos x+ (τ )y+ (τ ) cos ω (τ + τ0 )]dτ ≡ −k¯ I0+ − γ¯ I1+ + f¯ω2 I2+ cos ωτ0 where
I0+ = I1+ = I2+ =
+∞
−∞
+∞ −∞
+∞ −∞
x + ( τ ) y + ( τ )d τ = 2 π 2 , [ y + ( τ )] 2 d τ = 8 , y+ (τ ) cos x+ (τ ) cos ωτ dτ
πω
= 2[(π − 2 ) − 2ω2 ] sech
2
For − :
1 |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) f− < ≡ R∞ (ω ) f¯ | − π 2 k¯ + 4γ¯ | f+
(15)
(16)
(23)
f−
(24)
Obviously, for any ω > 0, R∞ (ω ) < R∞ (ω ), so the chaotic zone of − is wider than that of + . Therefore, as the increasing of the excitation amplitude, first the orbit − is chaotically excited, then both − and + are chaotically excited. For example, taking k¯ = 1, γ¯ = 1, the chaotic region in (1/ f¯, γ ) plane is shown as in Fig. 3. From Fig. 3 we find a new interesting phenomenon. The critical curve of chaos first decreases quickly to zero and then increases, at last it decreases to zero as the frequency ω increases from zero. f± There exists a zero point ω∗ = π 2−2 for R∞ (ω ). This means there exists a “controllable frequency” near ω∗ for both + and − (for example, taking ω = 0.7), excited at which chaotic motions do not take place no matter how large the excitation amplitude is. Next, fixing k¯ and f¯, then the conditions for chaos are as follows:
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
Fig. 3. The critical curves of f¯ for chaos.
273
Fig. 5. The critical curves of k¯ for chaos.
Lastly, assuming γ¯ and f¯ are fixed, then the conditions of chaotic oscillations are as follows: For + :
4γ¯ f¯ k¯ < 2 |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) − 2
π
(28)
π
For − :
max <
4γ¯
π2
f¯
π2
−
f¯
π2
|(π − 2 ) − 2ω |ω sech(π ω/2 ), 0 < k¯ 2
2
|(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) +
4γ¯
(29)
π2
Similarly, taking γ¯ = 1, f¯ = 1, since for any ω > 0,
f¯
π2
For + :
f¯ 4
π 2 k¯
max
π 2 k¯ 4
−
(25)
4
For − :
f¯ |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ), 0 4
π2
<0
(30)
4. Subharmonic bifurcations of the system In this section, subharmonic bifurcations of system (3) are investigated with the subharmonic Melnikov method [13].
< γ¯
π 2 k¯ f¯ < |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) + 4 4
(26)
4.1. Parameter conditions for subharmonic bifurcations
(27)
For any positive integers m and n, which are prime to each other, there exists a unique k ∈ (0, 1), such that T (k ) = 4K (k ) = m 2π m n T = nω . It can be computed that the subharmonic Melnikov function of system (3) along the periodic orbit + (k ) satisfying the resonance condition mT = nT (k ) is
For example, taking k¯ = 1, f¯ = 1, since for any ω > 0,
π 2 k¯ f¯ |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) − <0 4 4
4γ¯
On the other hand, the elastic coefficient k¯ > 0, so inequality (28) can not be satisfied. Therefore, the orbit + can not be excited in this case. The chaotic region of − in (k¯ , ω ) plane is shown as in Fig. 5. The critical curve is similar to the case of Fig. 4.
Fig. 4. The critical curves of γ¯ for chaos.
γ¯ < |(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) −
|(π − 2 ) − 2ω2 |ω2 sech(π ω/2 ) −
On the other hand, the damping coefficient γ¯ > 0, so inequality (25) can not be satisfied, therefore the orbit + can not be excited in this case. The chaotic region of − in (γ¯ , ω ) plane is shown as in Fig. 4. From Fig. 4 we can see that there exist a chaotic band for − in this case. Especially, the chaotic band becomes a point with the excitation frequency ω = π 2−2 . This means, the system can not be chaotically excited at this frequency no matter what the value of the damping is.
m/n M+ (τ0 ) = −k¯
mT
0
+ f¯ω2
x+ (τ , k )y+ (τ , k )dτ − γ¯
mT 0
mT 0
[ y + ( τ , k )] 2 d τ
cos x+ (τ , k )y+ (τ , k ) cos ω (τ + τ0 )dτ
≡ −k¯ I0+ (m, n ) − γ¯ I1+ (m, n ) + f¯ω2 I2+ (m, n ) cos ωτ0 (31)
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L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
where
Consequently,
I0+ (m, n ) = I1+
(m, n ) =
mT
0
0
mT
x + ( τ , k ) y + ( τ , k )d τ = 0 , [y+ (τ , k )]2 dτ = 16n(E − k2 K ),
mT
cos x+ (τ , k )y+ (τ , k ) cos ωτ dτ ⎧0 0, n = 1 or m is even; ⎪ ⎪ ⎨ 2 ( 8 k − 4 ) π K − π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] = sech(K ω ), ⎪ K2 ⎪ ⎩
I2+ (m, n ) =
n = 1 and m is odd
(32) and E(k) is the complete elliptic integral of the second kind, k2 = 1 − k2 . Consequently, m/n M+ ( τ0 )
⎧−16n(E − k2 K )γ¯ , n = 1 or m is even; ⎪ ⎪ ⎪ ⎪ ⎨−16(E − k2 K )γ¯ − (8k2 − 4 )π K − π [m2 π 2 − 4K 2 (1 − 2k2 )] K2 = ⎪ ¯ 2 ⎪ f ω sech(K ω ) cos ωτ0 , ⎪ ⎪ ⎩ n = 1 and m is odd
(33) Therefore when
( 2 k 2 − 1 ) π K − π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] 2 ω sech(K ω ) < 4 ( E − k2 K )K 2 f¯ 2 2 2 π ( 2k − 1 ) πω π (1 − 2k ) 2 ω sech(K ω ) = − + 4 ( E − k2 K )K E − k2 K E − k2 K
γ¯
≡ Rm + (ω )
subharmonic bifurcations of m (odd) orders arising from + (k ) will occur. The subharmonic Melnikov function of system (3) along the periodic orbit − (k ) satisfying the resonance condition mT = nTkl is
mT 0
+ f¯ω
x− (τ , k )y− (τ , k )dτ − γ¯
2 0
mT
mT
0
[ y − ( τ , k )] 2 d τ
I0− (m, n ) = I1− (m, n ) = I2−
(m, n ) =
0
0
mT
mT
Therefore when
π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] − ( 2 k 2 − 1 ) π K 2 ω sech(K ω ) < 4 ( E − k2 K )K 2 f¯ π ω2 π (2k2 − 1 ) π (1 − 2k2 ) 2 ω sech(K ω ) = − − 2 E−k K 4 ( E − k2 K )K E − k2 K ≡ Rm (38) − (ω )
γ¯
subharmonic bifurcations of m (odd) orders arising from − (k ) will occur. m From (34) and (38) one can obtain that Rm + (ω ) = R− (ω ). Furthermore, according to the difinition of elliptic integral, for any positive odd number m, it is easy to see that
( 2 k 2 − 1 ) π K − π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] < 0 , 4 ( E − k2 K )K 2 > 0, k ∈ ( 0, 1 )
(39)
So when
γ¯
π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] − ( 2 k 2 − 1 ) π K 2 ω sech(K ω ) 4 ( E − k2 K )K 2 f¯ π ( 2k2 − 1 ) π ( 1 − 2k2 ) 2 π ω2 = − − ω sech(K ω ) E − k2 K 4 ( E − k2 K )K E − k2 K ≡ Rm ( ω ) (40) <
4.2. The relation between subharmonic bifurcations and chaos Now we discuss the relation between subharmonic bifurcations and chaos. We just discuss the case that the elastic coefficient k¯ and the damping coefficient γ¯ is fixed. The discussions of other cases are similar. In the following discussions, we consider the case of k¯ = γ¯ = 1. First we give some propositions which will be used in the following discussions.
lim Rm (ω ) = π (1 + ω2 )sech(π ω/2 ).
cos x− (τ , k )y− (τ , k ) cos ω (τ + τ0 )dτ
m→∞
where mT
(37)
Proposition 1. For any fixed ω,
≡ −k¯ I0− (m, n ) − γ¯ I1− (m, n ) + f¯ω2 I2− (m, n ) cos ωτ0 (35)
n = 1 and m is odd
subharmonic bifurcations of m (odd) orders will occur.
(34)
m/n M− (τ0 ) = −k¯
( τ0 ) ⎧−16n(E − k2 K )γ¯ , n = 1 or m is even; ⎪ ⎪ ⎪ ⎪ ⎨−16(E − k2 K )γ¯ + (8k2 − 4 )π K − π [m2 π 2 − 4K 2 (1 − 2k2 )] K2 = ⎪ 2 ⎪ ¯ f ω sech(K ω ) cos ωτ0 , ⎪ ⎪ ⎩ m/n M−
Proof. For any fixed ω, when m → ∞, we have k → 1, and K → ∞, K → π2 , E → 1, k2 K → 0, so
( 2k2 − 1 )π π = lim = 0, k→1 4 ( E − k2 K )K k→1 4K π ( 1 − 2k2 ) π ω2 lim = −π , lim = π ω2 . 2 k→1 E − k K k→1 ( E − k2 K ) lim
x − ( τ , k ) y − ( τ , k )d τ = 0 , [y− (τ , k )]2 dτ = 16n(E − k2 K ),
(41)
therefore
cos x− (τ , k )y− (τ , k ) cos ωτ dτ
⎧0 0, n = 1 or m is even; ⎪ ⎪ ⎨ π [ m 2 π 2 − 4 K 2 ( 1 − 2 k 2 )] − ( 8 k 2 − 4 ) π K = sech(K ω ), ⎪ K2 ⎪ ⎩ n = 1 and m is odd
(36)
lim Rm (ω ) = π (1 + ω2 )sech(π ω/2 ) ≡ R∞ (ω ).
(42)
m→∞
Proposition 2. For any fixed ω f+
)−2ω2 | > 0, when π 2 k¯ + 4γ¯ < |(ππ−2 , (1+ω2 )
f−
we have R∞ (ω ) < R∞ (ω ) < R∞ (ω ).
Proof. The conclusion is obviously due to k¯ > 0, γ¯ > 0.
L. Zhou et al. / Chaos, Solitons and Fractals 99 (2017) 270–277
(a)
275
(b)
Fig. 6. The time history curves of system (2) for f = 0.1.
Fig. 7. The phase portraits of system (2) for f = 0.1.
Fig. 8. The Poincare´ sections of system (2) for f = 0.1.
According to the resonance condition mπ = 2K ω, we have K =
2 mπ
π ω . Since K ≥ 2 , so frequency condition for m order subhar2 monic bifurcation is ω ≥ π . 4m
It can be verified that while ω is small, R1 (ω) < R∞ (ω), for example taking ω = 3, we have R1 (ω ) − R∞ (ω ) = −0.3196, therefore when ω is small, 1 order subharmonic bifurcation implies chaos for this system. While ω is large, R1 (ω) > R∞ (ω), for example, taking ω = 5, we have R1 (ω ) − R∞ (ω ) = 0.2809; on the other hand for any ω > 0, Rm (ω) increases monotonically with m, together with Propositions 1 and 2, one can obtain that Theorem 1. When k¯ , γ¯ and ω satisfy
π 2 k¯ + 4γ¯ <
| ( π − 2 ) − 2ω 2 | , π (1 + ω 2 )
(43)
system (3) can be chaotically excited through finite subharmonic bifurcations. 5. Numerical simulations Using the fourth-order Runge-Kutta method, the phase portraits,time history curves and Poincare´ section of system (2) are obtained in this section. Choosing the system parameters k˜ = 0.1, γ = 0.1, f = 0.1, ω = 1.0, the initial value (x(0 ), y(0 )) = (3.1, 0.001 ), the time history
Fig. 9. The Lyapunov exponent spectrum of system (2) for f = 0.1.
curves and phase portraits are shown as in Figs. 6 and 7, respectively. The Poincare´ section is shown as in Fig. 8. Using the method of [15] and [21], the Lyapunov exponent spectrum is shown as in Fig. 9. The Lyapunov exponents are 0, −0.0056 and −0.0944, respectively. The maximum Lyapunov exponent is
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Fig. 10. The time history curves of system (2) for f = 2.
Fig. 11. The phase portraits of system (2) for f = 2.
Fig. 12. The Poincare´ sections of system (2) for f = 2.
0. The Kolmogorov-Sinai entropies in this case is K1 = 0. From Figs. 6–8 we can see that the system undergo periodic motions. Next, choosing f = 2.0, other system parameters are chosen as before, the time history curves and phase portraits are shown as in Figs. 10 and 11, respectively. The Poincare´ section is shown as in Fig. 12. The Lyapunov exponent spectrum is shown as in Fig. 13. The Lyapunov exponents are 0.1499, −0.1210 and −0.1289, respectively. The maximum Lyapunov exponent is 0.1499 > 0. The Kolmogorov–Sinai entropies in this case is K2 = 0.1499 > 0. From Figs. 10–12, one can see that the system is chaotically excited. Notice that the first choices of system parameters in above the critical thresholds computed by the Melnikov method, while the second one in below the thresholds computed by the Melnikov method, so numerical simulations agree with the analytical results. 6. Conclusions Using both analytical and numerical methods, the subharmonic bifurcations and chaotic motions of an inverted pendulum are investigated in this paper. The critical curves separating the chaotic and non-chaotic regions are obtained. It is shown that there exist caotic bands for this system. Especially, there exists a “controllable frequency” for this system. It is also proved that the system can be chaotically excited through finite subharmonic bifurcations. Abun-
Fig. 13. The Lyapunov exponent spectrum of system (2) for f = 2.
dant dynamical behaviors are presented. The results provide some inspiration and guidance for the analysis and dynamic design of this class of systems. For example, we can chose suitable value of the excitation frequency which is near the “controllable frequency”, so that the system can not undergo chaotic motions, et al.
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