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Dynamics, implementation and stability of a chaotic system with coexistence of hyperbolic and non-hyperbolic equilibria
Int. J. Electron. Commun. (AEÜ) 84 (2018) 199–205
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Int. J. Electron. Commun. (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Regular paper
Dynamics, implementation and stability of a chaotic system with coexistence of hyperbolic and non-hyperbolic equilibria
T
⁎
Chun-Lai Lia, , Hong-Min Lia, Wu Lib, Yao-Nan Tongb, Jing Zhangc, Du-Qu Weid, Fu-Dong Lie,f a
College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China School of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China c College of Mechanical Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China d College of Electronic Engineering, Guangxi Normal University, Guilin 541004, China e Office of Science and Technology Development, Peking University, Beijing 100871, China f Energy Research Institute, State Grid Corporation of China, Beijing 100761, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Chaotic system Non-hyperbolic equilibrium Circuital implementation Stabilization
Hyperbolic-type equilibrium requires that all the real parts of the corresponding eigenvalues are nonzero. In this paper, a three-dimensional autonomous chaotic system is introduced, and interestingly we find that one nonhyperbolic equilibrium point and two hyperbolic equilibrium points coexist in this system, which, according to the information we know, has not been previously reported. We first reveal the basic dynamics of the system through analyzing phase portrait, frequency spectrum, Poincaré map, bifurcation diagram and Lyapunov exponent. Then, based on the idea of the improved modular technology, we build an analog circuit to realize the chaotic system, which further verifies the theoretical results. Finally, we design a simple feedback controller on account of Lyapunov asymptotic stability theory, to globally suppress the system to its equilibrium points.
1. Introduction The equilibrium point of dynamical system is defined as the constant solution of its differential equation. The study on the equilibrium’s property of dynamical system has well served to explore the type of system [1–3], the shape of attractor [4–7], the amplitude variation of signal [8,9], and the practical engineering application [10–14]. If all the real parts of the corresponding eigenvalues are nonzero, the equilibrium is called to be hyperbolic [15]. For three-dimensional autonomous system, the hyperbolic equilibrium can be the type of stable or unstable saddle, node, saddle-focus or node-focus. Since the famous Lorenz model was found in 1963 [16], a number of Lorenz-type chaotic systems have been reported with two saddle-foci and one saddle [17–23]. They are hyperbolic-type system with nonzero real parts of characteristic roots, and the Šil’nikov homoclinic/heteroclinic theory is one commonly accepted criterion for proving the existence of chaos [24,25]. What’s more, these systems can generate two-scroll attractors alternatively swirling around the unstable saddle-focuses, which also imply the fact that the distribution of scrolls is determined by the number of saddle-focuses. The subsequent investigation along this fact naturally extended to construct (multi-directional) multi-scroll attractors through increase in the number of hyperbolic-type equilibrium
⁎
points with index-2, which can be realized by substituting the original nonlinear term with cubic function [26], polynomial function [27], multi-segment quadratic function [28], saw-tooth function [29], hysteresis function [30] and stair function [31]. At the same time, other few chaotic systems with non-hyperbolic equilibria, defined as holding eigenvalues with a real part equal to zero [32], are formulated and studied. For example, Li and Ou reported a chaotic system originated from Lorenz system and obtained the stability character of its non-hyperbolic equilibria by using the center manifold theorem [33]. Liu and Yang also investigated the condition of asymptotically stable of the non-hyperbolic equilibrium for a Lorenz-like chaotic system [34]. Wei and Yang constructed a new chaotic system coexistence with saddle-foci, non-hyperbolic and stable equilibria by varying one of system parameters [35]. As was found by Sprott [36,37], the saddle-focus equilibria doesn’t exist in the non-hyperbolic type of chaotic system. Therefore, it’s difficult for the Šil’nikov theorem to detect the chaos in these abnormal chaotic systems of non-hyperbolic type. Motivated by the above discussion, this paper reports our recent work of a three-dimensional autonomous chaotic system which, particularly interesting, holds three uncommon equilibrium points: a zero equilibrium point of non-hyperbolic type and two symmetrical
Corresponding author. E-mail addresses: [email protected] (C.-L. Li), [email protected] (H.-M. Li), [email protected] (W. Li), [email protected] (Y.-N. Tong), [email protected] (J. Zhang), [email protected] (D.-Q. Wei), [email protected] (F.-D. Li). https://doi.org/10.1016/j.aeue.2017.12.001 Received 14 June 2017; Accepted 1 December 2017 1434-8411/ © 2017 Elsevier GmbH. All rights reserved.
Int. J. Electron. Commun. (AEÜ) 84 (2018) 199–205
C.-L. Li et al.
equilibrium points of hyperbolic type. The finding is striking since it reveals that hyperbolic type and non-hyperbolic type equilibrium points can coexist in the same chaotic system, which, according to the information we know, has not been previously reported. And the finding will also constitute a stimulus to explore more undiscovered dynamics feature of chaotic system. What’s more, the proposed system will hold more complex topological form than stable manifold, unstable manifold and central manifold, compared to non-hyperbolic system, consequently it will enhance the potential application in chaotic cryptography and secure communication. To understand the complex behavior of the system, basic dynamical properties, such as phase portrait, frequency spectrum, Poincaré map, bifurcation diagram and Lyapunov exponent are studied. And to further verify the theoretical results, we build an analog circuit to realize the chaotic system based on the idea of improved modular technology. Finally, to globally suppress the system to its equilibrium points, we design a feedback control scheme based on the theory of Lyapunov asymptotic stability. This scheme is simple with only one linear controller yet impactful to suppress the system to its different equilibrium points. Consequently, the scheme is practicable in actual implementation, which is further illustrated by numerical simulations.
Thus, equilibrium point E0 is non-hyperbolic since the characteristic value λ2 equals to zero. However, equilibrium points E1 and E2 are unstable saddle-focus of index 2. And since all the real parts of the corresponding eigenvalues are nonzero, the two nonzero equilibrium points are hyperbolic. This finding reveals the unusual fact that hyperbolic type and non-hyperbolic type equilibrium points can simultaneously exist in one chaotic system. One might get the impression that the solutions of system (1) are actually bounded, and it has a globally attracting chaotic attractor. However, if one chooses x1(0) = x3(0) = 0 and x2(0) = c ≠ 0, then the exact solution of (1) is x1(t) = x3(t) = 0 and x2(t) = ceat. Hence the x2 axis is actually an unstable manifold for (1) when a > 0. As a result, if initial conditions are close to this axis, solution bound will increase. Hence, for the equilibrium point E0, if one chooses a g > 0, then there always exists initial conditions for which the inequality to determine g is violated. In other words, the stability result for E0 is only valid for a set of initial conditions, hence is local in nature.
2.3. Dynamics switch by parameter variation To reveal the process of dynamics switch for this system, we consider the parameter range 1 ≤ a ≤ 2.6, and the corresponding bifurcation diagram and spectrum of Lyapunov exponents by numerical calculation are depicted in Fig. 3. Preliminary analysis shows that the dynamical behaviors of system (1) switch among chaotic orbit and periodic orbit connected by inverse period-doubling bifurcation, with the increase of parameter a. Concretely, there exist five obvious periodic windows embedded in the chaotic region, with parameter a belonging to [1.195, 1.24], [1.547, 1.573], [1.686, 1.824], [2.152, 2.372] and [2.551, 2.6], respectively. As illustrated examples, we depict the periodic motion when a equal to 1.24 and 2.16 respective, seen in Fig. 4.
2. Model and dynamics of the reported system 2.1. The system model Our reported dynamical system is depicted in the following form 2 ⎧ x1̇ = −x1 + x2 x 3 x2̇ = ax2−x1 x3 ⎨ 3 ⎩ x3̇ = −x 3 + x1 x2
(1)
In system (1), x1, x2, x3 are the state variables, a is the system parameter with positive value. It’s known that system (1) is invariant under the transformation (x1, x2, x3) ↦ (−x1, −x2, x3) , thus we conclude that system (1) has reflection symmetry about x3-axis, and the nontrivial trajectories of x1 and x2 of system (1) hold a twin direction. When the only parameter a is equal to 2 and initial selection is set as (x1(0), x2(0), x3(0)) = (0.2, 0.1, 0.1), the three Lyapunov exponents of system (1) are calculated as (0.210312, 0.004775, −6.435465) by the orthogonal method with the simulation time T = 1000. As we can deduce that system (1) is chaotic since it holds one positive Lyapunov exponent. And the fractional Kaplan-Yorke dimension is subsequently derived to be 2.0319, which also confirms the chaotic behavior. The corresponding phase diagrams further verify the chaotic property for the reported system, as depicted in Fig. 1. In the frequency domain, we depict an apparently continuous broadband spectrum 20log|x2| of system (1) in Fig. 2(a). While in the time domain, we visualize the Poincaré map on x1–x3 plane with x2 = 0 in Fig. 2(b). It is clear that the Poincaré map is composed of virtually symmetrical branches and several nearly symmetrical twigs. What’s more, the attractor structure is displayed in the Poincaré map.
3. Circuit implementation of the reported system From the point of practical applications, the hardware realization of chaotic models is an important topic, especially realized by using commercially common electronic components [38–40]. Therefore, we will design an electronic circuit to realize the reported chaotic system in this section, by using the dimensionless state equations and the improved module-based technique [41]. First, to ensure the dynamic range of state variables determined by saturation value of active devices, and to guarantee the electronic circuit working effectively and capture the wave easily, the variable-scale reduction and time-scale transformation should be taken into account. Thus, when letting the proportional compression factors be (10, 2, 1) for variables (x1, x2, x3), and letting the time-scale transformation factor be 100, we derive the resulted state equation of system (1) with a = 2, as below 2
x ̇ = −100x1−20(−x2) x 3 ⎧ ⎪ 1 x2̇ = −200(−x2)−500x1 x3 ⎨ ⎪ x3̇ = −100x 33−2000x1 (−x2) ⎩
2.2. Analysis of equilibrium points By solving the equation set −x1 + x2 x 32 = 0 , ax2−x1 x3 = 0 , + x1 x2 = 0 , we find three equilibrium points of system (1) as E0 (0, 0, 0) , E1 (a5/6, a1/6, a1/3) , E2 (−a5/6, −a1/6, a1/3) . When selecting a = 2, we obtain the equilibrium points and the corresponding eigenvalues, as follows:
−x 33
(2)
Secondly, the improved module-based circuit diagram from state equation (2) can be derived by differential to integral conversion, as depicted in Fig. 5. In this design, we choose the operational amplifier LF353 chip and analog multiplier AD633 chip. It is worth mentioning that the AD633 may hold nonideal memory effect in actual implementation [42]. In the course of our implementation, the working condition is considered to be ideal for avoiding the ranges in which nonideal behavior is problematic. Thus, we obtain the circuit state equation from Fig. 5, as follows
E0(0, 0, 0): λ1 = −1, λ2 = 0, λ3 = 2. E1(1.7818, 1.1225, 1.2599): λ1 = 0.7826 + 1.7214i, λ2 = 0.7826– 1.7214i, λ3 = −5.3272. E2(−1.7818, −1.1225, 1.2599): λ1 = 0.7826 + 1.7214i, λ2 = 0.7826–1.7214i, λ3 = −5.3272. 200
Int. J. Electron. Commun. (AEÜ) 84 (2018) 199–205
C.-L. Li et al.
Fig. 1. Phase diagrams of system (1) with a = 2 and for the initial value (0.2, 0.1, 0.1) projected on (a) x1–x2 plane; (b) x2–x3 plane; (c) x1–x3 plane.
1
1
2 ⎧ x1̇ = − R1 C0 x1− 100R2 C0 (−x2) x 3 ⎪ 1 1 x2̇ = − R C (−x2)− 10R C x1 x3 3 0 4 0 ⎨ ⎪ x3̇ = − 1 x 3− 1 x1 (−x2) 3 10R5 C0 100R6 C0 ⎩
R1 = 100 kΩ, R2 = 5 kΩ, R3 = 50 kΩ, R 4 = 2 kΩ, R5 = 0.5 kΩ, R6 = 1 kΩ Based on the circuit diagram depicted in Fig. 5, the experimental circuit is built on a breadboard by using discrete electronic components, and the experimental setup is shown in Fig. 6. The experimental phase portraits captured with a digital camera are shown in Fig. 7. It’s found that the experimental results agree well with the numerical simulations.
(3)
Letting R7 = 10 kΩ and C0 = 100 nF, we get the values of other resistances by comparing Eq. (2) with Eq. (3), as below
Fig. 2. (a) Continuous frequency spectrum and (b) Poincare maps on x1–x3 plane with a = 2.
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Fig. 3. Dynamics switch between periodic and chaotic windows of system (1) interpreted by (a) Bifurcation and (b) Lyapunov exponents when 1 ≤ a ≤ 2.6.
Fig. 4. (a) Periodic orbit with a = 1.24; (b) double-periodic orbit with a = 2.16.
4. Stabilization for the reported system
x1
4.1. Control scheme 0.1x3
Although the behavior of chaotic system cannot be chronically forecasted, one can effectively utilize chaos by suppressing the future behavior with appropriate control technology. In the last several decades, the stability design for chaotic system has been extensively studied to show the importance of chaos control [43–52]. In this section, we add the single controller u = −g (x2−x e2) to the second equation of system (1), depicted by 2 ⎧ x1̇ = −x1 + x2 x 3 x2̇ = ax2−x1 x3 + u ⎨ 3 ⎩ x3̇ = −x 3 + x1 x2
2
R1
C0
−0.01x2 x32
− x2
x1
R2
R7
x2
R7
− x2
x1
R3
C0
x2
0.1x1 x3 R4
x3 (4)
− x2
For the controller u, g is the control gain to be determined, xe2 is the second component of the equilibrium point xe = (xe1, xe2, xe3) for the uncontrolled system (1). Assume that x3 is bounded satisfying x 32 ⩽ B3 and the value of B3 can be determined by the state trajectory of x3 for any initial conditions. For the case of controlling to zero equilibrium point, we set the 1 candidate Lyapunov function as V (x ) = 2 (x12 + x 22 + x 32) , and the time derivative of V (x ) is deduced by V̇ = −x12 + x1 x2 x 32 + (a−
x1 0.1x32
−0.1x1 x2
R5
0.01x33 R6
x3 Fig. 5. Circuit diagram for emulating system (1).
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C0
x3
Int. J. Electron. Commun. (AEÜ) 84 (2018) 199–205
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Fig. 6. Experimental setup for observation. Fig. 8. Control to zero equilibrium point with (a) g = 4; (b) g = 6.
g ) x 22−x 34 ⩽ −x12 + B3 |x1 ||x2 |−( g−a x2)2−x 34 . Thus, it yields that V̇ ⩽ − B2
(|x1 |− g−a |x2 |)2−x 34 ⩽ 0 with g ⩾ 43 + a , and the output variables of system (4) will converge to the zero equilibrium point asymptotically. For the stabilization of non-zero equilibrium points E1 and E2, we start with the controlled system
original system (1) can be stabilized to E1 and E2 by only using the single state feedback u = −g (x2−x e2) . 4.2. Numerical verification
x ̇ = −x1 + x2 x 32−g (x1−x e1) ⎧ ⎪ 1 x2̇ = ax2−x1 x3−g (x2−x e2) ⎨ ⎪ x3̇ = −x 33 + x1 x2−g (x3−x e3) ⎩
This section of the paper presents numerical simulations to demonstrate the effectiveness of the derived control scheme by adopting MATLAB software with ODE45 algorithm. For conveniently comparing the control process, we choose the system parameter a = 2, and the initial states (0.2, 0.1, 0.1). With this condition, system (4) is chaotic before executing control. We first control the system to its zero equilibrium point. In this process, the control gain is taken as g = 4 and g = 6 respectively, and the controller u is imposed at the fiftieth second. The control results are numerically described in Fig. 8(a) and (b). It’s found that one can stabilize the system to the fixed point as expected, and that a larger control gain is required to more quickly reach the control purpose. Now we consider the case of controlling to non-zero equilibrium points. Fig. 9(a) and (b) depicts the numerical result of controlling to E1(1.7818, 1.1225, 1.2599) with u = −6(x2 − 1.1225). And the numerical result of controlling to E2(−1.7818, −1.1225, 1.2599) is shown in Fig. 10(a) and (b) with u = −6(x2 + 1.1225). It is confirmed from Figs. 9 and 10 that one can also stabilize the system to the desired non-zero point E1 or E2, by selecting the corresponding controller.
(5)
Suppose that function fi (x ) is smooth in the neighborhood of x e and satisfies the Lipschitz condition ‖fi (x )−fi (x e )‖ ⩽ k‖x i−x ei ‖, i = 1,2,3; k is a positive constant. We introduce the candidate function 1 1 1 V (x ) = 2 (x1−x e1)2 + 2 (x2−x e2)2 + 2 (x3−x e3)2 which leads to 3 3 2 2 ̇ V = ∑i = 1 (x i−x ei ) fi (x )− ∑i = 1 g (x i−x ei ) ⩽ (3k−g )‖x −x e ‖ . Therefore, we obtain that V̇ ⩽ 0 when g ⩾ 3k , and thus the output variables of system (4) will converge to the non-zero equilibrium points asymptotically. If x2 convergences to xe2, we obtain the following two-dimensional subsystem of (1) by replacing x1, x2, x3 with x d1 = x1−x e1,x d2 = x2−x e2,x d2 = x3−x e3 .
⎧ x ḋ 1 = −x d1 3 ⎨ ⎩ x ḋ 3 = −xd3
(6)
which is uniformly exponentially stable about (xd1, xd3) = (0, 0), meaning that x1, x3 converge to xe1, xe3 exponentially. Thus, the
Fig. 7. Experimental phase portraits of system (1) projected on (a) x1–x2 plane; (b) x2–x3 plane; (c) x1–x3 plane.
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Fig. 9. Control to E1(1.7818, 1.1225, 1.2599) with u = −6(x2 − 1.1225).
Fig. 10. Control to E2(−1.7818, −1.1225, 1.2599) with u = −6(x2 + 1.1225).
5. Conclusion
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This paper reported a three-dimensional autonomous chaotic system holding three unusual equilibrium points: a zero equilibrium point of non-hyperbolic type and two symmetrical equilibrium points of hyperbolic type. The finding revealed the striking fact for the first time, that hyperbolic type and non-hyperbolic type equilibrium points can simultaneously coexist in one chaotic system. To understand the complex behavior of the system, basic dynamical properties are studied in detail. And to further verify the theoretical results, we build an analog circuit to realize the chaotic system, by using the dimensionless state equations and the improved module-based technique. Finally, a simple feedback controller is introduced to suppress the system to its equilibrium points, and numerical simulations further illustrated the enforceability of the theoretical scheme.
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Acknowledgments We first appreciate the anonymous reviewers for the valuable comments. This work was supported by Science and Technology Program of Hunan Province (No. 2016TP1021); the Research Foundation of Education Bureau of Hunan Province of China (Nos. 16B113, 16B114, 14A062); Hunan Provincial Natural Science Foundation of China (No. 2016JJ4036) and National Natural Science Foundation of China (Nos. 11562004, 61473118 and 11602084).
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Int. J. Electron. Commun. (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Regular paper
Dynamics, implementation and stability of a chaotic system with coexistence of hyperbolic and non-hyperbolic equilibria
T
⁎
Chun-Lai Lia, , Hong-Min Lia, Wu Lib, Yao-Nan Tongb, Jing Zhangc, Du-Qu Weid, Fu-Dong Lie,f a
College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China School of Information and Communication Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China c College of Mechanical Engineering, Hunan Institute of Science and Technology, Yueyang 414006, China d College of Electronic Engineering, Guangxi Normal University, Guilin 541004, China e Office of Science and Technology Development, Peking University, Beijing 100871, China f Energy Research Institute, State Grid Corporation of China, Beijing 100761, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Chaotic system Non-hyperbolic equilibrium Circuital implementation Stabilization
Hyperbolic-type equilibrium requires that all the real parts of the corresponding eigenvalues are nonzero. In this paper, a three-dimensional autonomous chaotic system is introduced, and interestingly we find that one nonhyperbolic equilibrium point and two hyperbolic equilibrium points coexist in this system, which, according to the information we know, has not been previously reported. We first reveal the basic dynamics of the system through analyzing phase portrait, frequency spectrum, Poincaré map, bifurcation diagram and Lyapunov exponent. Then, based on the idea of the improved modular technology, we build an analog circuit to realize the chaotic system, which further verifies the theoretical results. Finally, we design a simple feedback controller on account of Lyapunov asymptotic stability theory, to globally suppress the system to its equilibrium points.
1. Introduction The equilibrium point of dynamical system is defined as the constant solution of its differential equation. The study on the equilibrium’s property of dynamical system has well served to explore the type of system [1–3], the shape of attractor [4–7], the amplitude variation of signal [8,9], and the practical engineering application [10–14]. If all the real parts of the corresponding eigenvalues are nonzero, the equilibrium is called to be hyperbolic [15]. For three-dimensional autonomous system, the hyperbolic equilibrium can be the type of stable or unstable saddle, node, saddle-focus or node-focus. Since the famous Lorenz model was found in 1963 [16], a number of Lorenz-type chaotic systems have been reported with two saddle-foci and one saddle [17–23]. They are hyperbolic-type system with nonzero real parts of characteristic roots, and the Šil’nikov homoclinic/heteroclinic theory is one commonly accepted criterion for proving the existence of chaos [24,25]. What’s more, these systems can generate two-scroll attractors alternatively swirling around the unstable saddle-focuses, which also imply the fact that the distribution of scrolls is determined by the number of saddle-focuses. The subsequent investigation along this fact naturally extended to construct (multi-directional) multi-scroll attractors through increase in the number of hyperbolic-type equilibrium
⁎
points with index-2, which can be realized by substituting the original nonlinear term with cubic function [26], polynomial function [27], multi-segment quadratic function [28], saw-tooth function [29], hysteresis function [30] and stair function [31]. At the same time, other few chaotic systems with non-hyperbolic equilibria, defined as holding eigenvalues with a real part equal to zero [32], are formulated and studied. For example, Li and Ou reported a chaotic system originated from Lorenz system and obtained the stability character of its non-hyperbolic equilibria by using the center manifold theorem [33]. Liu and Yang also investigated the condition of asymptotically stable of the non-hyperbolic equilibrium for a Lorenz-like chaotic system [34]. Wei and Yang constructed a new chaotic system coexistence with saddle-foci, non-hyperbolic and stable equilibria by varying one of system parameters [35]. As was found by Sprott [36,37], the saddle-focus equilibria doesn’t exist in the non-hyperbolic type of chaotic system. Therefore, it’s difficult for the Šil’nikov theorem to detect the chaos in these abnormal chaotic systems of non-hyperbolic type. Motivated by the above discussion, this paper reports our recent work of a three-dimensional autonomous chaotic system which, particularly interesting, holds three uncommon equilibrium points: a zero equilibrium point of non-hyperbolic type and two symmetrical
Corresponding author. E-mail addresses: [email protected] (C.-L. Li), [email protected] (H.-M. Li), [email protected] (W. Li), [email protected] (Y.-N. Tong), [email protected] (J. Zhang), [email protected] (D.-Q. Wei), [email protected] (F.-D. Li). https://doi.org/10.1016/j.aeue.2017.12.001 Received 14 June 2017; Accepted 1 December 2017 1434-8411/ © 2017 Elsevier GmbH. All rights reserved.
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equilibrium points of hyperbolic type. The finding is striking since it reveals that hyperbolic type and non-hyperbolic type equilibrium points can coexist in the same chaotic system, which, according to the information we know, has not been previously reported. And the finding will also constitute a stimulus to explore more undiscovered dynamics feature of chaotic system. What’s more, the proposed system will hold more complex topological form than stable manifold, unstable manifold and central manifold, compared to non-hyperbolic system, consequently it will enhance the potential application in chaotic cryptography and secure communication. To understand the complex behavior of the system, basic dynamical properties, such as phase portrait, frequency spectrum, Poincaré map, bifurcation diagram and Lyapunov exponent are studied. And to further verify the theoretical results, we build an analog circuit to realize the chaotic system based on the idea of improved modular technology. Finally, to globally suppress the system to its equilibrium points, we design a feedback control scheme based on the theory of Lyapunov asymptotic stability. This scheme is simple with only one linear controller yet impactful to suppress the system to its different equilibrium points. Consequently, the scheme is practicable in actual implementation, which is further illustrated by numerical simulations.
Thus, equilibrium point E0 is non-hyperbolic since the characteristic value λ2 equals to zero. However, equilibrium points E1 and E2 are unstable saddle-focus of index 2. And since all the real parts of the corresponding eigenvalues are nonzero, the two nonzero equilibrium points are hyperbolic. This finding reveals the unusual fact that hyperbolic type and non-hyperbolic type equilibrium points can simultaneously exist in one chaotic system. One might get the impression that the solutions of system (1) are actually bounded, and it has a globally attracting chaotic attractor. However, if one chooses x1(0) = x3(0) = 0 and x2(0) = c ≠ 0, then the exact solution of (1) is x1(t) = x3(t) = 0 and x2(t) = ceat. Hence the x2 axis is actually an unstable manifold for (1) when a > 0. As a result, if initial conditions are close to this axis, solution bound will increase. Hence, for the equilibrium point E0, if one chooses a g > 0, then there always exists initial conditions for which the inequality to determine g is violated. In other words, the stability result for E0 is only valid for a set of initial conditions, hence is local in nature.
2.3. Dynamics switch by parameter variation To reveal the process of dynamics switch for this system, we consider the parameter range 1 ≤ a ≤ 2.6, and the corresponding bifurcation diagram and spectrum of Lyapunov exponents by numerical calculation are depicted in Fig. 3. Preliminary analysis shows that the dynamical behaviors of system (1) switch among chaotic orbit and periodic orbit connected by inverse period-doubling bifurcation, with the increase of parameter a. Concretely, there exist five obvious periodic windows embedded in the chaotic region, with parameter a belonging to [1.195, 1.24], [1.547, 1.573], [1.686, 1.824], [2.152, 2.372] and [2.551, 2.6], respectively. As illustrated examples, we depict the periodic motion when a equal to 1.24 and 2.16 respective, seen in Fig. 4.
2. Model and dynamics of the reported system 2.1. The system model Our reported dynamical system is depicted in the following form 2 ⎧ x1̇ = −x1 + x2 x 3 x2̇ = ax2−x1 x3 ⎨ 3 ⎩ x3̇ = −x 3 + x1 x2
(1)
In system (1), x1, x2, x3 are the state variables, a is the system parameter with positive value. It’s known that system (1) is invariant under the transformation (x1, x2, x3) ↦ (−x1, −x2, x3) , thus we conclude that system (1) has reflection symmetry about x3-axis, and the nontrivial trajectories of x1 and x2 of system (1) hold a twin direction. When the only parameter a is equal to 2 and initial selection is set as (x1(0), x2(0), x3(0)) = (0.2, 0.1, 0.1), the three Lyapunov exponents of system (1) are calculated as (0.210312, 0.004775, −6.435465) by the orthogonal method with the simulation time T = 1000. As we can deduce that system (1) is chaotic since it holds one positive Lyapunov exponent. And the fractional Kaplan-Yorke dimension is subsequently derived to be 2.0319, which also confirms the chaotic behavior. The corresponding phase diagrams further verify the chaotic property for the reported system, as depicted in Fig. 1. In the frequency domain, we depict an apparently continuous broadband spectrum 20log|x2| of system (1) in Fig. 2(a). While in the time domain, we visualize the Poincaré map on x1–x3 plane with x2 = 0 in Fig. 2(b). It is clear that the Poincaré map is composed of virtually symmetrical branches and several nearly symmetrical twigs. What’s more, the attractor structure is displayed in the Poincaré map.
3. Circuit implementation of the reported system From the point of practical applications, the hardware realization of chaotic models is an important topic, especially realized by using commercially common electronic components [38–40]. Therefore, we will design an electronic circuit to realize the reported chaotic system in this section, by using the dimensionless state equations and the improved module-based technique [41]. First, to ensure the dynamic range of state variables determined by saturation value of active devices, and to guarantee the electronic circuit working effectively and capture the wave easily, the variable-scale reduction and time-scale transformation should be taken into account. Thus, when letting the proportional compression factors be (10, 2, 1) for variables (x1, x2, x3), and letting the time-scale transformation factor be 100, we derive the resulted state equation of system (1) with a = 2, as below 2
x ̇ = −100x1−20(−x2) x 3 ⎧ ⎪ 1 x2̇ = −200(−x2)−500x1 x3 ⎨ ⎪ x3̇ = −100x 33−2000x1 (−x2) ⎩
2.2. Analysis of equilibrium points By solving the equation set −x1 + x2 x 32 = 0 , ax2−x1 x3 = 0 , + x1 x2 = 0 , we find three equilibrium points of system (1) as E0 (0, 0, 0) , E1 (a5/6, a1/6, a1/3) , E2 (−a5/6, −a1/6, a1/3) . When selecting a = 2, we obtain the equilibrium points and the corresponding eigenvalues, as follows:
−x 33
(2)
Secondly, the improved module-based circuit diagram from state equation (2) can be derived by differential to integral conversion, as depicted in Fig. 5. In this design, we choose the operational amplifier LF353 chip and analog multiplier AD633 chip. It is worth mentioning that the AD633 may hold nonideal memory effect in actual implementation [42]. In the course of our implementation, the working condition is considered to be ideal for avoiding the ranges in which nonideal behavior is problematic. Thus, we obtain the circuit state equation from Fig. 5, as follows
E0(0, 0, 0): λ1 = −1, λ2 = 0, λ3 = 2. E1(1.7818, 1.1225, 1.2599): λ1 = 0.7826 + 1.7214i, λ2 = 0.7826– 1.7214i, λ3 = −5.3272. E2(−1.7818, −1.1225, 1.2599): λ1 = 0.7826 + 1.7214i, λ2 = 0.7826–1.7214i, λ3 = −5.3272. 200
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Fig. 1. Phase diagrams of system (1) with a = 2 and for the initial value (0.2, 0.1, 0.1) projected on (a) x1–x2 plane; (b) x2–x3 plane; (c) x1–x3 plane.
1
1
2 ⎧ x1̇ = − R1 C0 x1− 100R2 C0 (−x2) x 3 ⎪ 1 1 x2̇ = − R C (−x2)− 10R C x1 x3 3 0 4 0 ⎨ ⎪ x3̇ = − 1 x 3− 1 x1 (−x2) 3 10R5 C0 100R6 C0 ⎩
R1 = 100 kΩ, R2 = 5 kΩ, R3 = 50 kΩ, R 4 = 2 kΩ, R5 = 0.5 kΩ, R6 = 1 kΩ Based on the circuit diagram depicted in Fig. 5, the experimental circuit is built on a breadboard by using discrete electronic components, and the experimental setup is shown in Fig. 6. The experimental phase portraits captured with a digital camera are shown in Fig. 7. It’s found that the experimental results agree well with the numerical simulations.
(3)
Letting R7 = 10 kΩ and C0 = 100 nF, we get the values of other resistances by comparing Eq. (2) with Eq. (3), as below
Fig. 2. (a) Continuous frequency spectrum and (b) Poincare maps on x1–x3 plane with a = 2.
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Fig. 3. Dynamics switch between periodic and chaotic windows of system (1) interpreted by (a) Bifurcation and (b) Lyapunov exponents when 1 ≤ a ≤ 2.6.
Fig. 4. (a) Periodic orbit with a = 1.24; (b) double-periodic orbit with a = 2.16.
4. Stabilization for the reported system
x1
4.1. Control scheme 0.1x3
Although the behavior of chaotic system cannot be chronically forecasted, one can effectively utilize chaos by suppressing the future behavior with appropriate control technology. In the last several decades, the stability design for chaotic system has been extensively studied to show the importance of chaos control [43–52]. In this section, we add the single controller u = −g (x2−x e2) to the second equation of system (1), depicted by 2 ⎧ x1̇ = −x1 + x2 x 3 x2̇ = ax2−x1 x3 + u ⎨ 3 ⎩ x3̇ = −x 3 + x1 x2
2
R1
C0
−0.01x2 x32
− x2
x1
R2
R7
x2
R7
− x2
x1
R3
C0
x2
0.1x1 x3 R4
x3 (4)
− x2
For the controller u, g is the control gain to be determined, xe2 is the second component of the equilibrium point xe = (xe1, xe2, xe3) for the uncontrolled system (1). Assume that x3 is bounded satisfying x 32 ⩽ B3 and the value of B3 can be determined by the state trajectory of x3 for any initial conditions. For the case of controlling to zero equilibrium point, we set the 1 candidate Lyapunov function as V (x ) = 2 (x12 + x 22 + x 32) , and the time derivative of V (x ) is deduced by V̇ = −x12 + x1 x2 x 32 + (a−
x1 0.1x32
−0.1x1 x2
R5
0.01x33 R6
x3 Fig. 5. Circuit diagram for emulating system (1).
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C0
x3
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Fig. 6. Experimental setup for observation. Fig. 8. Control to zero equilibrium point with (a) g = 4; (b) g = 6.
g ) x 22−x 34 ⩽ −x12 + B3 |x1 ||x2 |−( g−a x2)2−x 34 . Thus, it yields that V̇ ⩽ − B2
(|x1 |− g−a |x2 |)2−x 34 ⩽ 0 with g ⩾ 43 + a , and the output variables of system (4) will converge to the zero equilibrium point asymptotically. For the stabilization of non-zero equilibrium points E1 and E2, we start with the controlled system
original system (1) can be stabilized to E1 and E2 by only using the single state feedback u = −g (x2−x e2) . 4.2. Numerical verification
x ̇ = −x1 + x2 x 32−g (x1−x e1) ⎧ ⎪ 1 x2̇ = ax2−x1 x3−g (x2−x e2) ⎨ ⎪ x3̇ = −x 33 + x1 x2−g (x3−x e3) ⎩
This section of the paper presents numerical simulations to demonstrate the effectiveness of the derived control scheme by adopting MATLAB software with ODE45 algorithm. For conveniently comparing the control process, we choose the system parameter a = 2, and the initial states (0.2, 0.1, 0.1). With this condition, system (4) is chaotic before executing control. We first control the system to its zero equilibrium point. In this process, the control gain is taken as g = 4 and g = 6 respectively, and the controller u is imposed at the fiftieth second. The control results are numerically described in Fig. 8(a) and (b). It’s found that one can stabilize the system to the fixed point as expected, and that a larger control gain is required to more quickly reach the control purpose. Now we consider the case of controlling to non-zero equilibrium points. Fig. 9(a) and (b) depicts the numerical result of controlling to E1(1.7818, 1.1225, 1.2599) with u = −6(x2 − 1.1225). And the numerical result of controlling to E2(−1.7818, −1.1225, 1.2599) is shown in Fig. 10(a) and (b) with u = −6(x2 + 1.1225). It is confirmed from Figs. 9 and 10 that one can also stabilize the system to the desired non-zero point E1 or E2, by selecting the corresponding controller.
(5)
Suppose that function fi (x ) is smooth in the neighborhood of x e and satisfies the Lipschitz condition ‖fi (x )−fi (x e )‖ ⩽ k‖x i−x ei ‖, i = 1,2,3; k is a positive constant. We introduce the candidate function 1 1 1 V (x ) = 2 (x1−x e1)2 + 2 (x2−x e2)2 + 2 (x3−x e3)2 which leads to 3 3 2 2 ̇ V = ∑i = 1 (x i−x ei ) fi (x )− ∑i = 1 g (x i−x ei ) ⩽ (3k−g )‖x −x e ‖ . Therefore, we obtain that V̇ ⩽ 0 when g ⩾ 3k , and thus the output variables of system (4) will converge to the non-zero equilibrium points asymptotically. If x2 convergences to xe2, we obtain the following two-dimensional subsystem of (1) by replacing x1, x2, x3 with x d1 = x1−x e1,x d2 = x2−x e2,x d2 = x3−x e3 .
⎧ x ḋ 1 = −x d1 3 ⎨ ⎩ x ḋ 3 = −xd3
(6)
which is uniformly exponentially stable about (xd1, xd3) = (0, 0), meaning that x1, x3 converge to xe1, xe3 exponentially. Thus, the
Fig. 7. Experimental phase portraits of system (1) projected on (a) x1–x2 plane; (b) x2–x3 plane; (c) x1–x3 plane.
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Fig. 9. Control to E1(1.7818, 1.1225, 1.2599) with u = −6(x2 − 1.1225).
Fig. 10. Control to E2(−1.7818, −1.1225, 1.2599) with u = −6(x2 + 1.1225).
5. Conclusion
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This paper reported a three-dimensional autonomous chaotic system holding three unusual equilibrium points: a zero equilibrium point of non-hyperbolic type and two symmetrical equilibrium points of hyperbolic type. The finding revealed the striking fact for the first time, that hyperbolic type and non-hyperbolic type equilibrium points can simultaneously coexist in one chaotic system. To understand the complex behavior of the system, basic dynamical properties are studied in detail. And to further verify the theoretical results, we build an analog circuit to realize the chaotic system, by using the dimensionless state equations and the improved module-based technique. Finally, a simple feedback controller is introduced to suppress the system to its equilibrium points, and numerical simulations further illustrated the enforceability of the theoretical scheme.
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Acknowledgments We first appreciate the anonymous reviewers for the valuable comments. This work was supported by Science and Technology Program of Hunan Province (No. 2016TP1021); the Research Foundation of Education Bureau of Hunan Province of China (Nos. 16B113, 16B114, 14A062); Hunan Provincial Natural Science Foundation of China (No. 2016JJ4036) and National Natural Science Foundation of China (Nos. 11562004, 61473118 and 11602084).
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