scholarly journals How to Generate Chaos from Switching System: A Saddle Focus of Index 1 and Heteroclinic Loop-Based Approach

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Fang Bao ◽  
Simin Yu
Keyword(s):  
Linear System ◽  
Control Strategy ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Autonomous Systems ◽  
Switching Control ◽  
Heteroclinic Loop ◽  
Switching Control Strategy ◽  
Index 2

There exist two different types of equilibrium points in 3-D autonomous systems, named as saddle foci of index 1 and index 2, which are crucial for chaos generation. Although saddle foci of index 2 have been usually applied for creating double-scroll or double-wing chaotic attractors, saddle foci of index 1 are further considered for chaos generation in this paper. A novel approach for constructing chaotic systems is investigated by applying the switching control strategy and yielding a heteroclinic loop which connects two saddle foci of index 1. A basic 3-D linear system with an arbitrary normal direction of the eigenplane, possessing a saddle focus of index 1 whose corresponding eigenvalues satisfy the Shil'nikov inequality, is first introduced. Then a heteroclinic loop connecting two saddle foci of index 1 will be formed by applying the switching control strategy to the basic 3-D linear system. The heteroclinic loop consists of an unstable manifold, a stable manifold, and a heteroclinic point. Under the necessary conditions for forming the heteroclinic loop, the intended two-segmented piecewise linear system which exhibits the chaotic behavior in the sense of the Smale horseshoe can be finally constructed. An illustrative example is given, confirming the effectiveness of the proposed method.

Synthesis and Optimization of Transient Performances for Underdamped Second-Order Linear System Based on the Switching Control Strategy

2012 ◽  
Author(s):  
Qingyu Su ◽  
Georgi M. Dimirovski ◽  
Jun Zhao
Keyword(s):  
Linear System ◽  
Control Strategy ◽  
Optimization Problems ◽  
Switching Control ◽  
Second Order ◽  
Order Linear ◽  
Output Response ◽  
Special Cases ◽  
Switching Control Strategy ◽  
The Given

In this paper, the synthesis and optimization problems of the transient performances, such as overshoot and setting time, of the output response with respect to a step signal for an under-damped second-order linear system are studied. The traditional control strategy and the switching control strategy are used, respectively. Our control objective is twofold: Objective 1 is to ensure that the overshoot of the system output does not violate the given safety bound and objective 2 is to ensure that the settling time is smaller than the given allowable time. The obtained results indicate that the switching control strategy has main two evident advantages: firstly, the switching proportional controllers can take advantages of both the quick response and the small overshoot of the output response. Secondly, the switching controller is always available, while the single proportional controller is available in some special cases. Finally, an example of servomotor is given to demonstrate the effectiveness of the proposed method.

scholarly journals Generation Method of Multipiecewise Linear Chaotic Systems Based on the Heteroclinic Shil’nikov Theorem and Switching Control

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Chunyan Han ◽  
Fang Yuan ◽  
Xiaoyuan Wang
Keyword(s):  
Linear Systems ◽  
Circuit Design ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Switching Control ◽  
Heteroclinic Loop ◽  
Dynamical Characteristics ◽  
Fundamental Systems

Based on the heteroclinic Shil’nikov theorem and switching control, a kind of multipiecewise linear chaotic system is constructed in this paper. Firstly, two fundamental linear systems are constructed via linearization of a chaotic system at its two equilibrium points. Secondly, a two-piecewise linear chaotic system which satisfies the Shil’nikov theorem is generated by constructing heteroclinic loop between equilibrium points of the two fundamental systems by switching control. Finally, another multipiecewise linear chaotic system that also satisfies the Shil’nikov theorem is obtained via alternate translation of the two fundamental linear systems and heteroclinic loop construction of adjacent equilibria for the multipiecewise linear system. Some basic dynamical characteristics, including divergence, Lyapunov exponents, and bifurcation diagrams of the constructed systems, are analyzed. Meanwhile, computer simulation and circuit design are used for the proposed chaotic systems, and they are demonstrated to be effective for the method of chaos anticontrol.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

scholarly journals Novel 3-Scroll Chua’s Attractor with One Saddle-Focus and Two Stable Node-Foci

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Kaibin Chu ◽  
Zhengwei Zhu ◽  
Hui Qian ◽  
Huagan Wu
Keyword(s):  
Control Function ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Stable Node ◽  
The Novel ◽  
Analog Multiplier ◽  
Piecewise Linearity ◽  
Index 2 ◽  
Hardware Circuit ◽  
Zero Index

With new three-segment piecewise-linearity in the classic Chua’s system, two new types of 2-scroll and 3-scroll Chua’s attractors are found in this paper. By changing the outer segment slope of the three-segment piecewise-linearity as positive, the new 2-scroll Chua’s attractor has emerged from one zero index-1 saddle-focus and two symmetric stable nonzero node-foci. In particular, by newly introducing a piecewise-linear control function, an improved Chua’s system only with one zero index-2 saddle-focus and two stable nonzero node-foci is constructed, from which a 3-scroll Chua’s attractor is converged. Some remarks for Chua’s nonlinearities and the generating chaotic attractors are discussed, and the stabilities at the three equilibrium points are then analyzed, upon which the emerging mechanisms of the novel 2-scroll and 3-scroll Chua’s attractors are explored in depth. Furthermore, an analog electronic circuit built with operational amplifier and analog multiplier is designed and hardware circuit experiments are measured to verify the numerical simulations. These novel 2-scroll and 3-scroll Chua’s attractors reported in this paper are completely different from the classic Chua’s attractors, which will enrich the dynamics of the classic Chua’s system.

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