Line Of Equilibrium Points Research Articles

Mathematical modelling, bifurcation analysis, circuit design and FPGA implementation of a 5D hyperchaotic weather fluctuation model with a line of equilibrium points

Mathematical modelling, bifurcation analysis, circuit design and FPGA implementation of a 5D hyperchaotic weather fluctuation model with a line of equilibrium points

Mathematical modelling, bifurcation analysis, circuit design and FPGA implementation of a 5D hyperchaotic weather fluctuation model with a line of equilibrium points

Mathematical modelling, bifurcation analysis, circuit design and FPGA implementation of a 5D hyperchaotic weather fluctuation model with a line of equilibrium points

A New 3-D Multistable Chaotic System with Line Equilibrium: Dynamic Analysis and Synchronization

This work introduces a new 3-D chaotic system with a line of equilibrium points. We carry out a detailed dynamic analysis of the proposed chaotic system with five nonlinear terms. We show that the chaotic system exhibits multistability with two coexisting chaotic attractors. We apply integral sliding mode control for the complete synchronization of the new chaotic system with itself as leader-follower systems.

A New Double-Wing Chaotic System With Coexisting Attractors and Line Equilibrium: Bifurcation Analysis and Electronic Circuit Simulation

This research work reports a double-wing chaotic system with a line of equilibrium points and constructs an electronic circuit via MultiSIM for practical implementation. Explicitly, the new chaotic system has a total of six terms with two quadratic nonlinearities and absolute function nonlinearity. Using the phase plots in MATLAB, we demonstrate that the new chaotic system has double-wing chaotic attractor. We describe the Lyapunov exponents and the Kaplan-Yorke fractal dimension of the new chaotic system. A novel feature of the new chaotic system is that the system has rest points located on the z-axis as well as two rest points not on the z-axis. Thus, the new system has infinite number of rest points and hidden attractor. We also exhibit that the new double-wing chaotic system has multi-stability and we illustrate the coexistence of attractors for two different sets of initial conditions. Some interesting dynamical properties such as offset boosting are also presented. Finally, we build an electronic circuit of the new chaotic system and show that the theoretical model has practical feasibility for implementation.

Identificación de una Bifurcación de Hopf con o sin Parámetros

En este trabajo se presenta los lineamientos teóricos que identifica la existencia de una bifurcación genérica de Hopf, con o sin parámetros, a través de un punto o una línea de puntos de equilibrio para un sistema dinámico continúo. Con estos lineamientos definidos, se analiza la bifurcación de Hopf sin parámetros a través del estudio de contornos viscosos para un sistema de ecuaciones diferenciales parciales conformado por un término de reacción - difusión; con condición inicial constante a trozos y una línea de puntos de equilibrio que presenta un par de valores propios complejos conjugados con parte real nula en su linealización a medida que se aproxima al origen. En condiciones adecuadas, se distinguen dos casos para este tipo de bifurcación genérica de Hopf sin parámetros: hiperbólica y elíptica, genéricas en el sentido que es una bifurcación controlada en una línea de puntos de equilibrio.

Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability.

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

A novel four-dimensional conservative chaotic system without linear term, its analysis and adaptive control via integral sliding mode control

In this work, a novel conservative four-dimensional chaotic system without linear term is proposed. The fundamental qualitative properties of the novel chaotic system are described with details of volume preserving property, equilibrium points, symmetry, Lyapunov exponents and Kaplan-Yorke dimension. We show that the novel four-dimensional chaotic system has two planes and one line of equilibrium points. Next, an adaptive integral sliding mode controller is designed to stabilise the novel chaotic system with an unknown system parameter. Moreover, an adaptive integral sliding mode controller is designed to achieve global chaos synchronisation of the identical novel chaotic systems with an unknown system parameter. The adaptive control mechanism helps the control design by estimating the unknown parameter. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work.

A novel four-dimensional conservative chaotic system without linear term, its analysis and adaptive control via integral sliding mode control

In this work, a novel conservative four-dimensional chaotic system without linear term is proposed. The fundamental qualitative properties of the novel chaotic system are described with details of volume preserving property, equilibrium points, symmetry, Lyapunov exponents and Kaplan-Yorke dimension. We show that the novel four-dimensional chaotic system has two planes and one line of equilibrium points. Next, an adaptive integral sliding mode controller is designed to stabilise the novel chaotic system with an unknown system parameter. Moreover, an adaptive integral sliding mode controller is designed to achieve global chaos synchronisation of the identical novel chaotic systems with an unknown system parameter. The adaptive control mechanism helps the control design by estimating the unknown parameter. Numerical simulations using MATLAB are shown to illustrate all the main results derived in this work.

Recent new examples of hidden attractors

Hidden attractors represent a new interesting topic in the chaos literature. These attractors have a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. Oscillations in dynamical systems can be easily localized numerically if initial conditions from its open neighborhood lead to a long-time oscillation. This paper reviews several types of new rare chaotic flows with hidden attractors. These flows are divided into to three main groups: rare flows with no equilibrium, rare flows with a line of equilibrium points, and rare flows with a stable equilibrium. In addition we describe a novel system containing hidden attractors.