scholarly journals A Multistable Chaotic Jerk System with Coexisting and Hidden Attractors: Dynamical and Complexity Analysis, FPGA-Based Realization, and Chaos Stabilization Using a Robust Controller

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 569 ◽  
Author(s):  
Heng Chen ◽  
Shaobo He ◽  
Ana Dalia Pano Azucena ◽  
Amin Yousefpour ◽  
Hadi Jahanshahi ◽  
...  
Keyword(s):  
Complexity Analysis ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Constant Term ◽  
Successful Implementation ◽  
Hidden Attractors ◽  
Field Programmable ◽  
Jerk System

In the present work, a new nonequilibrium four-dimensional chaotic jerk system is presented. The proposed system includes only one constant term and has coexisting and hidden attractors. Firstly, the dynamical behavior of the system is investigated using bifurcation diagrams and Lyapunov exponents. It is illustrated that this system either possesses symmetric equilibrium points or does not possess an equilibrium. Rich dynamics are found by varying system parameters. It is shown that the system enters chaos through experiencing a cascade of period doublings, and the existence of chaos is verified. Then, coexisting and hidden chaotic attractors are observed, and basin attraction is plotted. Moreover, using the multiscale C0 algorithm, the complexity of the system is investigated, and a broad area of high complexity is displayed in the parameter planes. In addition, the chaotic behavior of the system is studied by field-programmable gate array implementation. A novel methodology to discretize, simulate, and implement the proposed system is presented, and the successful implementation of the proposed system on FPGA is verified through the simulation outcome. Finally, a robust sliding mode controller is designed to suppress the chaotic behavior of the system. To deal with unexpected disturbances and uncertainties, a disturbance observer is developed along with the designed controller. To show the successful performance of the designed control scheme, numerical simulations are also presented.

A New Four-Dimensional Non-Hamiltonian Conservative Hyperchaotic System

2020 ◽  
Vol 30 (16) ◽  
pp. 2050242
Author(s):  
Shuangquan Gu ◽  
Baoxiang Du ◽  
Yujie Wan
Keyword(s):  
Analog Circuit ◽  
Complexity Analysis ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Hyperchaotic System ◽  
Attraction Basin ◽  
Hidden Extreme Multistability ◽  
High Degree ◽  
First Time ◽  
Lyapunov Exponent Spectrum

This paper presents a new four-dimensional non-Hamiltonian conservative hyperchaotic system. In the absence of equilibrium points in the system, the phase trajectories generated by the system have hidden features. The conservative features that vary with the parameter have been analyzed in detail by Lyapunov exponent spectrum, bifurcation diagram, the sum of Lyapunov exponents, and the fractional dimensions, and during the analysis, multiple quasi-periodic four-dimensional tori as well as hyperchaotic attractors have been observed. The Poincaré sections confirm these dynamic behaviors. Amidst the process of studying the dynamical behavior of the system with initial values, the hidden extreme multistability, and the initial offset boosting behavior, the results have been witnessed for the very first time in a conservative chaotic system. The phase diagram and attraction basin also confirm this assertion, while two complex transient transition behaviors have been observed. Moreover, through the introduction of a spectral entropy algorithm, the complexity analysis of the time sequences generated by the system have been performed and compared with the existing literature. The results show that the system has a high degree of complexity. The design and construction of the analog circuit of the system for simulation, the circuit experimental results are consistent with the numerical simulation, further verifying the physical realizability of the newly proposed system. This lays a good foundation for its practical application in engineering.

An Autonomous Simple Chaotic Jerk System with Stable and Unstable Equilibria Using Reverse Sine Hyperbolic Functions

2020 ◽  
Vol 30 (05) ◽  
pp. 2050070 ◽  
Author(s):  
Manoj Joshi ◽  
Ashish Ranjan
Keyword(s):  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Phase Portraits ◽  
Hidden Attractors ◽  
Hyperbolic Functions ◽  
Simple Implementation ◽  
Route To Chaos ◽  
Jerk System

This article introduces a new simple jerk system with sine hyperbolic nonlinearity which gives the hidden attractor. An autonomous simple implementation of jerk system experiences an important and striking feature of hidden attractors with both stable equilibrium and unstable equilibrium using a reverse nonlinearity function with parametrically controlled approach. Some basic properties of the system are well studied and analyzed in terms of route to chaos, basins of attraction, Lyapunov exponent (LE), bifurcation sequences, coexistence of attractor and phase portraits. The chaotic behavior of the new system is investigated through numerical simulation and their equivalent electrical circuit implementation using single amplifier with few passive elements. The justification of theoretical observation of the proposed chaotic system is perfectly observed in PSPICE simulation and laboratory experiment.

DYNAMICS OF ELECTROSTATICALLY ACTUATED MICRO-ELECTRO-MECHANICAL SYSTEMS: SINGLE DEVICE AND ARRAYS OF DEVICES

2009 ◽  
Vol 19 (03) ◽  
pp. 1007-1022 ◽  
Author(s):  
V. Y. TAFFOTI YOLONG ◽  
P. WOAFO
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Mechanical Systems ◽  
Largest Lyapunov Exponent ◽  
Mems Devices ◽  
Micro Electro Mechanical Systems ◽  
Electrostatically Actuated ◽  
In Series ◽  
Electrostatic Coupling

The dynamical behavior of micro-electro-mechanical systems (MEMS) with electrostatic coupling is studied. A nonlinear modal analysis approach is applied to decompose the partial differential equation into a set of ordinary differential equations. The stability analysis of the equilibrium points is investigated. The amplitudes of the harmonic oscillatory states in the triple resonant states are obtained and discussed. Chaotic behavior is investigated using bifurcations diagram and the largest Lyapunov exponent. The dynamics of the MEMS with multiple functions in series is also investigated as well as the transitions boundaries for the complete synchronization state in a shift-invariant set of coupled MEMS devices.

Hyperchaos and Coexisting Attractors in a Modified van der Pol–Duffing Oscillator

2019 ◽  
Vol 29 (05) ◽  
pp. 1950067 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Abdul Jalil M. Khalaf ◽  
Zhouchao Wei ◽  
Viet-Thanh Pham ◽  
Ahmed Alsaedi ◽  
...  
Keyword(s):  
Discrete Model ◽  
Equilibrium Points ◽  
Duffing Oscillator ◽  
Dynamical Behavior ◽  
Coexisting Attractors ◽  
Van Der Pol ◽  
Euler’S Method ◽  
Gate Arrays ◽  
Field Programmable ◽  
Programmable Gate Arrays

This paper deals with a new modified hyperchaotic van der Pol–Duffing (MVPD) snap oscillator. Various dynamical properties of the proposed system are investigated with the help of Lyapunov exponents, stability analysis of the equilibrium points and bifurcation plots. The existence of the Hopf bifurcation is established by analyzing the corresponding characteristic equation. It is also proved that the MVPD oscillator shows multistability with coexisting attractors. Various numerical simulations are conducted and presented to show the dynamical behavior of the MVPD system. To show that the system is hardware realizable, we derive the discrete model of the MVPD system using the Euler’s method and using the hardware–software cosimulation, the proposed MVPD system is implemented in Field Programmable Gate Arrays. It is shown that the output of the digital implementations of the MVPD systems matches the numerical analysis.

Multistability and Hidden Attractors in the Dynamics of Permanent Magnet Synchronous Motor

2019 ◽  
Vol 29 (04) ◽  
pp. 1950056 ◽  
Author(s):  
Jay Prakash Singh ◽  
Binoy Krishna Roy ◽  
Nikolay V. Kuznetsov
Keyword(s):  
Permanent Magnet ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Permanent Magnet Synchronous Motor ◽  
Synchronous Motor ◽  
Adaptive Sliding Mode Control ◽  
Stable Node ◽  
Control Input ◽  
Hidden Attractors ◽  
Load Torque

This paper attempts to find some interesting and unique properties in the dynamics of a permanent magnet synchronous motor (PMSM). When compared with the existing literature on the dynamics of a PMSM, we find some interesting and unique behaviors which have not been reported so far. These are (i) occurrence of multistability, (ii) existence of hidden attractors and (iii) equilibrium point with two stable node-foci and one saddle point index-1. The above-said unique behaviors in the dynamics of a PMSM are not found in the literature to the best of authors’ knowledge. Three different cases with (a) [Formula: see text] (voltage/frequency) control, (b) constant load torque and (c) constant direct and quadrature-axis voltage, and load torque are considered to show the multistability in the dynamics of a PMSM. The multistability is confirmed by using the bifurcation analysis. In another case, when the load torque is selected as a feedback of quadrature-axis voltage, the system depicts hidden attractors (point, periodic and transient chaotic). An adaptive sliding mode control is designed to control the hidden transient chaotic behavior of the system. The simulation results confirm the suppression of the transient chaotic attractors with smaller stabilization time and chattering free control input.

Chaos suppression of Fractional order Willamowski–Rössler Chemical system and its synchronization using Sliding Mode Control

2016 ◽  
Vol 5 (3) ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Anitha Karthikeyan
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Practical Importance ◽  
Chemical System ◽  
Mode Control ◽  
Chaos Suppression ◽  
The Stability

AbstractMost of the Real systems shows chaotic behavior when they approach complex states. Especially in physical and chemical systems these behaviors define the character of the system. The control of these chaotic behaviors is of very high practical importance and hence mathematical models of these chaotic systems proves vital in deciding the control structures. One such model of chemical reactors is the Willamowski–Rössler system (WR). In this paper we derive a fractional order sliding mode control scheme where the states of the WR system are driven back to the defined equilibrium points. We have also synchronized master and slave fractional order WR system using sliding mode control. As the entire control law is defined in fractional order, we derived a new methodology to prove the stability of the controller. The numerical simulation and analysis are achieved with LabVIEW.

scholarly journals A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sundarapandian Vaidyanathan ◽  
Xiong Wang
Keyword(s):  
Chaotic System ◽  
Infinite Number ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Equilibrium Analysis ◽  
Chaotic Attractors ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Topological Horseshoes

Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control.

scholarly journals Stabilization of Port Hamiltonian Chaotic Systems with Hidden Attractors by Adaptive Terminal Sliding Mode Control

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 122 ◽  
Author(s):  
Ahmad Taher Azar ◽  
Fernando E. Serrano
Keyword(s):  
Chaotic Systems ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Chaotic Oscillator ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Terminal Sliding Mode ◽  
Sliding Mode Controllers ◽  
Empirical Tests ◽  
System States

In this study, the design of an adaptive terminal sliding mode controller for the stabilization of port Hamiltonian chaotic systems with hidden attractors is proposed. This study begins with the design methodology of a chaotic oscillator with a hidden attractor implementing the topological framework for its respective design. With this technique it is possible to design a 2-D chaotic oscillator, which is then converted into port-Hamiltonia to track and analyze these models for the stabilization of the hidden chaotic attractors created by this analysis. Adaptive terminal sliding mode controllers (ATSMC) are built when a Hamiltonian system has a chaotic behavior and a hidden attractor is detected. A Lyapunov approach is used to formulate the adaptive device controller by creating a control law and the adaptive law, which are used online to make the system states stable while at the same time suppressing its chaotic behavior. The empirical tests obtaining the discussion and conclusions of this thesis should verify the theoretical findings.

scholarly journals A Novel Chaotic System with Two Circles of Equilibrium Points: Multistability, Electronic Circuit and FPGA Realization

Electronics ◽  
2019 ◽  
Vol 8 (11) ◽  
pp. 1211 ◽  
Author(s):  
Sambas ◽  
Vaidyanathan ◽  
Tlelo-Cuautle ◽  
Zhang ◽  
Sukono ◽  
...  
Keyword(s):  
Chaotic System ◽  
Analog Circuit ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Digital Implementation ◽  
Gate Arrays ◽  
New Chaotic System ◽  
Field Programmable ◽  
Programmable Gate Arrays ◽  
Good Agreement

This paper introduces a new chaotic system with two circles of equilibrium points. The dynamical properties of the proposed dynamical system are investigated through evaluating Lyapunov exponents, bifurcation diagram and multistability. The qualitative study shows that the new system exhibits coexisting periodic and chaotic attractors for different values of parameters. The new chaotic system is implemented in both analog and digital electronics. In the former case, we introduce the analog circuit of the proposed chaotic system with two circles of equilibrium points using amplifiers, which is simulated in MultiSIM software, version 13.0 and the results of which are in good agreement with the numerical simulations using MATLAB. In addition, we perform the digital implementation of the new chaotic system using field-programmable gate arrays (FPGA), the experimental observations of the attractors of which confirm its suitability to generate chaotic behavior.

Different Families of Hidden Attractors in a New Chaotic System with Variable Equilibrium

2017 ◽  
Vol 27 (09) ◽  
pp. 1750138 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Sajad Jafari ◽  
Christos Volos ◽  
Tomasz Kapitaniak
Keyword(s):  
Chaotic System ◽  
Infinite Number ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
View Point ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
Complex Dynamical Behavior ◽  
New System

A new chaotic system having variable equilibrium is introduced in this paper. The presence of an infinite number of equilibrium points, a stable equilibrium, and no-equilibrium is observed in the system. Interestingly, this system is classified as a rare system with hidden attractors from the view point of computation. Complex dynamical behavior and a circuital implementation of the new system have been investigated in our work.

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