scholarly journals Coexistence and Dynamical Connections between Hyperchaos and Chaos in the 4D Rössler System: A Computer-Assisted Proof

2016 ◽  
Vol 15 (1) ◽  
pp. 356-390 ◽  
Author(s):  
Daniel Wilczak ◽  
Sergio Serrano ◽  
Roberto Barrio
Keyword(s):  
Computer Assisted ◽  
Computer Assisted Proof ◽  
System A ◽  
Rossler System ◽  
Rössler System

NUMERICAL EXPERIMENTS ON THE HYPERCHAOTIC CHEN SYSTEM BY THE ADOMIAN DECOMPOSITION METHOD

2008 ◽  
Vol 05 (03) ◽  
pp. 403-412 ◽  
Author(s):  
M. MOSSA AL-SAWALHA ◽  
M. S. M. NOORANI ◽  
I. HASHIM
Keyword(s):  
Numerical Experiments ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Dimensional System ◽  
Chen System ◽  
System A ◽  
Adomian Decomposition ◽  
Rossler System ◽  
Quadratic Nonlinearities ◽  
Rössler System

The aim of this paper is to investigate the accuracy of the Adomian decomposition method (ADM) for solving the hyperchaotic Chen system, which is a four-dimensional system of ODEs with quadratic nonlinearities. Comparisons between the decomposition solutions and the fourth order Runge–Kutta (RK4) solutions are made. We look particularly at the accuracy of the ADM as the hyperchaotic Chen system has higher Lyapunov exponents than the hyperchaotic Rössler system. A comparison with the hyperchaotic Rössler system is given.

HOPF BIFURCATION CONTROL USING NONLINEAR FEEDBACK WITH POLYNOMIAL FUNCTIONS

2004 ◽  
Vol 14 (05) ◽  
pp. 1683-1704 ◽  
Author(s):  
PEI YU ◽  
GUANRONG CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Control ◽  
Hopf Bifurcations ◽  
Nonlinear Feedback ◽  
Lorenz Equation ◽  
System A ◽  
Rossler System ◽  
Rössler System ◽  
The Lorenz System ◽  
The Stability

A general explicit formula is derived for controlling bifurcations using nonlinear state feedback. This method does not increase the dimension of the system, and can be used to either delay (or eliminate) existing bifurcations or change the stability of bifurcation solutions. The method is then employed for Hopf bifurcation control. The Lorenz equation and Rössler system are used to illustrate the application of the approach. It is shown that a simple control can be obtained to simultaneously stabilize two symmetrical equilibria of the Lorenz system, and keep the symmetry of Hopf bifurcations from the equilibria. For the Rössler system, a control is also obtained to simultaneously stabilize two nonsymmetric equilibria and meanwhile stabilize possible Hopf bifurcations from the equilibria. Computer simulation results are presented to confirm the analytical predictions.

scholarly journals Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

2021 ◽  
Vol 11 (15) ◽  
pp. 6955
Author(s):  
Andrzej Rysak ◽  
Magdalena Gregorczyk
Keyword(s):  
Optimal Parameter ◽  
Differential Transform Method ◽  
Bifurcation Diagrams ◽  
Transform Method ◽  
Rossler System ◽  
Differential Transform ◽  
Rössler System ◽  
Fractional System ◽  
Parameter Values

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

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