NEW RESULTS ABOUT ROUTE TO CHAOS IN ROSSLER SYSTEM

2004 ◽  
Vol 14 (01) ◽  
pp. 293-308 ◽  
Author(s):  
SVETOSLAV NIKOLOV ◽  
VALKO PETROV
Keyword(s):  
Bifurcation Analysis ◽  
Stability Region ◽  
Analytical Formula ◽  
Stability Loss ◽  
Transition To Chaos ◽  
Rossler System ◽  
Rössler System ◽  
Route To Chaos ◽  
Boundary Of Stability

In this paper, the theory of Lyapunov–Andronov is applied to investigate the route to chaos in Rossler system. On the base of a new analytical formula for the first Lyapunov value at the boundary of stability region, we make a detailed bifurcation analysis of this system. From the obtained results the following new conclusions are made: Transition to chaos in the Rossler's system takes place at soft stability loss in the form of a cascade of periodic self-oscillations. Then the occurrence of chaotic self-oscillations in this system takes place under hard stability loss.

FIRST LYAPUNOV VALUE AND BIFURCATION BEHAVIOR OF SPECIFIC CLASS OF THREE-DIMENSIONAL SYSTEMS

2004 ◽  
Vol 14 (08) ◽  
pp. 2811-2823 ◽  
Author(s):  
SVETOSLAV NIKOLOV
Keyword(s):  
Dynamic Systems ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Analytical Formula ◽  
Specific Class ◽  
Third Order ◽  
Cubic Polynomial ◽  
Route To Chaos ◽  
Boundary Of Stability ◽  
Bifurcation Behavior

This paper presents a study of the behavior of a special class of 3D dynamic systems (i.e. RHS of the third-order equation is a cubic polynomial), using Lyapunov–Andronov's theory. Considering the general case, we find a new analytical formula for the first Lyapunov's value at the boundary of stability. It enables one to study in detail the bifurcation behavior (and the route to chaos, in particular) of dynamic systems of the above type. We check the validity of our analytical results on the first Lyapunov's value by studying the route to chaos of two 3D dynamic systems with proved chaotic behavior. These are Chua's and Rucklidge's systems. Considering their route to chaos, we find new results.

scholarly journals Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control

2016 ◽  
Vol 46 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Svetoslav Nikolov ◽  
Valentin Nedev
Keyword(s):  
Dynamic System ◽  
Bifurcation Analysis ◽  
Inverted Pendulum ◽  
Dynamic Behaviour ◽  
Nonlinear Differential Equations ◽  
Analytical Formula ◽  
Bounded Control ◽  
Pivot Point ◽  
Bifurcation Behaviour ◽  
Boundary Of Stability

Abstract This paper presents an investigation on the behaviour of con- ventional inverted pendulum with an inertia disk in its free extreme. The system is actuated by means of torques applied to the disk by a DC mo- tor, mounted on the pendulum’s arm. Thus, the system is underactuated since the pendulum can rotate freely around its pivot point. The dynam- ical model is given with three ordinary nonlinear differential equations. Using Poincare-Andronov-Hopf’s theory, we find a new analytical formula for the first Lyapunov’s value at the boundary of stability. It enables one to study in detail the bifurcation behaviour of the above dynamic system. We check the validity of our analytical results on the first Lyapunov’s value by numerical simulations. Hence, we find some new results.

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.

A SIMPLE OBSERVER OF THE HYPERCHAOTIC RÖSSLER SYSTEM

2010 ◽  
Vol 24 (22) ◽  
pp. 4325-4331
Author(s):  
XING-YUAN WANG ◽  
JUN-MEI SONG
Keyword(s):  
Dynamical System ◽  
Time Domain ◽  
Global Exponential Stability ◽  
Observer Design ◽  
The Other ◽  
State Observation ◽  
Rossler System ◽  
Error System ◽  
Rössler System ◽  
The Time Domain

This paper studies the hyperchaotic Rössler system and the state observation problem of such a system being investigated. Based on the time-domain approach, a simple observer for the hyperchaotic Rössler system is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design that it does not need to transmit all signals of the dynamical system. It is proved theoretically, and numerical simulations show the effectiveness of the scheme finally.

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