scholarly journals Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 352
Author(s):  
Svetoslav G. Nikolov ◽  
Vassil M. Vassilev
Keyword(s):  
Chaotic Behavior ◽  
Global Analysis ◽  
Analytical Formula ◽  
Period Doubling ◽  
The Other ◽  
Regular Behavior ◽  
Focal Value ◽  
Automatic Regulator ◽  
Boundary Of Stability ◽  
Period Doubling Bifurcations

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

CHAOTIC BEHAVIOR INDUCED BY THERMAL MODULATION IN A MODEL OF THE CONVECTIVE FLOW OF A FLUID MIXTURE IN A POROUS MEDIUM

1999 ◽  
Vol 09 (02) ◽  
pp. 383-396 ◽  
Author(s):  
J.-M. MALASOMA ◽  
P. WERNY ◽  
C.-H. LAMARQUE
Keyword(s):  
Porous Medium ◽  
Convective Flow ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Type I ◽  
Fluid Mixture ◽  
Numerical Investigations ◽  
Period Doubling Bifurcations

Numerical investigations of the global behavior of a model of the convective flow of a binary mixture in a porous medium are reported. We find a complex behavior characterized by the presence of coexisting periodic, quasiperiodic and chaotic attractors. Bifurcations of periodic solutions and routes to chaos via type-I intermittency and period-doubling bifurcations are described. Boundary crises and band merging crises have also been observed.

EFFECT OF DISSIPATION ON THE HAMILTONIAN CHAOS IN COUPLED OSCILLATOR SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 859-867 ◽  
Author(s):  
V. SHEEJA ◽  
M. SABIR
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Dissipation Function ◽  
Hamiltonian Chaos ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Route To Chaos ◽  
Period Doubling Bifurcations ◽  
External Driving

We study the effect of linear dissipative forces on the chaotic behavior of coupled quartic oscillators with two degrees of freedom. The effect of quadratic Rayleigh dissipation functions, one with diagonal coefficients only and the other with nondiagonal coefficients as well are studied. It is found that the effect of Rayleigh Dissipation function with diagonal coefficients is to suppress chaos in the system and to lead the system to its equilibrium state. However, with a dissipation function with nondiagonal elements, other types of behaviors — including fixed point attractor, periodic attractors and even chaotic attractors — are possible even when there is no external driving. In such a system the route to chaos is through period-doubling bifurcations. This result contradicts the view that linear dissipation always causes decay of oscillations in oscillator models.

DYNAMICS AND MODULATED CHAOS FOR TWO COUPLED OSCILLATORS

2004 ◽  
Vol 14 (01) ◽  
pp. 337-346 ◽  
Author(s):  
QINSHENG BI
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
The Other ◽  
Van Der Pol ◽  
Parametrically Excited ◽  
Van Der Pol Oscillators ◽  
Period Doubling Bifurcations

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

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.

Electronic Circuit Experiments and SPICE Simulation of Double Covering Bifurcation of 2-Torus Quasi-Periodic Flow in Phase-Locked Loop Circuit

2016 ◽  
Vol 26 (07) ◽  
pp. 1630017
Author(s):  
Kyohei Kamiyama ◽  
Tetsuro Endo ◽  
Isao Imai ◽  
Motomasa Komuro
Keyword(s):  
Electronic Circuit ◽  
Period Doubling ◽  
Phase Locked Loop ◽  
Double Covering ◽  
The Other ◽  
Periodic Flow ◽  
Spice Simulation ◽  
Frequency Signal ◽  
Discrete Map ◽  
Period Doubling Bifurcations

Double covering (DC) bifurcation of a 2-torus quasi-periodic flow in a phase-locked loop circuit was experimentally investigated using an electronic circuit and via SPICE simulation; in the circuit, the input radio-frequency signal was frequency modulated by the sum of two asynchronous sinusoidal baseband signals. We observed both DC and period-doubling bifurcations of a discrete map on two Poincaré sections, which were realized by changing the sample timing from one baseband sinusoidal signal to the other. The results confirm the DC bifurcation of the original flow.

Period Doubling and Chaos in Unsymmetric Structures Under Parametric Excitation

1989 ◽  
Vol 56 (4) ◽  
pp. 947-952 ◽  
Author(s):  
W. Szemplin´ska-Stupnicka ◽  
R. H. Plaut ◽  
J.-C. Hsieh
Keyword(s):  
Limit Cycles ◽  
Nonlinear Oscillations ◽  
Chaotic Behavior ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Period Doubling ◽  
Analytical Techniques ◽  
Excited System ◽  
Period Doubling Bifurcations ◽  
Quadratic And Cubic Nonlinearities

Nonlinear oscillations of a single-degree-of-freedom, parametrically-excited system are considered. The stiffness involves quadratic and cubic nonlinearities and models a shallow arch or buckled mechanism. The excitation frequency is assumed to be close to twice the natural frequency of the system. Numerical integration is used to obtain phase plane portraits, power spectra, and Poincare´ maps for large-time motions. Period-doubling bifurcations and several types of limit cycles and chaotic behavior are observed. Approximate analytical techniques are applied to analyze some of the limit cycles and transitions of behavior. The results are used to estimate the parameter region in which chaos may occur.

SEQUENCES OF CYCLES AND TRANSITIONS TO CHAOS IN A MODIFIED GOODWIN'S GROWTH CYCLE MODEL

2007 ◽  
Vol 17 (06) ◽  
pp. 1911-1932 ◽  
Author(s):  
GIORGIO COLACCHIO ◽  
MARCO SPARRO ◽  
CLAUDIO TEBALDI
Keyword(s):  
Chaotic Behavior ◽  
Nonlinear Modeling ◽  
Period Doubling ◽  
Growth Cycle ◽  
Economic Systems ◽  
Volterra Models ◽  
Wage Share ◽  
Goodwin Model ◽  
Relevant Role ◽  
Period Doubling Bifurcations

The model introduced by Goodwin [1967] in "A Growth Cycle" represents a milestone in the nonlinear modeling of economic dynamics. On the basis of a few simple assumptions, the Goodwin Model (GM) is formulated exactly as the well-known Lotka–Volterra system, in terms of the two variables "wage share" and "employment rate". A number of extensions have been proposed with the aim to make the model more robust, in particular, to obtain structural stability, lacking in GM original formulation. We propose a new extension that: (a) removes the limiting hypothesis of "Harrod-neutral" technical progress: (b) on the line of Lotka–Volterra models with adaptation, introduces the concept of "memory", which plays a relevant role in the dynamics of economic systems. As a consequence, an additional equation appears, the validity of the model is substantially extended and a rich phenomenology is obtained, in particular, transition to chaotic behavior via period-doubling bifurcations.

Nonlinear Behavior of a Novel Switching Jerk System

2020 ◽  
Vol 30 (14) ◽  
pp. 2050202
Author(s):  
Hany A. Hosham
Keyword(s):  
Chaotic Behavior ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Continuous System ◽  
Qualitative Behavior ◽  
Bifurcation Problem ◽  
Sliding Motion ◽  
Discontinuity Surfaces ◽  
Jerk System ◽  
Period Doubling Bifurcations

This paper proposes a novel chaotic jerk system, which is defined on four domains, separated by codimension-2 discontinuity surfaces. The dynamics of the proposed system are conveniently described and analyzed through a generalization of the Poincaré map which is constructed via an explicit solution of each subsystem. This provides an approach to formulate a robust bifurcation problem as a nonlinear inhomogeneous eigenvalue problem. Also, we establish some criteria for the existence of a period-doubling bifurcation and discuss some of the interesting categories of complex behavior such as multiple period-doubling bifurcations and chaotic behavior when the trajectory undergoes a segment of sliding motion. Our results emphasize that the sharp switches in the behavior are mainly responsible for generating new and unique qualitative behavior of a simple linear system as compared to the nonlinear continuous system.

Robust Estimation-Based Control of Chaotic Behavior in an Oscillator with Inertial and Elastic Symmetric Nonlinearities

2003 ◽  
Vol 9 (6) ◽  
pp. 665-684 ◽  
Author(s):  
A. A. Al-Qaisia ◽  
A. M. Harb ◽  
A. A. Zaher ◽  
M. A. Zohdy
Keyword(s):  
Robust Estimation ◽  
Nonlinear Oscillator ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Physical Parameters ◽  
Bifurcation And Chaos ◽  
Transition To Chaos ◽  
Simulation Results ◽  
Forced Nonlinear Oscillator ◽  
Period Doubling Bifurcations

In this paper, we study the dynamics of a forced nonlinear oscillator with inertial and elastic symmetric nonlinearities using modern nonlinear, bifurcation and chaos theories. The results for the response have shown that, for a certain combination of physical parameters, this oscillator exhibits a chaotic behavior or a transition to chaos through a sequence of period doubling bifurcations. The main objective of this paper is to control the chaotic behavior for this type of oscillator. A nonlinear estimation-based controller is proposed and the transient performance is investigated. The design of the parameter update mechanism is analyzed while discussing ways to extend its performance to further account for other types of uncertainties. We examine robustness problems as well as ways to tune the controller parameters. Simulation results are presented for the uncontrolled and controlled cases, verifying the effectiveness and the capability of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.

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