Transition to Chaotic Behavior via a Reproducible Sequence of Period-Doubling Bifurcations

1981 ◽  
Vol 47 (4) ◽  
pp. 243-246 ◽  
Author(s):  
Marzio Giglio ◽  
Sergio Musazzi ◽  
Umberto Perini
Keyword(s):  
Chaotic Behavior ◽  
Period Doubling ◽  
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.

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.

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.

scholarly journals Controlling Chaos through Period-Doubling Bifurcations in Attitude Dynamics for Power Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Power Systems ◽  
Control Method ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Lyapunov Stability Theory ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Controlling Chaos ◽  
Period Doubling Bifurcations

This paper addresses the complex nonlinear dynamics involved in controlling chaos in power systems using bifurcation diagrams, time responses, phase portraits, Poincaré maps, and frequency spectra. Our results revealed that nonlinearities in power systems produce period-doubling bifurcations, which can lead to chaotic motion. Analysis based on the Lyapunov exponent and Lyapunov dimension was used to identify the onset of chaotic behavior. We also developed a continuous feedback control method based on synchronization characteristics for suppressing of chaotic oscillations. The results of our simulation support the feasibility of using the proposed method. The robustness of parametric perturbations on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

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