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.

Two Groups of Chaotic Vibrations in a Single-Degree-of-Freedom Maglev System

1997 ◽  
Author(s):  
Zhixiang Xu ◽  
Hideyuki Tamura
Keyword(s):  
Magnetic Levitation ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Moving Base ◽  
Period Doubling Bifurcations

Abstract In this paper, a single-degree-of-freedom magnetic levitation dynamic system, whose spring is composed of a magnetic repulsive force, is numerically analyzed. The numerical results indicate that a body levitated by magnetic force shows many kinds of vibrations upon adjusting the system parameters (viz., damping, excitation amplitude and excitation frequency) when the system is excited by the harmonically moving base. For a suitable combination of parameters, an aperiodic vibration occurs after a sequence of period-doubling bifurcations. Typical aperiodic vibrations that occurred after period-doubling bifurcations from several initial states are identified as chaotic vibration and classified into two groups by examining their power spectra, Poincare maps, fractal dimension analyses, etc.

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.

scholarly journals Sequence of Routes to Chaos in a Lorenz-Type System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fangyan Yang ◽  
Yongming Cao ◽  
Lijuan Chen ◽  
Qingdu Li
Keyword(s):  
Limit Cycles ◽  
Chaotic Attractor ◽  
Type System ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Main Route ◽  
Route To Chaos ◽  
Period Doubling Bifurcations

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

scholarly journals NOISE-INDUCED CHAOS AND BACKWARD STOCHASTIC BIFURCATIONS IN THE LORENZ MODEL

2013 ◽  
Vol 23 (05) ◽  
pp. 1350092 ◽  
Author(s):  
IRINA BASHKIRTSEVA ◽  
LEV RYASHKO ◽  
PAVEL STIKHIN
Keyword(s):  
Limit Cycles ◽  
Noise Intensity ◽  
Period Doubling ◽  
Threshold Value ◽  
Comparative Investigation ◽  
Function Technique ◽  
Stochastic Bifurcation ◽  
Lorenz Model ◽  
Stochastic Bifurcations ◽  
Period Doubling Bifurcations

We study the phenomena of stochastic D- and P-bifurcations of randomly forced limit cycles for the Lorenz model. As noise intensity increases, regular multiple limit cycles of this model in a period-doubling bifurcations zone are deformed to be stochastic attractors that look chaotic (D-bifurcation) and their multiplicity is reduced (P-bifurcation). In this paper for the comparative investigation of these bifurcations, the analysis of Lyapunov exponents and stochastic sensitivity function technique are used. A probabilistic mechanism of backward stochastic bifurcations for cycles of high multiplicity is analyzed in detail. We show that for a limit cycle with multiplicity two and higher, a threshold value of the noise intensity which marks the onset of chaos agrees with the first backward stochastic bifurcation.

SHRIMP STRUCTURE AND ASSOCIATED DYNAMICS IN PARAMETRICALLY EXCITED OSCILLATORS

2006 ◽  
Vol 16 (12) ◽  
pp. 3567-3579 ◽  
Author(s):  
Y. ZOU ◽  
M. THIEL ◽  
M. C. ROMANO ◽  
J. KURTHS ◽  
Q. BI
Keyword(s):  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Two Dimensional ◽  
Recurrence Plots ◽  
Dimensional Parameter ◽  
Parametrically Excited ◽  
Excited System ◽  
Stable Solutions ◽  
Bifurcation Structures

We investigate the bifurcation structures in a two-dimensional parameter space (PS) of a parametrically excited system with two degrees of freedom both analytically and numerically. By means of the Rényi entropy of second order K2, which is estimated from recurrence plots, we uncover that regions of chaotic behavior are intermingled with many complex periodic windows, such as shrimp structures in the PS. A detailed numerical analysis shows that the stable solutions lose stability either via period doubling, or via intermittency when the parameters leave these shrimps in different directions, indicating different bifurcation properties of the boundaries. The shrimps of different sizes offer promising ways to control the dynamics of such a complex system.

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.

THE RESPONSE OF A NONLINEAR SYSTEM WITH A NONSEMISIMPLE ONE-TO-ONE RESONANCE TO A COMBINATION PARAMETRIC RESONANCE

1995 ◽  
Vol 05 (04) ◽  
pp. 971-982 ◽  
Author(s):  
C. CHIN ◽  
A. H. NAYFEH ◽  
D. T. MOOK
Keyword(s):  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Power Spectra ◽  
Period Doubling ◽  
Supersonic Stream ◽  
Equilibrium Solutions ◽  
Shooting Technique ◽  
First Order ◽  
Period Doubling Bifurcations ◽  
Combination Parametric Resonance

The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.

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