Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-Of-Freedom Vibro-Impact System

2000 ◽  
Vol 68 (4) ◽  
pp. 670-674 ◽  
Author(s):  
G. L. Wen and ◽  
J. H. Xie
Keyword(s):  
Finite Number ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Period Doubling Bifurcation ◽  
Periodic Response ◽  
Impact System ◽  
Route To Chaos ◽  
Two Degree Of Freedom ◽  
Impact Motion ◽  
Period Doubling Bifurcations

A nontypical route to chaos of a two-degree-of-freedom vibro-impact system is investigated. That is, the period-doubling bifurcations, and then the system turns out to the stable quasi-periodic response while the period 4-4 impact motion fails to be stable. Finally, the system converts into chaos through phrase locking of the corresponding four Hopf circles or through a finite number of times of torus-doubling.

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 DYNAMICS OF A SIMPLE PARAMETRICALLY DRIVEN DISSIPATIVE CIRCUIT

2011 ◽  
Vol 21 (07) ◽  
pp. 1927-1933 ◽  
Author(s):  
P. PHILOMINATHAN ◽  
M. SANTHIAH ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
S. RAJASEKAR
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classic Period ◽  
Control Signal ◽  
Period Doubling Bifurcation ◽  
Chaotic Circuit ◽  
Route To Chaos ◽  
Parameter Values

We introduce a simple parametrically driven dissipative second-order chaotic circuit. In this circuit, one of the circuit parameters is varied by an external periodic control signal. Thus by tuning the parameter values of this circuit, classic period-doubling bifurcation route to chaos is found to occur. The experimentally observed phenomena is further validated through corresponding numerical simulation of the circuit equations. The periodic and chaotic dynamics of this model is further characterized by computing Lyapunov exponents.

EXPERIMENTAL INVESTIGATION OF A TWO-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM

2012 ◽  
Vol 22 (05) ◽  
pp. 1250110 ◽  
Author(s):  
GUILIN WEN ◽  
HUIDONG XU ◽  
LU XIAO ◽  
XIAOPING XIE ◽  
ZHONG CHEN ◽  
...  
Keyword(s):  
Dynamical Behavior ◽  
Degree Of Freedom ◽  
Modeling And Analysis ◽  
Strongly Nonlinear ◽  
Impact System ◽  
Dynamical Behaviors ◽  
Nonsmooth Systems ◽  
Experimental Apparatus ◽  
Thick Steel ◽  
Two Degree Of Freedom

Vibro-impact systems with intermittent contacts are strongly nonlinear. The discontinuity of impact can give rise to rich nonlinear dynamic behaviors and bring forth challenges in the modeling and analysis of this type of nonsmooth systems. The dynamical behavior of a two-degree-of-freedom vibro-impact system is investigated experimentally in this paper. The experimental apparatus is composed of two spring-linked oscillators moving on a lead rail. One of the two oscillators connected to an excitation system intermittently impacts with a spherical obstacle fixed on the thick steel wall. With different gap sizes between the impacting oscillator and the obstacle, the dynamical behaviors are investigated by changing the excitation frequencies. The experimental results show periodic, grazing and chaotic dynamical behaviors of the vibro-impact system.

Nonlinear Oscillations of a Particle in the Plane Under Longitudinal End Excitation

2003 ◽  
Author(s):  
Rajesh K. Jha ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Oscillations ◽  
Low Frequency ◽  
Period Doubling ◽  
Lumped Parameter Model ◽  
Lumped Parameter ◽  
Parametric Resonances ◽  
Jump Phenomena ◽  
Route To Chaos ◽  
Chaotic Nature ◽  
Two Degree Of Freedom

We study the forced vibrations of a two degree of freedom lumped parameter model of a belt span under longitudinal excitation. The belt inertia is modelled as a particle and the belt elasticity is modelled by two identical linear springs. Numerical integration is used to calculate free responses and perform frequency and amplitude sweeps. Frequency sweep results indicate parametric resonances, jump phenomena, sub- and super-harmonic responses, quasiperiodicity and chaos. Amplitude sweep at a low frequency shows bifurcations of limit cycles and the period doubling route to chaos. Poincare sections are computed to show the chaotic nature of the responses.

Routes to Chaos in Squeeze-Film Dampers

2004 ◽  
Author(s):  
Jawaid I. Inayat-Hussain ◽  
Njuki W. Mureithi
Keyword(s):  
Period Doubling ◽  
Squeeze Film ◽  
Saddle Node Bifurcation ◽  
Chaotic Vibrations ◽  
Saddle Node ◽  
Highly Nonlinear ◽  
Squeeze Film Dampers ◽  
Route To Chaos ◽  
Type 3 ◽  
Period Doubling Bifurcations

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document