Period-doubling route to chaos in a two-degree-of-freedom flexibly-mounted rigid plate placed in water

2015 ◽  
Vol 57 ◽  
pp. 375-390 ◽  
Author(s):  
Pariya Pourazarm ◽  
Matthew Lackner ◽  
Yahya Modarres-Sadeghi
Keyword(s):  
Period Doubling ◽  
Degree Of Freedom ◽  
Rigid Plate ◽  
Route To Chaos ◽  
Two Degree Of Freedom

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.

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.

PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

1995 ◽  
Vol 05 (03) ◽  
pp. 725-740 ◽  
Author(s):  
C. CHIN ◽  
A.H. NAYFEH
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Degree Of Freedom ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities ◽  
One To One ◽  
Two Degree Of Freedom

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

Neimark-Sacker Bifurcations Near Degenerate Grazing Point in a Two Degree-of-Freedom Impact Oscillator

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
Shan Yin ◽  
Jinchen Ji ◽  
Shuning Deng ◽  
Guilin Wen
Keyword(s):  
Periodic Motion ◽  
Small Neighborhood ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Grazing Impact ◽  
Mapping Technique ◽  
Impact Oscillator ◽  
Parameter Continuation ◽  
The Impact ◽  
Two Degree Of Freedom

Saddle-node or period-doubling bifurcations of the near-grazing impact periodic motions have been extensively studied in the impact oscillators, but the near-grazing Neimark-Sacker bifurcations have not been discussed yet. For the first time, this paper uncovers the novel dynamic behavior of Neimark-Sacker bifurcations, which can appear in a small neighborhood of the degenerate grazing point in a two degree-of-freedom impact oscillator. The higher order discontinuity mapping technique is used to determine the degenerate grazing point. Then, shooting method is applied to obtain the one-parameter continuation of the elementary impact periodic motion near degenerate grazing point and the peculiar phenomena of Neimark-Sacker bifurcations are revealed consequently. A two-parameter continuation is presented to illustrate the relationship between the observed Neimark-Sacker bifurcations and degenerate grazing point. New features that differ from the reported situations in literature can be found. Finally, the observed Neimark-Sacker bifurcation is verified by checking the existence and stability conditions in line with the generic theory of Neimark-Sacker bifurcation. The unstable bifurcating quasi-periodic motion is numerically demonstrated on the Poincaré section.

Periodic-Doubling and Hopf Bifurcations of a Two-Degree-of-Freedom System with Elastic Constraints and Clearances

2014 ◽  
Vol 1006-1007 ◽  
pp. 285-289
Author(s):  
Xi Feng Zhu ◽  
Quan Fu Gao
Keyword(s):  
Hopf Bifurcation ◽  
Period Doubling ◽  
Simulation Method ◽  
Degree Of Freedom ◽  
Local Bifurcations ◽  
Stiffness And Damping Coefficient ◽  
The Stability ◽  
Elastic Constraints ◽  
Stiffness And Damping ◽  
Two Degree Of Freedom

Based on the study of a dual component system with elastic constraints, the stability and local bifurcations of the soft-impacts system, such as piecewise property and singularity, was analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and period-doubling bifurcation, were investigated exactly. In the large constraint stiffness case, the period-doubling and Hopf bifurcation exist in the two-degree-of-freedom system with elastic constraints and clearances. The clearances of the system, stiffness and damping coefficient of the elastic constraints is the main reasons for influencing the chaotic motion. The steady 1-1-1 period orbits or 2-1-1 period orbits will exist within a wideband frequency range and the value of velocity will be higher when appropriate system parameters are chosen.

Amplitude Dynamics of an Autoparametric Two Degree-of-Freedom System

1991 ◽  
Author(s):  
Anil K. Bajaj ◽  
Joseph M. Johnson ◽  
Seo Il Chang
Keyword(s):  
Hopf Bifurcation ◽  
Nonlinear Oscillations ◽  
Resonance Condition ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Coupled Mode ◽  
Averaged Equations ◽  
Steady Solutions ◽  
Two Degree Of Freedom

Abstract Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric system are studied for resonant excitations. The method of averaging is used to obtain first order approximations to the response of the system. In the subharmonic case of internal and external resonance, where the external excitation is in the neighborhood of the higher natural frequency, a complete bifurcation analysis of the averaged equations is undertaken. The “locked pendulum” mode of response bifurcates to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, though it requires mistuning from the exact internal resonance condition. The Hopf bifurcation sets are constructed and dynamic steady solutions of the amplitude or averaged equations are investigated using software packages AUTO and KAOS. It is shown that both super- and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.

Bifurcations and Chaotic Dynamics in an Electrostatically Actuated Impact Microactuator: A Numerical Exploration

2003 ◽  
Author(s):  
Xiaopeng Zhao ◽  
Chevva Konda Reddy ◽  
Ali H. Nayfeh
Keyword(s):  
Chaotic Dynamics ◽  
Period Doubling ◽  
Careful Study ◽  
Electrostatically Actuated ◽  
System A ◽  
Numerical Exploration ◽  
Route To Chaos ◽  
Consistent Performance ◽  
The Impact ◽  
Two Degree Of Freedom

We study the dynamics of an electrostatically driven impact actuator. As the name suggests, the impact actuator uses impacts between its moving elements to produce nano-displacements. While on one hand, impact actuators provide a way to produce small displacements with moderate actuation voltages, on the other hand impacts make the underlying dynamics nonsmooth. Impacts are a source of nonlinearity and a careful study of the dynamics is essential in order to ensure a consistent performance of the device. We model the impact microactuator reported by Mita and associates using a two-degree-of-freedom system. A simple impact law based on the coefficient of restitution is used. Our results show that the dynamics can be very complex as the system parameters are varied. Namely, as the amplitude and frequency of excitation are varied, the system exhibits period doubling and grazing bifurcations onto the route to chaos.

Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems

2018 ◽  
Vol 28 (04) ◽  
pp. 1850049 ◽  
Author(s):  
H. X. Fu ◽  
Y. H. Qian
Keyword(s):  
Periodic Solutions ◽  
Homotopy Analysis Method ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Algebraic Equations ◽  
Analysis Method ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Homotopy Analysis ◽  
Two Degree Of Freedom

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

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