Stochastic Dynamics of Impact Oscillators

2004 ◽  
Vol 72 (6) ◽  
pp. 862-870 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
Jun H. Park
Keyword(s):  
Model Reduction ◽  
Stochastic Dynamics ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Exit Times ◽  
Single Degree Of Freedom ◽  
Impact Oscillators ◽  
Saddle Type ◽  
Dissipative Hamiltonian System ◽  
Stochastic Bifurcations

The purpose of this work is to develop an averaging approach to study the dynamics of a vibro-impact system excited by random perturbations. As a prototype, we consider a noisy single-degree-of-freedom equation with both positive and negative stiffness and achieve a model reduction, i.e., the development of rigorous methods to replace, in some asymptotic regime, a complicated system by a simpler one. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative Hamiltonian system with either center type or saddle type fixed points. We achieve the model-reduction through stochastic averaging. Examination of the reduced Markov process on a graph yields mean exit times, probability density functions, and stochastic bifurcations.

Noisy Impact Oscillators

2004 ◽  
Author(s):  
Jun H. Park ◽  
N. Sri Namachchivaya
Keyword(s):  
Model Reduction ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Exit Times ◽  
Single Degree Of Freedom ◽  
Impact Oscillators ◽  
Single Degree ◽  
Saddle Type ◽  
Dissipative Hamiltonian System ◽  
Stochastic Bifurcations

The purpose of this work is to develop an averaging approach to study the dynamics of a vibro-impact system excited by random perturbations. As a prototype, we consider a noisy single-degree-of-freedom equation with both positive and negative stiffness and achieve a model reduction; i.e., the development of rigorous methods to replace in some asymptotic regime, a complicated system by a simpler one. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative Hamiltonian system with either center type or saddle type fixed points. We achieve the model-reduction through stochastic averaging. Examination of the reduced Markov process on a graph yields mean exit times, probability density functions, and stochastic bifurcations.

Unified Approach for Noisy Nonlinear Mathieu-Type Systems

2001 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
Richard B. Sowers ◽  
H. J. Van Roessel
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Mathieu Equation ◽  
Type Systems ◽  
Stochastic Averaging ◽  
Degree Of Freedom ◽  
Unified Approach ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Low Dimensional

Abstract The purpose of this work is to develop a unified approach to study the dynamics of a single degree of freedom system excited by both periodic and random perturbations. We consider the noisy Duffing-van der Pol-Mathieu equation as a prototypical single degree of freedom system and achieve a reduction by developing rigorous methods to replace, in some limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

Co-dimension Two Bifurcations in the Presence of Noise

1991 ◽  
Vol 58 (1) ◽  
pp. 259-265 ◽  
Author(s):  
N. Sri Namachchivaya
Keyword(s):  
Normal Form ◽  
Diffusion Process ◽  
Stochastic Averaging ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Exit Times ◽  
Double Zero ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stochastic Bifurcations

Some results pertaining to co-dimension two stochastic bifurcations are presented. The normal form associated with non-semi-simple double-zero eigenvalues is considered. The method of stochastic averaging applicable for singularly perturbed stochastic differential equations is used to further reduce the problem to a one dimensional diffusion process. Probability density, most probable values, stability conditions in probability, and mean exit times are obtained for the reduced system.

UNIFIED APPROACH FOR NOISY NONLINEAR MATHIEU-TYPE SYSTEMS

2001 ◽  
Vol 01 (03) ◽  
pp. 405-450 ◽  
Author(s):  
N. SRI NAMACHCHIVAYA ◽  
RICHARD B. SOWERS
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Mathieu Equation ◽  
Type Systems ◽  
Stochastic Averaging ◽  
Unified Approach ◽  
Van Der Pol ◽  
Periodic Hamiltonian System ◽  
Time Periodic ◽  
Stochastic Bifurcations

The purpose of this work is to develop a unified approach to study the dynamics of a single-degree-of-freedom system excited by both periodic and random perturbations. As a prototype, we consider the noisy Duffing–van der Pol–Mathieu equation and achieve a reduction by developing rigorous methods to replace, in a limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a lower-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

An Averaging Approach for Noisy Strongly Nonlinear Periodically Forced Systems

2002 ◽  
Author(s):  
Seunggil Choi ◽  
N. Sri Namachchivaya
Keyword(s):  
Stochastic Averaging ◽  
External Noise ◽  
Degree Of Freedom ◽  
Periodic Force ◽  
Single Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Single Degree ◽  
Periodically Driven ◽  
Statistical Measures ◽  
Stochastic Bifurcations

The purpose of this work is to develop a unified approach to study the dynamics of single degree of freedom systems excited by both periodic and random perturbations. The near resonant motion of such systems is not well understood. We will study this problem in depth with the aim of discovering a common geometric structure in the phase space, and to determine the effects of noisy perturbations on the passage of trajectories through the resonance zone. We consider the noisy, periodically driven Duffing equation as a prototypical single degree of freedom system and achieve a model-reduction through stochastic averaging. Depending on the strength of the noise, reduced Markov process takes its values on a line or on graph with certain gluing conditions at the vertex of the graph. The reduced model will provide a framework for computing standard statistical measures of dynamics and stability, namely, mean exit times, probability density functions, and stochastic bifurcations. This work will also explain a counter-intuitive phenomena of stochastic resonance, in which a weak periodic force in a nonlinear system can be enhanced by the addition of external noise.

scholarly journals Second-order differential equations with random perturbations and small parameters

2017 ◽  
Vol 147 (4) ◽  
pp. 763-779
Author(s):  
M. Kamenskii ◽  
S. Pergamenchtchikov ◽  
M. Quincampoix
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Second Order ◽  
Random Perturbations ◽  
Existence And Uniqueness Theorem ◽  
Second Order Differential Equations ◽  
Averaging Theorem ◽  
Small Parameters

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.

Non-Standard Reduction of Noisy Structural Systems: Shallow Arches

2000 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Exit Time ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Shallow Arches ◽  
Higher Dimensional ◽  
Mean Exit Time ◽  
Stationary Probability Density Function ◽  
Low Dimensional

Abstract The dynamics of a shallow arch subjected to small random external and parametric excitation is invegistated in this work. We develop rigorous methods to replace, in some limiting regime, the original higher dimensional system of equations by a simpler, constructive and rational approximation – a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results such as mean exit time, stationary probability density function.

Stochastic dynamics for the solutions of a modified Holling–Tanner model with random perturbation

2014 ◽  
Vol 25 (11) ◽  
pp. 1450105 ◽  
Author(s):  
Zhenjie Liu
Keyword(s):  
Stochastic System ◽  
Stochastic Dynamics ◽  
Deterministic Model ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Environmental Forcing ◽  
Predator Prey ◽  
Predator Prey Model

In this paper, we consider a stochastic nonautonomous predator–prey model with modified Leslie–Gower and Holling II schemes in the presence of environmental forcing. The deterministic model is the modified Holling–Tanner model which is an extension of the classical Leslie–Gower model. We show that there is a unique positive solution to the stochastic system for any positive initial value. Sufficient conditions for strong persistence in mean and extinction to the stochastic system are established.

Chattering and related behaviour in impact oscillators

1994 ◽  
Vol 347 (1683) ◽  
pp. 365-389 ◽  
Keyword(s):  
Initial Data ◽  
Systematic Study ◽  
Infinite Number ◽  
Periodic Motion ◽  
Finite Time ◽  
Impact Oscillator ◽  
Single Degree Of Freedom ◽  
Impact Oscillators ◽  
Single Degree ◽  
Ball Bouncing

One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.

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