scholarly journals Stochastic Averaging of Dynamical Systems with Multiple Time Scales Forced with $\alpha$-Stable Noise

2015 ◽  
Vol 13 (4) ◽  
pp. 1194-1223 ◽  
Author(s):  
W. F. Thompson ◽  
R. A. Kuske ◽  
A. H. Monahan
Keyword(s):  
Dynamical Systems ◽  
Time Scales ◽  
Stochastic Averaging ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Stable Noise

Non-Stationary Oscillations at Slow Transitions Across Instabilities

2007 ◽  
Author(s):  
Rudolf R. Pusˇenjak ◽  
Maks M. Oblak ◽  
Jurij Avsec
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Time Scales ◽  
Primary Resonance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Excitation Parameter ◽  
Linear Dynamical Systems ◽  
Weakly Nonlinear ◽  
Non Linear

The paper presents the study of non-stationary oscillations, which is based on extension of Lindstedt-Poincare (EL-P) method with multiple time scales for non-linear dynamical systems with cubic non-linearities. The generalization of the method is presented to discover the passage of weakly nonlinear systems through the resonance as a control or excitation parameter varies slowly across points of instabilities corresponding to the appearance of bifurcations. The method is applied to obtain non-stationary resonance curves of transition across points of instabilities during the passage through primary resonance of harmonically excited oscillators of Duffing type.

Extended Lindstedt-Poincare Method With Multiple Time Scales for Nonstationary Oscillations

2006 ◽  
Author(s):  
Rudi R. Pusenjak ◽  
Jurij Avsec ◽  
Maks M. Oblak
Keyword(s):  
Dynamical Systems ◽  
Time Scales ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Excitation Frequency ◽  
Van Der Pol Oscillator ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Response Curves ◽  
Nonstationary Oscillations

The paper presents the extended Lindstedt-Poincare (ELP) method, which applies multiple time scales to treat nonstationary oscillations arising in dynamical systems with cubic non-linearities. The passage through the resonance is conducted to study deviations from the stationary response. The method is applied to the dynamical systems such as Duffing oscillator and van der Pol oscillator, whereat effects of varying the excitation frequency and varying the excitation amplitudes, respectively are studied. It is shown that application of multiple scales benefits to find more accurate expressions of stationary responses in comparison to the conventional Lindstedt-Poincare method and consequently contributes to the versatile and effective calculation of the nonstationary frequency response curves.

scholarly journals A NOISY SYSTEM WITH A FLATTENED HAMILTONIAN AND MULTIPLE TIME SCALES

2003 ◽  
Vol 03 (01) ◽  
pp. 1-54 ◽  
Author(s):  
NATELLA V. O'BRYANT
Keyword(s):  
Time Scales ◽  
Martingale Problem ◽  
Stochastic Averaging ◽  
Multiple Time Scales ◽  
Time Averaging ◽  
Multiple Time ◽  
Skew Brownian Motion ◽  
Stratified Space ◽  
Reduced Space ◽  
And Diffusion

We consider a two-dimensional weakly dissipative dynamical system with time-periodic drift and diffusion coefficients. The average of the drift is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. Using the martingale problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficients, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of time scales, are similar to the conditions on the domain of skew Brownian motion and are related to the description of spider martingales.

scholarly journals COMPUTATIONAL MODELING OF DYNAMICAL SYSTEMS

2005 ◽  
Vol 15 (03) ◽  
pp. 471-481 ◽  
Author(s):  
JOHAN JANSSON ◽  
CLAES JOHNSON ◽  
ANDERS LOGG
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Computational Modeling ◽  
Time Scales ◽  
Time Scale ◽  
Short Note ◽  
Multiple Time Scales ◽  
Time Interval ◽  
Multiple Time ◽  
Fast Time

In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10–9 computed over the time interval [0,100], and a lattice consisting of large and small point masses.

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