DYNAMICS OF RELAXATION OSCILLATIONS

2001 ◽  
Vol 11 (05) ◽  
pp. 1471-1482 ◽  
Author(s):  
PAUL E. PHILLIPSON ◽  
PETER SCHUSTER
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Time Span ◽  
Singular Perturbation Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Relaxation Oscillations ◽  
Van Der Pol

Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker–Haag piecewise-linear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker–Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to three-dimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos.

BIFURCATIONS OF RELAXATION OSCILLATIONS NEAR FOLDED SADDLES

2005 ◽  
Vol 15 (11) ◽  
pp. 3411-3421 ◽  
Author(s):  
JOHN GUCKENHEIMER ◽  
KATHLEEN HOFFMAN ◽  
WARREN WECKESSER
Keyword(s):  
Perturbation Theory ◽  
Dynamical Systems ◽  
Singular Perturbation ◽  
Time Scale ◽  
Singular Limit ◽  
Singular Perturbation Theory ◽  
Multiple Time ◽  
Multiple Time Scale ◽  
Relaxation Oscillations ◽  
Geometric Methods

Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow–fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one-parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, in which relaxation oscillations become homoclinic to a folded saddle.

On the Existence of Limit Cycles and Relaxation Oscillations in a 3D van der Pol-like Memristor Oscillator

2017 ◽  
Vol 27 (07) ◽  
pp. 1750102
Author(s):  
Marcelo Messias ◽  
Anderson L. Maciel
Keyword(s):  
Phase Space ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Relaxation Oscillations ◽  
Van Der Pol ◽  
Piecewise Linear System ◽  
Parameter Values

We study a van der Pol-like memristor oscillator, obtained by substituting a Chua’s diode with an active controlled memristor in a van der Pol oscillator with Chua’s diode. The mathematical model for the studied circuit is given by a three-dimensional piecewise linear system of ordinary differential equations, depending on five parameters. We show that this system has a line of equilibria given by the [Formula: see text]-axis and the phase space [Formula: see text] is foliated by invariant planes transverse to this line, which implies that the dynamics is essentially two-dimensional. We also show that in each of these invariant planes may occur limit cycles and relaxation oscillations (that is, nonsinusoidal repetitive (periodic) solutions), depending on the parameter values. Hence, the oscillator studied here, constructed with a memristor, is also a relaxation oscillator, as the original van der Pol oscillator, although with a main difference: in the case of the memristor oscillator, an infinity of oscillations are produced, one in each invariant plane, depending on the initial condition considered. We also give conditions for the nonexistence of oscillations, depending on the position of the invariant planes in the phase space.

scholarly journals PERIODIC SOLUTIONS AND SLOW MANIFOLDS

2007 ◽  
Vol 17 (08) ◽  
pp. 2533-2540 ◽  
Author(s):  
FERDINAND VERHULST
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
Relaxation Oscillation ◽  
Slow Manifold ◽  
Logistic Equation ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Van Der Pol Equation ◽  
Van Der Pol

After reviewing a number of results from geometric singular perturbation theory, we give an example of a theorem for periodic solutions in a slow manifold. This is illustrated by examples involving the van der Pol-equation and a modified logistic equation. Regarding nonhyperbolic transitions we discuss a four-dimensional relaxation oscillation and also canard-like solutions emerging from the modified logistic equation with sign-alternating growth rates.

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

On voltage stability from the viewpoint of singular perturbation theory

1994 ◽  
Vol 16 (6) ◽  
pp. 409-417 ◽  
Author(s):  
N. Yorino ◽  
H. Sasaki ◽  
Y. Masuda ◽  
Y. Tamura ◽  
M. Kitagawa ◽  
...  
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Voltage Stability ◽  
Singular Perturbation Theory
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