Abstract—Relaxation oscillators can usually be represented as a feedback system with hysteresis. The relay relaxation oscillator consists of relay hysteresis and a linear system in feedback. The objective of this work is to study the existence of periodic orbits and the dynamics of coupled relay oscillators. In particular, we give a complete analysis for the case of unimodal periodic orbits, and illustrate the presence of degenerate and asymmetric orbits. We also discuss how complex orbits can arise from bifurcation of unimodal orbits. Finally, we focus on oscillators with an integrator as the linear component, and study the entrainment under external forcing, and phase locking when such oscillators are coupled in a ring. I.
|
268
|
Theory of Matrices
– Gantmacher
- 1959
|
|
72
|
The geometry of biological time
– Winfree
- 1980
|
|
42
|
Rapid synchronization through fast threshold modulation
– Somers, Kopell
- 1993
|
|
38
|
Systems with Hysteresis
– Krasnosel’skii, Pokrovskii
- 1989
|
|
28
|
Perspectives of nonlinear dynamics
– Jackson
- 1990
|
|
23
|
1993] Coupled nonlinear oscillators and the symmetries of animal gaits
– Collins, Stewart
|
|
22
|
Coupled oscillators and biological synchronization
– Strogatz, Stewart
- 1993
|
|
19
|
Mathematical models for hysteresis
– Macki, Nistri, et al.
- 1993
|
|
15
|
Theory of Oscillators
– Andronov, Vitt, et al.
- 1966
|
|
14
|
Asymptotic methods for relaxation oscillations and applications
– Grasman
- 1987
|
|
10
|
Anti-phase solutions in relaxation oscillators coupled through excitatory interactions
– Kopell, Somers
- 1995
|
|
6
|
Relay Control Systems
– Tsypkin
- 1984
|
|
6
|
Nonlinear resonance in systems with hysteresis
– Bliman, Krasnosel’skii, et al.
- 1996
|
|
6
|
Global stability of relay feedback systems
– Goncalves, Megretski, et al.
- 2001
|
|
5
|
Asymptotically stable oscillations in systems with hysteresis nonlinearities
– Brokate, Pokrovskii
- 1998
|
|
3
|
Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology
– Grasman, Jansen
- 1979
|
|
3
|
Nonlinear chemical dynamics: oscillations, patterns and chaos
– Epstein, Showalter
- 1996
|
|
3
|
Qualitative analysis of the periodically forced relaxation oscillator
– Levi
- 1981
|
|
3
|
Oscillations in systems with relay feedback,” Adapt
– Åström
- 1995
|
|
3
|
Autotuning of PID Controllers: Relay Feedback Approach
– Yu
- 1999
|
|
3
|
On periodic solutions of a thermostat equation
– Gripenberg
- 1987
|
|
3
|
Fast switches in relay feedback systems
– Johansson, Rantzer, et al.
- 1999
|
|
3
|
Stroboscopic phase portrait of a forced nonlinear oscillator,” Progr
– Kai, Tomita
|
|
2
|
Oscillations, Waves and Chaos in Chemical Kinetics
– Scott
- 1994
|
|
2
|
Periodic solutions of linear systems coupled with relay
– Bliman, Krasnosel’skiĭ
- 1997
|
|
2
|
Resonance in oscillatory and excitable systems
– DeYoung, Othmer
- 1990
|
|
2
|
Self-synchronization of coupled oscillators with hysteretic responses
– Tanaka, Lichtenberg, et al.
- 1997
|
|
2
|
Sliding orbits and their bifurcations in relay feedback systems
– Bernardo, Johansson, et al.
- 1999
|
|
2
|
A second variational theory for optimal periodic processes
– Speyer, Evans
- 1984
|
|
2
|
Optimal periodic control and periodic systems analysis: An overview
– Bittanti, Guardabassi
- 1986
|
|
1
|
Relaxation oscillations in mathematical models of ecology
– Kolesov, Kolesov
- 1995
|
|
1
|
Dynamics of the Van der Pol equation
– Guckenheimer
- 1980
|
|
1
|
Aström et al., “Recent advances in relay feedback methods-a survey
– J
- 1995
|
|
1
|
Global stability of oscillations induced by a relay feedback
– Megretski
- 1997
|
|
1
|
Modeling of oscillatory dynamics of a simple enzyme-diffusion system with hysteresis the case of lumped permeabilities
– Zou, Siegel
- 1999
|
|
1
|
Endogenous and Exogenous Regulation and Control of Physiological Systems
– Northrop
- 2000
|
|
1
|
Periodic solutions of n-th order equations with relay control,” Automat
– Kolesov
- 1969
|
|
1
|
Dror et al., “A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators
– O
- 1999
|
|
1
|
Ermentrout, “n:m phase-locking of weakly coupled oscillators
– B
- 1981
|
|
1
|
of A. Subbarao Varigonda was born in Eluru, India, in 1975. He received the Bachelors degree in chemical engineering from the Indian Institute of Technology, Kanpur, India in 1996 and the Masters degree in electrical engineering in 1999 from the Universit
– Chaos, Scientific
- 1998
|
