Control of Oscillation Patterns in a Symmetric Coupled Biological Oscillator System

Takamatsu, Atsuko; Tanaka, Reiko; Yamamoto, Takatoki; Fujii, Teruo

doi:10.1063/1.1612218

Cited by 2 publications

(2 citation statements)

References 9 publications

(13 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This paper deals with two important solutions of equivariant ODE systems with D 4 symmetry: heteroclinic cycles and spatiotemporal/spatial symmetric periodic motions. Such dynamical behaviors have been observed in practice, for example, in biological systems [Takamatsu et al, 2003]. Two techniques have been developed to study heteroclinic cycles in D n -equivariant systems.…”

Section: Introductionmentioning

confidence: 99%

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

Murza

2016

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

show abstract

Section: Introductionmentioning

confidence: 99%

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

Murza

2016

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

show abstract

“…For example, in [10], the oscillation patterns were analyzed by the symmetric Hopf bifurcation theory applying group theory. Recently, in [11], the oscillation patterns of the bifurcating periodic oscillations of three coupled Van der Pol oscillators were studied.…”

Section: Introductionmentioning

confidence: 99%

Active Control of Oscillation Patterns in the Presence of Multiarmed Pitchfork Structure of the Critical Manifold of Singularly Perturbed System

Vrabel

Abas

Kopcek

et al. 2013

Mathematical Problems in Engineering

View full text Add to dashboard Cite

We analyze the possibility of control of oscillation patterns for nonlinear dynamical systems without the excitation of oscillatory inputs. We propose a general method for the partition of the space of initial states to the areas allowing active control of the stable steady-state oscillations. Furthermore, we show that the frequency of oscillations can be controlled by an appropriately positioned parameter in the mathematical model. This paper extends the knowledge of the nature of the oscillations with emphasis on its consequences for active control. The results of the analysis are numerically verified and provide the feedback for further design of oscillator circuits.

show abstract

Control of Oscillation Patterns in a Symmetric Coupled Biological Oscillator System

Cited by 2 publications

References 9 publications

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

Active Control of Oscillation Patterns in the Presence of Multiarmed Pitchfork Structure of the Critical Manifold of Singularly Perturbed System

Contact Info

Product

Resources

About