Synchronization Phenomena of Impulsively-Coupled van der Pol Oscillators Yoko Uwate and Yoshifumi Nishio and Ruedi Stoop UNI / ETH Zurich, Switzerland Tokushima Univeristy, Japan
Introduction (1) Coupled Oscillators Synchronization, Bifurcation Various higher-dimensional nonlinear phenomena Example: Large scale networks Information processing mechanism Society networks Behavior of human relationship Good models!! Modeling by real circuits!! Big Nishio dream!! Laboratory
Previous Work van der Pol oscillator Coupled by TVR Simulated Result Coexistence!! (1.44 < ω < 1.58) TVR Very interesting!!
Point!! Coupled oscillators Generally System in two oscillators coupled by a resistor Only one synchronization state: stable!! (even if plural states exist ) Coexistence of attractors in coupled oscillatory systems Applying :novel parallel information processing ex. embedded information during learning process
Coupling oscillatory systems!! Positive resistor Anti-phase Negative resistor In-phase? Time-Varying resistor In-phase Anti-phase coexistence Coupling element Important role for synchronization phenomena of coupled oscillatory systems.
Purpose In this study, we investigate synchronization phenomena of impulsively-coupled van der Pol oscillators. r Amp - r
Circuit Model Two coupled van der Pol oscillators Circuit Model characteristics irk v k
Time Varying Resistor Characteristics of TVR Time Varying Resistor: TVR The function representing the variation of the TVR is the square wave. p The angular frequency and the Nishio duty Laboratory ratio. ω t
Standard TVR Duty ratio: p=0.5 r W Amp - r W T ω t t W = 50% T
Impulsively TVR W = 25% T W = 5% T W W Amp Amp ωt ωt
Circuit Equations Normalized circuit equations Circuit Equations By changing the variables and the parameters,
Synchronization Coexistence Synchronization Phenomena In-phase Anti-phase y1 y2 x 2 y 1 y 2 x 2 x1 x2 x 1 x 1 x 2 x 1 x 1 x 1 x 2 x 2 τ τ Parameter ε = 2.0, γ = ± 0.1, p = 0.5, ω = 1.5
Phase Difference Standard TVR (W=50% T) Phase Difference [degree] Anti-phase Coexistence In-phase ω ε = 2.0, γ = ± 0.1
Phase Difference Impulsively TVR (W=25% T) Phase Difference [degree] Anti-phase Coexistence In-phase ω ε = 2.0, γ = ± 0.1
Phase Difference Impulsively TVR (W=8% T) Phase Difference [degree] Anti-phase Coexistence In-phase Non-synchronization ω ε = 2.0, γ = ± 0.1
Phase Difference Impulsively TVR (W=5% T) Phase Difference [degree] Coexistence Non-synchronization ω ε = 2.0, γ = ± 0.1
Comparison W=50% T W=25% T ω ω W=8% T W=5% T ω ω
Bifurcation Diagram By changing of W (width of TVR) ε = 2.0, γ = ± 0.1 ω In-phase Coexistence W=50% Anti-phase W [%] Non- synchronization W=5%
Bifurcation Diagram Dependence on Amplitude of TVR ε = 2. 0 Amplitude: r = 0.04 Amplitude: r = 0.16 In-phase In-phase Anti-phase Anti-phase W [%] W [%] Relationship: Amplitude Coexistence area
Discussion (1-1) Reducing width of TVR decreasing amplitude of TVR W=20%, r = 0.1 W=10%, r = 0.2 r r Amp - r - r
Discussion (1-2) Phase difference W=20%, r = 0.1 W=10%, r = 0.2 Coexistence (1.54 < ω < 1.59) Coexistence (1.53 Nishio < ω Laboratory < 1.59)
Bifurcation Diagram Dependence on Nonlinearity of Oscillator γ = ±0. 1 ε = 1.0 ε= 3.0 In-phase Anti-phase In-phase Anti-phase W [%] W [%] Relationship: Nonlinearity - Frequency
Discussion (2) Nonlinearity of Oscillator ε= 0.2 Frequency Fast ε= 4.0 Slow
Conclusions In this study: we investigate synchronization phenomena of impulsively-coupled van der Pol oscillators. By computer simulations: Amplitude of TVR Area of coexistence area Reducing width of TVR Decreasing amplitude of TVR Nonlinearity of oscillator Frequency of coexistence area
Future Works 1. Mechanism breakdown synchronization 2. Bifurcation diagram????? ω ω W [%]