Hisa Aki Tanaka

Self-synchronization of coupled oscillators with hysteretic responses

Abstract We analyze a large system of nonlinear phase oscillators with sinusoidal nonlinearity, uniformly distributed natural frequencies and global all-to-all coupling, which is an extension of Kuramotos model to second-order systems. For small coupling, the system evolves to an incoherent state with the phases of all the oscillators distributed uniformly. As the coupling is increased, the system exhibits a discontinuous transition to the coherently synchronized state at a pinning threshold.of the coupling strength, or to a partially synchronized oscillation coherent state at a certain threshold below the pinning threshold. If the coupling is decreased from a strong coupling with all the oscillators synchronized coherently, this coherence can persist until the depinning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the oscillators. We obtain analytically both the pinning and depinning threshold and also expalin the discontinuous transition at the thresholds for the underdamped case in the large system size limit. Numerical exploration shows the oscillatory partially coherent state bifurcates at the depinning threshold and also suggests that this state persists independent of the system size. The system studied here provides a simple model for collective behaviour in damped driven high-dimensional Hamiltonian systems which can explain the synchronous firing of certain fireflies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems.

Geometric structure of mutually coupled phase-locked loops

Dynamical properties such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLLs) are problems of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLLs incorporating lag filters and triangular phase detectors. The fourth-order ordinary differential equation (ODE) governing the mutually coupled PLLs is reduced to the equivalent third-order ODE due to the symmetry, where the system is analyzed in the context of nonlinear dynamical system theory. An understanding as to how and when lock-in can be obtained or out-of-lock behavior persists, is provided by the geometric structure of the invariant manifolds generated in the vector field from the third-order ODE. In addition, a connection to the recently developed theory on chaos and bifurcations from degenerated homoclinic points is also found to exist. The two-parameter diagrams of the one-homoclinic orbit are obtained by graphical solution of a set of nonlinear (finite dimensional) equations. Their graphical results useful in determining whether the system undergoes lock-in or continues out-of-lock behavior, are verified by numerical simulations.

Chaos from orbit-flip homoclinic orbits generated in real systems

A new class of chaotic systems is discovered that are generated in a practical, nonlinear, mutually coupled phase-locked loop (PLL) circuit. Presented theoretical results make it possible to understand experimental results of mutually coupled PLLs on the onset of chaos using the geometry of the invariant manifolds, while the resultant simple geometry and complex dynamics is expected to have applications in other areas, e.g., power systems or interacting bar magnets. Motivated by the numerical study of this system, the topological horseshoe is proven to be generated in the codimension 3 unfolding of a degenerated orbit-flip homoclinic point for this system. Qualitatively different type of bifurcation phenomena are also observed to appear depending on the phase detector (PD) characteristics.

Synchronization limit and chaos onset in mutually coupled phase-locked loops

Dynamical property such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLLs) is a problem of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLLs incorporating lag filters and triangular phase detectors. The system is analysed in the context of nonlinear dynamical system theory. The symmetry of the mutually coupled PLLs system reduces the original 4th order ordinary differential equation (ODE) that governs the phase dynamics of the voltage-controlled oscillators (VCO) outputs to the 3rd order ODE, for which the geometric structure of the invariant manifolds provides an understanding as to how and when lock-in can be obtained or out-of-lock behavior persists. In addition, two-parameter diagrams of the one-homoclinic orbit are obtained by solving a set of nonlinear (finite dimensional) equations. This graphical results are confirmed to be useful in determining whether the system undergoes lock-in or continues out-of-lock behavior by numerical simulations. Presented theoretical results make it possible to understand experimental results of mutually coupled PLLs on the onset of chaos using the geometry of the invariant manifolds, where the resultant dynamical chaotic phenomena is postulated to represent an unfolding of the orbit-flip homoclinic point. Motivated by the numerical study of the system generated invariant manifolds, the topological horseshoe is proven to be generated even in the unfolding of a degenerated orbit-flip homoclinic point for the piecewise linear system under consideration.