Stochastic and Collective Properties of Nonlinear Oscillators

Citation

Kogan, Oleg Boris (2009) Stochastic and Collective Properties of Nonlinear Oscillators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/93R8-TJ70. https://resolver.caltech.edu/CaltechETD:etd-06012009-145134

Abstract

Two systems of nonlinear oscillators are considered: (a) a single periodically driven nonlinear oscillator interacting with a heat bath, which may operate in the regime of bistability or monostability, and (b) a one-dimensional chain of self-sustaining phase oscillators with nearest-neighbor interaction.

For a single oscillator we analyze the scaling crossovers in the thermal activation barrier between the two stable states. The rate of metastable decay in nonequilibrium systems is expected to display scaling behavior: the logarithm of the decay rate should scale as a power of the distance to a bifurcation point where the metastable state disappears. We establish the range where different scaling behavior is displayed and show how the crossover between different types of scaling occurs. Using the instanton method, we map numerically the entire parameter range of bistability and find the regions where the scaling exponents are 1 or 3/2, depending on the damping. The exponent 3/2 is found to extend much further from the bifurcation then where it would be expected to hold as a result of an overdamped soft mode. Additionally, we uncover a new scaling behavior with exponent of ≈1.3 that extends beyond the close vicinity of the bifurcation point.

We also study the pattern of fluctuational trajectories in the monostable regime. For nonequilibrium systems, fluctuational and relaxational trajectories are not simply related by time-reversibility, as is the case in thermal equilibrium. One of the consequences of this is the onset of singularities in the pattern of fluctuational trajectories, where most probable paths to neighboring states are far away from each other. This also creates nonsmoothness in the probability distribution of the system in its phase space. We discover that the pattern of optimal paths in equilibrium systems is fragile with respect to the driving strength F, and investigate how the singularities occur as the system is driven away from equilibrium. As the strength of the driving F approaches zero, the cusp of the spiral caustic system recedes to larger radius R and the angle of the cusp also decreases. The dependence of R on F displays two scaling laws with crossovers, where the scaling exponents depend on the damping.

For the one-dimensional chain of nearest-neighbor coupled phase oscillators, we develop a renormalization group method to investigate synchronization clusters. We apply it numerically to Lorentzian distributions of intrinsic frequencies and couplings and investigate the statistics of the resultant cluster sizes and frequencies. We find that the distributions of sizes of frequency clusters are exponential, with a characteristic length. The dependence of this length upon parameters of these Lorentzian distributions develops an asymptotic power law with an exponent of 0.48 ± 0.02. The findings obtained with the renormalization group are compared with numerical simulations of the equations of motion of the chain, with an excellent agreement in all the aforementioned quantities.

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