Dynamics, Synchronization and Pattern Formation in Coupled Thomas Oscillators

Abstract

This thesis work is mainly about dynamics, synchronization and pattern formation in coupled Thomas oscillators in the chaotic regime. Two mutually coupled identical oscillators as well as oscillators in a network are considered. The synchronizability of Thomas oscillators on various networks are established via master stability function formalism under linear coupling scheme for identical setting of oscillators. For pattern formation, local, nonlocal, global coupling schemes on a ring are considered. The special nature of Thomas oscillators and its connection to active Brownian particles are established via numerical simulations. The study of dynamics and synchronization of two mutually coupled oscillators are based on the calculations of Lyapunov exponents, Bifurcation diagram, phase portrait, Transverse Lyapunov exponents, Pearson coefficients, Transverse distance and similarity index. Two different values of system parameter are used in the chaotic regime under linear and nonlinear coupling schemes. In both cases the coupled system undergoes a period of transient chaos. Three different types of initial conditions are used to study the transients and synchronization. For low value of coupling strength, the system shows weak forms of synchronization. For linear coupling, the nature of synchronization agrees with the predictions of the general observation found in prototypical Rössler system and Lorenz system. The nature of synchronization is much more complex in the case of nonlinear coupling. The system bifurcates to lag or anti-lag synchronization even after achieving complete synchronization. It also shows space-lag(swarming) and multistability with nonlinear coupling. The emergence of lag or anti-lag is confirmed with similarity index calculation. Our calculation of largest transverse Lyapunov exponent for nonlinear coupling exactly matches with the predictions of Pearson coefficient and Transverse distance which would have been lost on any linearization of the transverse perturbation equation for the coupled system. The variables in our system are components of velocity of a particle moving in a force field and indeed, there are velocity-velocity correlations like in coupled active Brownian particles. Therefore, we claim that the stochastic dynamics of active Brownian particles can be modeled by chaotic dynamics of Thomas system. We found the important results that lag / anti-lag and space lag(swarming) synchronization within the regime of complete synchronization. The synchronization properties of Thomas oscillators in a network is studied for identical oscillators with linear coupling scheme via master stability function formulation. For the set of system parameters in the chaotic regime, they show type-I and type-II behavior of MSF. The synchronizability for various network architectures are also studied. For the study of pattern formation, we considered hundred Thomas oscillators on a
ring with nonlocal coupling with nonlinear coupling function. We could achieve chimera states for a certain range of intermediate coupling constants. The Chimera states are quantified by means of strength of incoherence and discontinuity measure calculation. The system shows cluster, chimera, multi-chimera as the coupling is increased. We could also achieve chimera states for nearly local coupling with nonlinear coupling functions. The global coupling shows complete synchronization of oscillators. The obtained result of pattern formation in the network is useful to understand the dynamics of active Brownian particles. For active Brownian particles, the probability distribution of velocities resembles the present observation of the chimera states. The discontinuous jump observed in the case of zero system parameter corresponds to a first order phase transition and matches with the statistical model of self-propelled particles.