Aspects of Temporal Perturbation in Dynamical Systems

Bhadra, Nivedita (2019) Aspects of Temporal Perturbation in Dynamical Systems. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Temporally perturbed dynamical systems, even in isolation, can exhibit several rich and intriguing features. In this thesis, we have explored a few aspects of such temporally perturbed systems which include dynamic stabilization, noise-induced transition, effect of time-varying network in long-range systems, etc. We have started our study with a simple gravity pendulum with a vibrating suspension point, also known as Kapitza pendulum. Above certain threshold values of system parameters, the inverted position of this system becomes a stable configuration – a phenomenon known as dynamic stabilization. This phenomenon can be understood in terms of the effective potential the system is subject to due to the vertical drive at the point of suspension. We investigate the effect of several parameters on the stability region in the system. We observe that the interaction between two such pendula has noticeable effects on the stability margin. In addition, we have explored how the network structure plays significant effects on the stability in the many-body version of this system. We have explored the effect of time-varying network in a system with long-range interaction, known as Hamiltonian mean-field model. Due to the presence of the temporal modulation of the coupling, the critical point of phase transition undergoes a shift which is a function of frequency and amplitude of the modulating function. In a Josephson junction with both fundamental and second harmonic in it current phase relation, we have introduced a temporal modulation. The effective potential the system experiences due to the modulation shows that ϕ = π can be stabilised, which is unstable in the absence of this modulation. We have performed a dynamic system analysis in a cosmological model. Our study matches well with the current cosmological observaions.

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