Investigation of Non-Linear Dynamical Systems Exhibiting Mean Field Interaction and Multi-Stability

Neeraj, . (2025) Investigation of Non-Linear Dynamical Systems Exhibiting Mean Field Interaction and Multi-Stability. PhD thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

Beginning with intricate financial systems and progressing all the way up to the nonlinear condensed phase of matter, this thesis is devoted to the investigation of a wide variety of dynamical systems. Within the framework of Bosonic interaction, we go into depth regarding the investigation of quantum soliton solutions that resemble kinks. It is possible to transition between the equilibrium phase and the planar droplet with these solutions without any interruptions. They are inherently dependent on a constant background in order to continue existing in the quantum realm. It is interesting to note that the constant background is exactly equivalent to one-third of the average amplitude of the uniform condensate. In contrast to kink/anti-kink, dark, and kink-like solitons, these kink-like solitons are distinctively different. Given that they display an asymptotic connection to the normal state at one end, grey solitons are particularly interesting for this reason. Each of the parameters for the self-trapped flat-top method has a value that is identical to the chemical potential. An investigation into the possibility of a dynamical phase transition occurring in a photonic condensate system is presented in the following published work. Within the context of this discussion, we take into consideration a Fabry-Perot cavity system that is cylindrically centric. Utilising a thermalising dye medium to fill the space results in a decrease in the optical activity of the space. Behaviour of the system can be characterised using the nonlinear Schrodinger equation (NLSE) with a source term. Evidently, the dominant polarisation component of a condensate has the ability to coherently drive the other subdominant polarisation component, as demonstrated by the findings presented in this chapter. In contrast to the dominant component, which is typically in a uniform superfluid phase, the subdominant component of the condensate exhibits two distinct phases that are separated by a dynamical phase transition. A superfluid droplet phase and a periodically modulated superfluid phase are the phases that are being discussed here about. Next we introduce classical data driven systems, take recourse to instantaneous phase space approach to analyse EEG data, similar to the nonlinear model used to study moving LCR circuits. This approach explains many brain behaviours. The observation of oscillatory behaviour on a large scale, including limit cycles, bi-stability, and wave packet collapse/revival from unstable periodic orbits, offers new insights into brain dynamics and confirms previously hypothesised hypotheses. Some characteristics show greater strength in the epileptogenic zone and during seizures, suggesting a potential use for instantaneous medical prediction and intervention. Oscillator dynamics analysis in phase-space reveals increased potential energy contribution in epileptogenic zone signals and patients before seizures. The collapse and revival of coherence wave packet dynamics are observed in this study. Fourier domain analysis of power spectra in phase space indicates this phenomenon occurs around 18Hz. The financial systems investigated comprise of stock market dynamics of BSE and NYSE representing a developing and a developed economies, respectively. The period of study includes the duration of financial crisis and subsequent recovery, which makes the study interesting. A range of tools starting from Random Matrix Theory to Wavelet transforms to Grange causality are used for this study. The counter intuitive finding, small market like India reflecting the dynamical behaviour of fully developed economy like US at a higher frequency with a time lag.Since fluctuation being the universal component of financial system, it manifest in bigger economies with a time lag compared to smaller economy. It is well understood that low frequency fluctuations invoke correlations among different sectors of an economy, on other hand the high frequency excites different sector of it, hence this provides a natural setting for simulating a bigger economy using smaller economy at higher frequency .

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