Spectral signatures of bifurcations

Guha, Debajyoti (2026) Spectral signatures of bifurcations. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Debajyoti Guha (19RS013)) 19RS013.pdf - Submitted Version Restricted to Repository staff only Download (50MB)

Abstract

Traditional bifurcation analysis focuses on the evolution of steady states in phase space as system parameters vary. Another way to characterize a dynamical system is through its frequency content. This thesis explores the use of ‘spectral bifurcation diagrams’ as a complementary tool to understand dynamical transitions in nonlinear systems with an eye on discrete maps. By computing the Fourier spectrum of the discrete time series as the parameter varies, one obtains a representation that highlights the appearance, disappearance, diffusion, and distortion of dominant frequencies. It has been shown earlier that periodic orbits appear as sharp discrete peaks, quasiperiodicity brings about frequency pairs, and chaotic dynamics is characterized by a wideband spectrum. We extend the scope of this technique to obtain newer insights into various classical local bifurcations — including saddle-node, period-doubling, pitchfork, transcritical and Neimark–Sacker bifurcations — as well as global phenomena such as crises, intermittency, and border-collision bifurcations in piecewise-smooth maps. For each scenario, the associated spectral bifurcation diagrams are constructed and compared with standard bifurcation diagrams. Spectral bifurcation diagrams offer an alternative and, in some cases, a more intuitive means of observing nonlinear transitions. The results suggest that spectral analysis can serve as a powerful visual and diagnostic tool in the study of nonlinear dynamics.Traditional bifurcation analysis focuses on the evolution of steady states in phase space as system parameters vary. Another way to characterize a dynamical system is through its frequency content. This thesis explores the use of ‘spectral bifurcation diagrams’ as a complementary tool to understand dynamical transitions in nonlinear systems with an eye on discrete maps. By computing the Fourier spectrum of the discrete time series as the parameter varies, one obtains a representation that highlights the appearance, disappearance, diffusion, and distortion of dominant frequencies. It has been shown earlier that periodic orbits appear as sharp discrete peaks, quasiperiodicity brings about frequency pairs, and chaotic dynamics is characterized by a wideband spectrum. We extend the scope of this technique to obtain newer insights into various classical local bifurcations — including saddle-node, period-doubling, pitchfork, transcritical and Neimark–Sacker bifurcations — as well as global phenomena such as crises, intermittency, and border-collision bifurcations in piecewise-smooth maps. For each scenario, the associated spectral bifurcation diagrams are constructed and compared with standard bifurcation diagrams. Spectral bifurcation diagrams offer an alternative and, in some cases, a more intuitive means of observing nonlinear transitions. The results suggest that spectral analysis can serve as a powerful visual and diagnostic tool in the study of nonlinear dynamics.

Actions (login required)

View Item