Abstract
We provide an introduction to some basic concepts of nonlinear dynamics, focusing on those required in understanding noise effects in nonlinear systems dealt with later in the book. We start with defining a nonlinear dynamical system and discussing the notion of fixed points, namely, values of dynamical variables such that if the dynamics is initiated at these values, the variables do not evolve in time. We next turn to types of fixed points in one and two-dimensional systems. A particular type of fixed point in two dimensions is the so-called limit cycle, which is next discussed. It may happen as one tunes the dynamical parameters that the nature of the fixed points changes, new fixed points are generated, or existing fixed points are destroyed. This phenomenon of bifurcation and its various prototypical types in one dimension are then discussed. We proceed to detail characterization of one limit cycle as well as two and more than two interacting limit cycles. As an example of the latter, we discuss the celebrated Kuramoto model of interacting limit-cycle oscillators of distributed frequencies and discuss spontaneous synchronization, whereby a macroscopic number of the oscillators exhibits collective oscillations at a common frequency in the stationary state.
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Notes
- 1.
In textbooks on classical mechanics, the number of degrees of freedom of a system is defined to be the number of generalized coordinates. In this case, one needs to specify all the generalized coordinates and the generalized velocities (or momenta) to completely specify the state of the system. In this setting of dynamical systems, however, all the variables needed to specify completely the state of the system are collectively called the degrees of freedom.
- 2.
The Van der Pol oscillator model was originally proposed by Balthasar van der Pol in 1920. He was then working as an electrical engineer for the multinational conglomerate Philips in Netherlands. The model describes a nonlinear electrical circuit used in early radios.
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Das, D., Gupta, S. (2023). Nonlinear Dynamics. In: Facets of Noise. Fundamental Theories of Physics, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-031-45312-0_5
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