Signatures of Routes to Chaos

Chaos is ubiquitous in nonlinear dynamical systems. It is a hidden state of deterministic disorder in certain regions of parameters and initial conditions of a nonlinear system. The path taken by the system to reach chaos is known as its route to chaos. The state of the system can drastically change from a stable fixed point or a periodic solution to chaos due to the effect of local or global bifurcation. The available tools, such as bifurcation diagrams, Lyapunov spectra, Poincaré maps, phase plots, and time series plots, explore the routes for such qualitative changes. These are tiresome procedures. For more efficient verifications, this work offers a guide or catalog for several routes to chaos (RTC) through which a continuous-time dynamical system enters or exits from chaos. Additionally, this review paper highlights various signatures associated with common RTC to make identification easier. The main emphasis is on enabling a reader to assess these RTC by merely observing the explained signatures in the bifurcation diagrams and the Poincaré maps.