Dear Pranav,

Crisis induced route to chaos and quasi-periodic route to chaos are phenomenologically different. The first one is a global route chaos. Crisis basically refers to the collision of a chaotic attractor with an unstable invariant set, typically an unstable periodic orbit (Grebogi et al. 1983; Ott 2002). In this regard, transient chaos and reverse crisis is also very interesting.

On the other hand, the quasi-periodic route is a local route to chaos which gives birth to chaos through multiple Hopf bifurcations. In this case, first, a quasi-periodic attractor is born from limit-cycle oscillations. Thereafter, the quasi-periodic torus is broken to a strange attractor through subsequent Hopf bifurcations.

To identify the route to chaos, one can employ the time series analysis tools to establish the different dynamical states present in the system.

You can refer to my recent publication for details:

Chandan Bose, Sunetra Sarkar (2018) : Investigating chaotic wake dynamics past a flapping airfoil and the role of vortex interactions behind the chaotic transition, Physics of Fluids, Volume 30, Issue 4, 047101. DOI: 10.1063/1.5019442.

Best regards,

Chandan

Dear Pranav,

Depending on different characteristics of a dynamical system, there exist several routes to chaos that can be identified by the response of the system. To start with, you may look at the time histories of the states. Next, Fourier analysis may be utilized. Then, you should plot the phase maps between different states. In the next stage, you can plot poincare plots for the states of a system to see which route is taken by the system to go chaotic. You may see some of my publications, most importantly, the most recent one in Journal of Fluids and Structures (2018) where we have used this methodology to find the route to chaos using the unstaeady aerodynamic force signals to investigate the system dynamics.

Regards

Muhammad