On Shilnikov attractors of three-dimensional flows and maps

Abstract

We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of three-dimensional flows and maps. The presented results are illustrated with various specific examples.

Keywords:

• Chaotic dynamics
• spiral chaos
• strange attractor
• homoclinic orbit
• saddle-focus

2020 Mathematics Subject Classifications:

• 37G35
• 37G20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Their list is quite large; we will indicate only some of the most known three-dimensional models. Thus, spiral chaos was found in radio-electronic devices, such as the Chua circuit [17], the Anishchenko-Astakhov generator [3], in the optical laser systems [4,5,41,53], in chemical systems [6,11], in a certain class of models describing the behaviour of neurons [18], in biophysical experiments [40], in electromechanical systems [15,35], in electrochemical processes [13,39], in nonlinear convection in magnetic fields [42], in mechanical systems [33], etc.

2 Of course, when inside the whirlpool, there are no other attractors – for example, local bifurcations may lag behind global bifurcations and then, it can happen that both the limit cycle $L_{\mu }$ is stable and a homoclinic loop exists.

3 Such type of bifurcation of doubling of invariant curve is called a component-doubling bifurcation, see more details in Gonchenko et al. [28].

Additional information

Funding

This paper was carried out in the framework of the Russian Ministry of Science and Education [grant number 0729-2020-0036]. A. Gonchenko was supported by the RSciF [grant number 20-71-00079] (Section 4 and 5). Yu. Bakhanova and A. Kazakov were supported by the RSciF [grant number 19-71-10048] (Section 3). S. Gonchenko, A. Kazakov and E. Samylina thank the Theoretical Physics and Mathematics Advancement Foundation ‘BASIS’ [grant number 20-7-1-36-5], for support of scientific investigations.