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.
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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 [Citation17], the Anishchenko-Astakhov generator [Citation3], in the optical laser systems [Citation4,Citation5,Citation41,Citation53], in chemical systems [Citation6,Citation11], in a certain class of models describing the behaviour of neurons [Citation18], in biophysical experiments [Citation40], in electromechanical systems [Citation15,Citation35], in electrochemical processes [Citation13,Citation39], in nonlinear convection in magnetic fields [Citation42], in mechanical systems [Citation33], 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 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. [Citation28].
L.O. Chua, M. Komuro, and T. Matsumoto, The double scroll family, Circuits Syst. IEEE Trans. 33(11) (1986), pp. 1072–1118. V.S. Anishchenko, Complex oscillations in simple systems, (1990), (in Russian). F.T. Arecchi, R. Meucci, and W. Gadomski, Laser dynamics with competing instabilities, Phys. Rev. Lett. 58(21) (1987), pp. 2205. F.T. Arecchi, A. Lapucci, R. Meucci, J.A. Roversi, and P.H. Coullet, Experimental characterization of Shil'nikov chaos by statistics of return times, EPL (Europhysics Letters) 6(8) (1988), pp. 677. A.N. Pisarchik, R. Meucci, and F.T. Arecchi, Theoretical and experimental study of discrete behavior of Shilnikov chaos in a CO2 laser, Eur. Phys. J. D-At. Mol. Opt. Phys. 13(3) (2001), pp. 385–391. C.S. Zhou, J. Kurths, E. Allaria, S. Boccaletti, R. Meucci, and F.T. Arecchi, Constructive effects of noise in homoclinic chaotic systems, Phys. Rev. E. 67(6) (2003), pp. 066220. F. Argoul, A. Arneodo, and P. Richetti, Experimental evidence for homoclinic chaos in the Belousov-Zhabotinskii reaction, Phys. Lett. A 120(6) (1987), pp. 269–275. A. Arneodo, F. Argoula, J. Elezgarayab, and P. Richettia, Homoclinic chaos in chemical systems, Phys. D: Nonlinear Phenom. 62(1) (1993), pp. 134–169. U. Feudel, A. Neiman, X. Pei, W. Wojtenek, H. Braun, M. Huber, and F. Moss, Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons, Chaos: Interdiscip. J. Nonlinear Sci. 10(1) (2000), pp. 231–239. D. Parthimos, D.H. Edwards, and T.M. Griffith, Shilnikov homoclinic chaos is intimately related to type-III intermittency in isolated rabbit arteries: role of nitric oxide, Phys. Rev. E. 67(5) (2003), pp. 051922. J.C. Chedjou, P. Woafo, and S. Domngang, Shilnikov chaos and dynamics of a self-sustained electromechanical transducer, J. Vib. Acoust. 123(2) (2001), pp. 170–174. M.T.M. Koper, P. Gaspard, and J.H. Sluyters, Mixed-mode oscillations and incomplete homoclinic scenarios to a saddle focus in the indium/thiocyanate electrochemical oscillator, J. Chem. Phys. 97(11) (1992), pp. 8250–8260. M.R. Bassett and J.L. Hudson, Shilnikov chaos during copper electrodissolution, J. Physical Chemistry 92(24) (1988), pp. 6963–6966. T. Noh, Shilnikov chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode, Electrochim. Acta. 54(13) (2009), pp. 3657–3661. A.M. Rucklidge, Chaos in a low-order model of magnetoconvection, Phys. D: Nonlinear Phenom. 62(1) (1993), pp. 323–337. M.E. Henderson, M. Levi, and F. Odeh, The geometry and computation of the dynamics of coupled pendula, Int. J. Bifur. Chaos 1(01) (1991), pp. 27–50. A.S. Gonchenko, S.V. Gonchenko, and D.V. Turaev, Doubling of invariant curves and chaos in three-dimensional diffeomorphisms, Chaos 31 (2021), pp. 113130. 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.