Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-17T02:27:33.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Shilnikov chaos, Filippov sliding and boundary equilibrium bifurcations

Published online by Cambridge University Press:  13 June 2018

P. A. GLENDINNING
P. A. GLENDINNING*
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK email: p.a.glendinning@manchester.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the 1960s, L.P. Shilnikov showed that certain homoclinic orbits for smooth families of differential equations imply the existence of chaos, and there are complicated sequences of bifurcations near the parameter value at which the homoclinic orbit exists. We describe how this analysis is modified if the differential equations are piecewise smooth and the homoclinic orbit has a sliding segment. Moreover, we show that the Shilnikov mechanism appears naturally in the unfolding of boundary equilibrium bifurcations in $\mathbb{R}^3$.

Keywords

Bifurcation theoryhomoclinic orbitspiecewise smooth dynamics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 5: Theory and applications of nonsmooth dynamical systems , October 2018 , pp. 757 - 777
Copyright
Copyright © Cambridge University Press 2018 

References

[1] Arnéodo, A., Coullet, P. & Tresser, C. (1981) Possible new strange attractors with spiral structure. Commun. Math. Phys. 79 (4), 573–579.Google Scholar
[2] Arnéodo, A., Coullet, P. & Tresser, C. (1982) Oscillators with chaotic behavior: An illustration of a theorem by Shil'nikov. J. Stat. Phys. 27 (1), 171–182.Google Scholar
[3] Belitskii, G. R. (1973) Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Funct. Anal. Appl. 7 (4), 268–277.Google Scholar
[4] Chen, K. T. (1963) Equivalence and decomposition of vector fields about an elementary critical point. Amer. J. Math. 85 (4), 693–722.Google Scholar
[5] Colombo, A. & Jeffrey, M. R. (2013) The two-fold singularity: Leading order dynamics in n-dimensions. Physica D 263, 1–10.Google Scholar
[6] di Bernardo, M., Budd, C., Champneys, A. R. & Kowalczyk, P. (2008) Piecewise-Smooth Dynamical Systems, Springer, London.Google Scholar
[7] di Bernardo, M., Pagano, D. J. & Ponce, E. (2008) Nonhyperbolic boundary equilibrium bifurcations in planar Filippov systems: A case study approach. Int. J. Bifurcation Chaos 18 (5), 1377, DOI: 10.1142/S0218127408021051.Google Scholar
[8] de Carvalho, T. & Tonon, D.J. (2014) Normal forms for codimension one planar piecewise smooth vector fields. Int. J. Bifurcation Chaos 24 (7), 1450090, DOI: 10.1142/S0218127414500904.Google Scholar
[9] Dercole, F., Della Rossa, F., Colombo, A. & Kuznetsov, Y. A. (2011) Two degenerate boundary equilibrium bifurcations in planar Filippov systems. SIADS 10 (4), 1525–1553.Google Scholar
[10] Dieci, L. & Lopez, L. (2009) Sliding motion in Filippov differential equations: Theoretical results and a computational approach. SIAM J. Numer. Anal. 47 (3), 2023–2051.Google Scholar
[11] Feroe, J. A. (1993) Homoclinic orbits in a parametrized saddle-focus system. Physica D 62 (1–4), 254–262.Google Scholar
[12] Filippov, A. F. (1988) Differential Equations With Discontinuous Right Hand Sides, Kluwer, Netherlands.Google Scholar
[13] Gaspard, P., Kapral, R. & Nicolis, G. (1984) Bifurcation phenomena near homoclinic systems: A two-parameter analysis. J. Stat. Phys. 35 (5–6), 697–727.Google Scholar
[14] Glendinning, P. (1997) Differential equations with bifocal homoclinic orbits. Int. J. Bifurcation Chaos 7 (1), 27–37.Google Scholar
[15] Glendinning, P. (2016) Classification of boundary equilibrium bifurcations of planar Filippov systems. Chaos 26, 013108.Google Scholar
[16] Glendinning, P. & Sparrow, C. (1984) Local and global behavior near homoclinic orbits. J. Stat. Phys. 35 (5–6), 645–696.Google Scholar
[17] Guardia, M., Seara, T. M. & Teixeira, M. A. (2011) Generic bifurcations of low codimension of planar Filippov Systems. J. Diff. Equ. 250 (4), 1967–2023.Google Scholar
[18] Hogan, S. J., Homer, M. E., Jeffrey, M. R. & Szalai, R. (2016) Piecewise smooth dynamical systems theory: The case of the missing boundary equilibrium bifurcations. J. Nonl. Sci. 26 (5), 1161–1173.Google Scholar
[19] Hös, C. & Champneys, A. R. (2012) Grazing bifurcations and chatter in a pressure relief valve model. Physica D 241 (22), 2068–2076.Google Scholar
[20] Jeffrey, M. R. & Colombo, A. (2009) The two-fold singularity of discontinuous vector fields. SIADS 8 (2), 624–640.Google Scholar
[21] Kuznetsov, Y. A., Rinaldi, S. & Gragnani, A. (2003) One-parameter bifurcations in planar Filippov systems. Int. J. Bifurcation Chaos 13 (8), 2157–2188.Google Scholar
[22] Llibre, J., Ponce, E. & Teruel, A. E. (2007) Horseshoes near homoclinic orbits for piecewise linear differential systems in $\mathbb{R}^3$. Int. J. Bifurcation Chaos 17 (4), 1171–1184.Google Scholar
[23] Matsumoto, T., Chua, L. O. & Komuro, M. (1985) The double scroll. IEEE Trans. Circ. Syst. 32 (8), 797–818.Google Scholar
[24] Novaes, D. D. & Teixeira, M. A. (2015) Shilnikov problem in Filippov dynamical systems, arXiv.Google Scholar
[25] Nusse, H. E. & Yorke, J. A. (1992) Border-collision bifurcation including ‘period two to period three’ for piecewise smooth systems. Physica D 57 (1–2), 39–57.Google Scholar
[26] Piiroinen, P. & Kuznetsov, Y. A. (2008) An event-driven method to simulate Filippov systems with accurate computing of sliding motions. ACM Trans. Math. Softw. 34 (3), Art. 13.Google Scholar
[27] Orsay (2012) Scilab: Free and Open Source Software. URL: http://www.scilab.org (last accessed 4 June 2018).Google Scholar
[28] Sell, G. R. (1985) Smooth linearization near a fixed point. Amer. J. Math. 107 (5), 1035–1091.Google Scholar
[29] Shilnikov, L. P. (1965) A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6, 163–166.Google Scholar
[30] Shilnikov, L. P. (1970) A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type. Math. USSR Sb. 10 (1), 91–102.Google Scholar
[31] Sternberg, S. (1959) The structure of local homoeomorphisms III. Amer. J. Math. 81 (3), 578–604.Google Scholar
[32] Tresser, C. (1983) Un théorème de Sil'nikov en C 1, 1. C.R.Acad. Sci. Série I 296, 545–548.Google Scholar
[33] Tresser, C. (1984) About some theorems by L.P. Sil'nikov. Ann. de l'IHP 40 (4), 441–461.Google Scholar