Torus breakdown

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Valentin S. Afraimovich (2007), Scholarpedia, 2(10):1933. doi:10.4249/scholarpedia.1933 revision #91881 [link to/cite this article]
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An onset from regular to chaotic regimes through the break-down of a two-dimensional invariant torus has been observed in many natural and man-made systems including the Rayleigh-Benard convection (Jensen et al. 1985), the Taylor-Couette flow (Marques et al. 2001), and electrical circuits (Anishchenko et al. 1993, Baptista and Caldas, 1998).

The detailed numerical experiment on a specific family of maps was performed by Aronson et al. (1982), see also (Curry and Yorke, 1978), . A mathematical description of such a phenomenon in the resonance case was first suggested by Afraimovich and Shilnikov (1983). Previously, Fenichel (1971) showed that the torus always loses its smoothness before the break-down.


Loss of Smoothness

The torus loses its smoothness and eventually disintegrates in three different principal ways. The three scenarios can be described as follows. Assume that for \(0<\varepsilon<\varepsilon_c \) the dynamical system \(f_{\varepsilon}\) has a smooth attracting two-dimensional torus \(T_{\varepsilon} \ ,\) containing an even number of periodic orbits, half of which are stable and half are unstable. The orbits that are unstable on the torus are of saddle-type in the whole phase space of the system. Thus, the torus \(T_{\varepsilon} \) is the closure of the unstable manifolds to the saddle orbits. For the sake of simplicity, assume that there are only two periodic (stable and saddle) orbits on the torus. Let \(\varepsilon_c \) be a bifurcation value. The three scenarios are related to the following bifurcations:

  1. The saddle-node bifurcation of a periodic orbit in a critical case, i.e. the unstable set of the saddle-node periodic orbit forms a non-smooth manifold, homeomorphic to the torus.
  2. The period-doubling bifurcation of the periodic orbit that is stable for \(\varepsilon<\varepsilon_c \ .\) Note that apart from the perioid-doubling bifurcation, the stable periodic orbit can also undergo the Neimark-Sacker bifurcation, codimension-2 bifurcations, etc.
  3. The appearance of a homoclinic orbit is due to the tangency of the stable and unstable manifolds of the saddle periodic orbit.


We illustrate these bifurcations by studying a map of an annulus, that models the Poincare map in a neighborhood of a destroyed torus. The specific map is immaterial; for example one may consider the Zaslavsky map in the form \[(x,\theta)\rightarrow (e^{-r}(x+a\sin \theta ),(\theta+r+x+a\sin \theta) \mbox{ mod } 2\pi) \] of the annulus \(0<x \le x_0\ ,\) where \( r >1, a\ge 0, x_0>a/(1-e^{-r})\) (Afraimovich and Hsu 2003).

Figure 1: The bifurcation diagram of the two-parameter family of Zaslavsky maps. Different mechanisms of loss of smoothness and the breakdown of the closed invariant curve are shown in its different parts.

It is easy to verify that for any integer \(k\) on the plane of parameters \((r,a)\ ,\) the curve\[ B^+: a=\pm(2\pi k-r)(1-e^{-r})\] is a bifurcation curve corresponding to a saddle-node bifurcation of the fixed point. The curve\[ B^-: a^2=(2\pi k-r)^2(1-e^{-r})^2+4(1+e^{-r})^2\] corresponds to the period-doubling bifurcation. There are also bifurcation curves \(B_{1,2}\) corresponding to the tangency of the unstable and stable manifolds of a fixed saddle point from different sides. The bifurcation diagram is shown in Figure 1. For the values of parameters in the pentagonal region bounded by these curves, the map has an invariant curve homeomorphic (but not necessarily diffeomorphic) to the circle.

The paths through \(B^+\) and \(B_{12}\) from the inside are accompanied by the appearance of strange attractors containing infinitely many periodic orbits. The bifurcation on \( B^- \) is the first one in the infinite sequence of period-doubling bifurcations that eventually lead to the appearance of the Feigenbaum attractor and chaotic regimes.


V. Afraimovich and S.-B. Hsu, Lectures on Chaotic Dynamical Systems, AMS/IP Studies in Advanced Mathematics, International Press, 2003.

V.S. Afraimovich and L.P. Shilnikov, On invariant two-dimensional tori, their breakdown and stochasticity in: Methods of the Qualitative Theory of Differential Equations, Gor'kov. Gos. University, (1983), 3-26. Translated in: Amer. Math. Soc. Transl., (2), vol. 149 (1991), 201-212.

V.S. Anishchenko, M.A. Safonova and L.O. Chua, Confirmation of the Afraimovich-Shilnikov torus-breakdown theorem via torus circuits, IEEE Transactions on Circuits and Systems, vol. 40 (1993), 792-800.

D.G. Aronson, M.A. Chory, R.P. Mcgehee and G.R. Hall, Bifurcation from an invariant circle for two-parameter families of maps of the plane, Commun. Math. Phys., vol.83 (1982), 303-354.

M. S. Baptista and I. L. Caldas Dynamics of the two-frequency torus breakdown in the driven double scroll circuit, "Phys. Rev. E" vol. 58 (1998), 4413--4420.

J.H. Curry, J.A. Yorke, A transition from Hopf bifurcation to chaos: computer experiments with maps in R 2 , in: J.C. Martin, N.G. Markley, W. Perrizo (Eds.), The Structure of Attractors in Dynamical Systems, Springer Notes in Mathematics, Vol. 668, Springer, Berlin, (1978), pp. 48--66.

N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., vol.21 (1971), 193-226.

M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia and J. Stevens, Global universality at the onset of chaos: results of forced Raylegh-Benard experiment, Phys. Rev. Lett., vol.55 (1985), 439-441.

F. Marques, J.M. Lopes and J. Chen, A periodically forced flow displaying symmetry breaking via a three-tori gluing bifurcations and two-tori resonances, Physica D, vol.156 (2001), 81-97.

Internal references

See Also

Attractor, Bifurcations, Bubbling Transition, Crises, Dynamical Systems, Periodic Orbit, Stability, Unstable Periodic Orbits

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