Routes to chaos

This phrase refers to the process by which a simple attracting set for a dynamical system (like a fixed point or periodic orbit) becomes chaotic as an external parameter is varied.

One considers a one-parameter family of differential equations

(a1)

or difference equations (mappings)

(a2)

where is a smooth function of the real parameter and the point , which belongs to some finite-dimensional phase space (like ); is an initial condition. (Other domains are sometimes of interest, for example, mappings of the circle or the torus .) Suppose that for a fixed value of , the initial conditions in some open set in the phase space approach a compact set which exhibits sensitive dependence on initial conditions. (Such a set is also called an attractor. However, there is no universally satisfactory definition of attractor (see e.g. [a1]). In this article it means an invariant set for the mapping or flow that attracts almost-all initial conditions in some open neighbourhood, that cannot be split into disjoint non-trivial closed invariant subsets, and on which the motion is recurrent. See [a2] for a discussion of sensitive dependence in one-dimensional mappings; cf. also Repelling set; Strange attractor.) Roughly speaking, if and are two nearby initial conditions on , their trajectories remain close only for a short time. Then the trajectories separate at an exponential rate until their subsequent motion appears uncorrelated. (The rate of separation eventually saturates since is bounded.) The attractor is chaotic whenever exhibits sensitive dependence on initial conditions (cf. also Chaos). This article discusses four ways in which chaotic attractors for (a1) and (a2) can arise as the parameter is varied. These are the most common known routes to chaos, and there is ample confirmation of their existence in physical experiments. Other routes to chaos have been found (see e.g. [a3]), and more undoubtedly remain undiscovered, particularly in high-dimensional systems. See [a4] for a review article and bibliography.

Period doubling route to chaos.

In this route to chaos, a fixed point loses stability (in a pitchfork bifurcation, cf. Bifurcation) to an attracting period-2 orbit as the parameter passes some critical value. At some later parameter value, the period-2 orbit loses stability (in a pitchfork bifurcation) to a period-4 orbit, etc. The parameter values at which these period doublings occur form an increasing sequence converging to some finite value , at which the original fixed point is replaced by an aperiodic attractor (which may be chaotic for ).

M.J. Feigenbaum

originally studied period doubling in the difference equation

(a3)

also called the quadratic mapping. When , has a non-zero fixed point at which is stable for , since

When , the derivative at is . For slightly larger values of , the derivative is larger than in absolute value, and is unstable: almost-all initial conditions in are attracted to a period-2 orbit, , . A similar derivative evaluation for , , shows that each is stable for . At each loses its stability because the derivative . As before, each is replaced by a pair of attracting points , such that , , . The points correspond to an attracting period-4 orbit for when is slightly larger than . This process continues indefinitely; at each bifurcation, the periodic orbit is replaced by a new attracting periodic orbit of twice the period. The parameter values at which the bifurcations occur form an increasing sequence that is bounded above by a number with the property that

(a4)

The number is the Feigenbaum constant. It is a remarkable fact that is independent of the details of the mapping as long as satisfies certain general hypotheses; see , [a6] for details. Chaos occurs in the quadratic mapping for many values of . In fact, M. Jakobson [a7] proved that the set of such parameter values has positive measure.

Similar results hold in higher dimensions, i.e., for mappings where [a8]. In [a9] it is shown how period-doubling cascades arise in the formation of "horseshoes" as the parameter varies. [a10] contains a collection of papers describing the existence of period doubling in a variety of physical situations.

Intermittency route to chaos.

Y. Pomeau and P. Manneville [a11] describe how an attracting periodic orbit (like a fixed point) for can disappear and be replaced by a chaotic attractor for . For slightly larger than , initial conditions near the periodic orbit remain there for a long time. This regular behaviour is interrupted by a "burst" in which the trajectory moves away from a neighbourhood of the periodic orbit and exhibits possibly irregular behaviour. Eventually, the trajectory re-enters the region near the periodic orbit and the process is repeated. The dynamical behaviour for is characterized by an infinite sequence of intervals of nearly periodic (laminar) motion followed by bursts. The length of the laminar regions scales as for near [a11].

Three types of intermittency are distinguished, depending on the eigenvalues of the associated Jacobian matrix of partial derivatives evaluated at the periodic orbit. Type- intermittency occurs when a stable and an unstable periodic orbit that coexist for collide at (the Jacobian matrix at the resulting periodic orbit has eigenvalue ) and disappear for (i.e., there is a saddle-node bifurcation at ). An example using the difference equation

(a5)

is given in [a4]. For , has one stable and one unstable period-3 orbit. They collide at , and for slightly less than , the iterates exhibit a nearly period-3 motion interrupted by bursts. Type- intermittency has been found in Poincaré mappings of the Lorenz equations (cf. Lorenz attractor, [a11]) and in experiments on oscillating chemical reactions [a12].

Type- intermittency occurs when the eigenvalues of the associated Jacobian matrix are complex and cross the unit circle away from the real line. In type- intermittency, the eigenvalues pass through . Heuristic arguments and numerical evidence suggest that the Lyapunov characteristic exponent of the chaotic attractor created when passes scales as , at least in the case of type- and type- intermittency [a11].

Ruelle–Takens–Newhouse route to chaos.

Suppose that for there is an attracting fixed point for (a1) that loses its stability in a Hopf bifurcation at (i.e., the fixed point is replaced by an attracting periodic orbit). In addition, suppose that at there is another Hopf bifurcation to a quasi-periodic orbit (i.e., the attractor is now a torus ). A subsequent Hopf bifurcation at creates a quasi-periodic -torus. However, S.E. Newhouse, D. Ruelle and F. Takens [a13] showed that for every constant vector field on the torus can be perturbed by an arbitrarily small amount to a new vector field with a chaotic attractor. Thus, in an experiment, one may see a bifurcation from a -frequency quasi-periodic flow to a chaotic attractor; see, for example, [a14], . This route to turbulence is in contrast to the classical theory of L. Landau and E. Lifshitz [a16], which says that turbulence arises from the successive addition of incommensurate frequences as a parameter is increased.

Crisis route to chaos.

The previous examples showed how an existing attractor can change its character as a parameter is varied. In a crisis, there is initially only a chaotic transient for , i.e., initial conditions in some region of the phase space approach what appears to be a chaotic attractor and may spend a long time near it. Eventually, however, the orbit leaves for another part of the phase space, never to return. The time spent near the chaotic transient becomes longer as . When , the chaotic transient becomes a chaotic attractor: initial conditions that once escaped now remain forever.

A simple example of a crisis is given by the quadratic mapping (a3). For , almost every initial condition in the interval generates a trajectory that bounces chaotically in for a time. Eventually, some iterate falls to the left of , and the orbit tends to . At , the transient is converted to an attractor: almost every initial condition in approaches a chaotic attractor.

In this example, the chaotic attractor is contained in when . The crisis occurs at when an interior point of is mapped to , which is part of the stable manifold of the unstable fixed point at . For , a portion of the interval is mapped into the basin of attraction (cf. Chaos) for , so the chaotic attractor is replaced by a chaotic transient. The mean time that typical initial conditions spend in before escaping to scales as with the parameter [a17].

Similar results hold in higher dimensions. [a17] discusses the Hénon mapping

where a -piece and a -piece chaotic attractor coexist for . As , the -piece attractor approaches the stable manifold of a period-6 saddle orbit that forms part of the boundary of the basin of attraction for the -piece attractor. When , part of the -piece attractor crosses this stable manifold. Thus, for the -piece attractor becomes a transient — eventually some iterate maps into the basin of the -piece attractor. Numerical evidence suggests that the mean time spent near the chaotic transient is proportional to , where is a "critical exponent" that can be expressed in terms of the eigenvalues associated with the periodic orbit involved in the crisis [a18]. The existence of crises has also been demonstrated for the Lorenz equations and mappings of the torus [a17].

References

[a1]	J. Milnor, "On the concept of attractor" Commun. Math. Phys. , 99 (1985) pp. 177–195
[a2]	J. Guckenheimer, "Sensitive dependence to initial conditions for one-dimensional maps" Commun. Math. Phys. , 70 (1979) pp. 133–160
[a3]	E.N. Lorenz, Physica D , 35 (1989) pp. 299–317
[a4]	J.-P. Eckmann, "Roads to turbulence in dissipative dynamical systems" Rev. Mod. Phys. , 53 (1981) pp. 643–654
[a5a]	M.J. Feigenbaum, "Qualitative universality for a class of nonlinear transformations" J. Stat. Phys. , 19 (1978) pp. 25–52
[a5b]	M.J. Feigenbaum, "The universal metric properties of nonlinear transformations" J. Stat. Phys. , 21 (1979) pp. 669–706
[a6]	P. Collet, J.-P. Eckmann, O.E. Lanford, "Universal properties of maps on an interval" Commun. Math. Phys. , 76 (1980) pp. 211–254
[a7]	M. Jakobson, "Absolutely continuous invariant measures for one-parameter families of one-dimensional maps" Commun. Math. Phys. , 81 (1981) pp. 39–88
[a8]	P. Collet, J.-P. Eckmann, H. Koch, "Period doubling bifurcations for families of maps on " J. Stat. Phys. , 25 (1981) pp. 1–14
[a9]	J.A. Yorke, K.T. Alligood, "Period doubling cascades of attractors: a prerequisite for horseshoes" Commun. Math. Phys. , 101 (1985) pp. 305–321
[a10]	P. Cvitanović (ed.) , Universality in chaos , A. Hilger (1989)
[a11]	Y. Pomeau, P. Manneville, "Intermittent transition to turbulence in dissipative dynamical systems" Commun. Math. Phys. , 74 (1980) pp. 189–197
[a12]	F. Argoul, A. Arneodo, P. Richetti, J.C. Roux, H.L. Swinney, Acct. Chem. Res. , 20 (1987) pp. 436–442
[a13]	S.E. Newhouse, D. Ruelle, F. Takens, "Occurrence of strange axiom attractors near quasiperiodic flow on , " Commun. Math. Phys. , 64 (1978) pp. 35–40
[a14]	M. Giglio, S. Musazzi, U. Perini, "Transition to chaotic behaviour via a reproducible sequence of period-doubling bifurcations" Phys. Rev. Lett. , 47 (1981) pp. 243–246
[a15a]	P.R. Fenstermacher, H.L. Swinney, J.P. Gollub, "Dynamical instabilities and transition to chaotic Taylor vortex flow" J. Fluid Mech. , 94 (1979) pp. 103–128
[a15b]	A. Brandstäter, H.L. Swinney, Phys. Rev. A , 35 (1987) pp. 2207–2220
[a16]	L.D. Landau, E.M. Lifshitz, "Fluid mechanics" , Pergamon (1959) (Translated from Russian)
[a17]	C. Grebogi, E. Ott, J.A. Yorke, "Crises, sudden changes in chaotic attractors, and transient chaos" Physica D , 7 (1983) pp. 181–200
[a18]	C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. Lett. , 57 (1986) pp. 1284–1287