# 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

$$\tag{a1 } \frac{d }{dt } x( t) = F _ \lambda ( x( t)),\ x( 0) = x _ {0} ,$$

or difference equations (mappings)

$$\tag{a2 } x _ {n+ 1 } = F _ \lambda ( x _ {n} ) ,$$

where $F$ is a smooth function of the real parameter $\lambda$ and the point $x$, which belongs to some finite-dimensional phase space (like $\mathbf R ^ {n}$); $x _ {0}$ is an initial condition. (Other domains are sometimes of interest, for example, mappings of the circle $S ^ {1}$ or the torus $T ^ {2}$.) Suppose that for a fixed value of $\lambda$, the initial conditions in some open set in the phase space approach a compact set $A$ which exhibits sensitive dependence on initial conditions. (Such a set $A$ 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 $x _ {0}$ and $y _ {0}$ are two nearby initial conditions on $A$, 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 $A$ is bounded.) The attractor $A$ is chaotic whenever $A$ 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 $\lambda$ 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 $\lambda _ \infty$, at which the original fixed point is replaced by an aperiodic attractor (which may be chaotic for $\lambda > \lambda _ \infty$).

M.J. Feigenbaum

originally studied period doubling in the difference equation

$$\tag{a3 } x _ {n+ 1 } = F _ \lambda ( x _ {n} ) = \lambda x _ {n} ( 1 - x _ {n} ),$$

also called the quadratic mapping. When $\lambda > 1$, $F _ \lambda$ has a non-zero fixed point at $x _ {f} = 1 - 1/ \lambda$ which is stable for $1 < \lambda < 3$, since

$$\left | \frac{dF _ \lambda }{dx } ( x _ {f} ) \right | = | 2- \lambda | < 1.$$

When $\lambda = \lambda _ {1} = 3$, the derivative at $x _ {f}$ is $- 1$. For slightly larger values of $\lambda$, the derivative is larger than $1$ in absolute value, and $x _ {f}$ is unstable: almost-all initial conditions in $[ 0, 1]$ are attracted to a period-2 orbit, $x _ {1} = F _ \lambda ( x _ {2} )$, $x _ {2} = F _ \lambda ( x _ {1} )$. A similar derivative evaluation for $F _ \lambda ^ { 2 } ( x _ {i} ) = F _ \lambda ( F _ \lambda ( x _ {i} )) = x _ {i}$, $i= 1,2$, shows that each $x _ {i}$ is stable for $\lambda _ {1} < \lambda < \lambda _ {2} \approx 3.449$. At $\lambda _ {2}$ each $x _ {i}$ loses its stability because the derivative $dF _ {\lambda _ {2} } ^ { 2 } ( x _ {i} )/dx = - 1$. As before, each $x _ {i}$ is replaced by a pair of attracting points $x _ {i1 }$, $x _ {i2 }$ such that $x _ {i1 } = F _ \lambda ^ { 2 } ( x _ {i2 } )$, $x _ {i2 } = F _ \lambda ^ { 2 } ( x _ {i1 } )$, $i= 1, 2$. The points $x _ {ij }$ correspond to an attracting period-4 orbit for $F _ \lambda$ when $\lambda$ is slightly larger than $\lambda _ {2}$. 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 $\{ \lambda _ {k} \}$ that is bounded above by a number $\lambda _ \infty \approx 3.5699$ with the property that

$$\tag{a4 } \lim\limits _ {k \rightarrow \infty } \frac{\lambda _ {k} - \lambda _ {k- 1 } }{\lambda _ {k+ 1 } - \lambda _ {k} } = \delta \approx 4.6692.$$

The number $\delta$ is the Feigenbaum constant. It is a remarkable fact that $\delta$ is independent of the details of the mapping $F _ \lambda$ as long as $F _ \lambda$ satisfies certain general hypotheses; see , [a6] for details. Chaos occurs in the quadratic mapping for many values of $\lambda < 4$. 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 $\mathbf R ^ {n} \rightarrow \mathbf R ^ {n}$ where $n > 1$[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 $\lambda < \lambda _ {c}$ can disappear and be replaced by a chaotic attractor for $\lambda > \lambda _ {c}$. For $\lambda$ slightly larger than $\lambda _ {c}$, 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 $\lambda > \lambda _ {c}$ is characterized by an infinite sequence of intervals of nearly periodic (laminar) motion followed by bursts. The length of the laminar regions scales as $( \lambda - \lambda _ {c} ) ^ {- 1/2 }$ for $\lambda$ near $\lambda _ {c}$[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- $1$ intermittency occurs when a stable and an unstable periodic orbit that coexist for $\lambda < \lambda _ {c}$ collide at $\lambda = \lambda _ {c}$( the Jacobian matrix at the resulting periodic orbit has eigenvalue $1$) and disappear for $\lambda > \lambda _ {c}$( i.e., there is a saddle-node bifurcation at $\lambda _ {c}$). An example using the difference equation

$$\tag{a5 } x _ {n+ 1 } = F _ \lambda ( x _ {n} ) = 1 - \lambda x _ {n} ^ {2}$$

is given in [a4]. For $\lambda > 1.75$, $F _ \lambda ^ { 3 }$ has one stable and one unstable period-3 orbit. They collide at $\lambda = 1.75$, and for $\lambda$ slightly less than $1.75$, the iterates exhibit a nearly period-3 motion interrupted by bursts. Type- $1$ intermittency has been found in Poincaré mappings of the Lorenz equations (cf. Lorenz attractor, [a11]) and in experiments on oscillating chemical reactions [a12].

Type- $2$ intermittency occurs when the eigenvalues of the associated Jacobian matrix are complex and cross the unit circle away from the real line. In type- $3$ intermittency, the eigenvalues pass through $- 1$. Heuristic arguments and numerical evidence suggest that the Lyapunov characteristic exponent of the chaotic attractor created when $\lambda$ passes $\lambda _ {c}$ scales as $( \lambda - \lambda _ {c} ) ^ {- 1/2 }$, at least in the case of type- $1$ and type- $3$ intermittency [a11].

## Ruelle–Takens–Newhouse route to chaos.

Suppose that for $\lambda < \lambda _ {1}$ there is an attracting fixed point for (a1) that loses its stability in a Hopf bifurcation at $\lambda = \lambda _ {1}$( i.e., the fixed point is replaced by an attracting periodic orbit). In addition, suppose that at $\lambda = \lambda _ {2} > \lambda _ {1}$ there is another Hopf bifurcation to a quasi-periodic orbit (i.e., the attractor is now a torus $T ^ {2}$). A subsequent Hopf bifurcation at $\lambda = \lambda _ {3} > \lambda _ {2}$ creates a quasi-periodic $3$- torus. However, S.E. Newhouse, D. Ruelle and F. Takens [a13] showed that for $n \geq 3$ every constant vector field on the torus $T ^ {n}$ 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 $2$- 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 $\lambda > \lambda _ {c}$, 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 $\lambda \rightarrow \lambda _ {c}$. When $\lambda \leq \lambda _ {c}$, 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 $\lambda > 4$, almost every initial condition in the interval $I=[ 0, 1]$ generates a trajectory that bounces chaotically in $I$ for a time. Eventually, some iterate falls to the left of $0$, and the orbit tends to $- \infty$. At $\lambda = 4$, the transient is converted to an attractor: almost every initial condition in $I$ approaches a chaotic attractor.

In this example, the chaotic attractor is contained in $( 0, 1)$ when $\lambda < 4$. The crisis occurs at $\lambda = 4$ when an interior point of $( 0, 1)$ is mapped to $1$, which is part of the stable manifold of the unstable fixed point at $0$. For $\lambda > 4$, a portion of the interval $I$ is mapped into the basin of attraction (cf. Chaos) for $- \infty$, so the chaotic attractor is replaced by a chaotic transient. The mean time that typical initial conditions spend in $I$ before escaping to $- \infty$ scales as $( \lambda - 4) ^ {- 1/2 }$ with the parameter [a17].

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

$$x _ {n+ 1 } = 1 - \lambda x _ {n} ^ {2} + y _ {n} ,$$

$$y _ {n+ 1 } = 0.3 x _ {n} ,$$

where a $6$- piece and a $4$- piece chaotic attractor coexist for $1.0624 < \lambda < \lambda _ {c} \approx 1.08074$. As $\lambda \rightarrow \lambda _ {c}$, the $6$- 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 $4$- piece attractor. When $\lambda > \lambda _ {c}$, part of the $6$- piece attractor crosses this stable manifold. Thus, for $\lambda > \lambda _ {c}$ the $6$- piece attractor becomes a transient — eventually some iterate maps into the basin of the $4$- piece attractor. Numerical evidence suggests that the mean time spent near the chaotic transient is proportional to $( \lambda - \lambda _ {c} ) ^ {- \gamma }$, where $\gamma$ 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

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How to Cite This Entry:
Routes to chaos. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Routes_to_chaos&oldid=48594
This article was adapted from an original article by E.J. Kostelich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article