Heteroclinic Orbit
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the
phase portrait A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point. Phase portraits are an invaluable tool in studying dyn ...
of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different
equilibrium point In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The point \tilde\in \mathbb^n is an equilibrium point for the differential equation :\frac = \ma ...
s. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit. Consider the continuous dynamical system described by the
ODE An ode (from grc, ᾠδή, ōdḗ) is a type of lyric poetry. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three majo ...
::\dot x=f(x) Suppose there are equilibria at x=x_0 and x=x_1, then a solution \phi(t) is a heteroclinic orbit from x_0 to x_1 if ::\phi(t)\rightarrow x_0\quad \mathrm\quad t\rightarrow-\infty and ::\phi(t)\rightarrow x_1\quad \mathrm\quad t\rightarrow+\infty This implies that the orbit is contained in the
stable manifold In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repello ...
of x_1 and the
unstable manifold In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repe ...
of x_0.


Symbolic dynamics

By using the
Markov partition A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete ...
, the long-time behaviour of
hyperbolic system In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
can be studied using the techniques of
symbolic dynamics In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (e ...
. In this case, a heteroclinic orbit has a particularly simple and clear representation. Suppose that S=\ is a
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...
of ''M'' symbols. The dynamics of a point ''x'' is then represented by a
bi-infinite string In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called th ...
of symbols :\sigma =\{(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in S \; \forall k \in \mathbb{Z} \} A
periodic point In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
of the system is simply a recurring sequence of letters. A heteroclinic orbit is then the joining of two distinct periodic orbits. It may be written as :p^\omega s_1 s_2 \cdots s_n q^\omega where p= t_1 t_2 \cdots t_k is a sequence of symbols of length ''k'', (of course, t_i\in S), and q = r_1 r_2 \cdots r_m is another sequence of symbols, of length ''m'' (likewise, r_i\in S). The notation p^\omega simply denotes the repetition of ''p'' an infinite number of times. Thus, a heteroclinic orbit can be understood as the transition from one periodic orbit to another. By contrast, a homoclinic orbit can be written as :p^\omega s_1 s_2 \cdots s_n p^\omega with the intermediate sequence s_1 s_2 \cdots s_n being non-empty, and, of course, not being ''p'', as otherwise, the orbit would simply be p^\omega.


See also

* Heteroclinic connection *
Heteroclinic cycle In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching t ...
*
Heteroclinic bifurcation Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. ...
* Homoclinic orbit *
Traveling wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...


References

*
John Guckenheimer John Mark Guckenheimer (born 1945) joined the Department of Mathematics at Cornell University in 1985. He was previously at the University of California, Santa Cruz (1973-1985). He was a Guggenheim fellow in 1984, and was elected president of the ...
and
Philip Holmes Philip John Holmes (born May 24, 1945) is the Eugene Higgins Professor of Mechanical and Aerospace Engineering at Princeton University. As a member of the Mechanical and Aerospace Engineering department, he formerly served as the interim chair ...
, ''Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields'', (Applied Mathematical Sciences Vol. 42), Springer Dynamical systems