Homoclinic Bif
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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 ...
, a homoclinic orbit is a trajectory of a flow 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 ...
which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of 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 ...
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 an equilibrium. 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 is an equilibrium at x=x_0, then a solution \Phi(t) is a homoclinic orbit if :\Phi(t)\rightarrow x_0\quad \mathrm\quad t\rightarrow\pm\infty If the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
has three or more dimensions, then it is important to consider the
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
of the unstable manifold of the saddle point. The figures show two cases. First, when the stable manifold is topologically a cylinder, and secondly, when the unstable manifold is topologically a
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
; in this case the homoclinic orbit is called ''twisted''.


Discrete dynamical system

Homoclinic orbits and homoclinic points are defined in the same way for
iterated function In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
s, as the intersection of the stable set and
unstable set 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 repel ...
of some fixed point or
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. We also have the notion of homoclinic orbit when considering discrete dynamical systems. In such a case, if f:M\rightarrow M is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
M, we say that x is a homoclinic point if it has the same past and future - more specifically, if there exists a fixed (or periodic) point p such that :\lim_f^n(x)=p.


Properties

The existence of one homoclinic point implies the existence of an infinite number of them. This comes from its definition: the intersection of a stable and unstable set. Both sets are
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
by definition, which means that the forward iteration of the homoclinic point is both on the stable and unstable set. By iterating N times, the map approaches the equilibrium point by the stable set, but in every iteration it is on the unstable manifold too, which shows this property. This property suggests that complicated dynamics arise by the existence of a homoclinic point. Indeed, Smale (1967) showed that these points leads to
horseshoe map In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavi ...
like dynamics, which is associated with chaos.


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 a
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 homoclinic 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 =\ 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 In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end o ...
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 orbit In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end o ...
*
Homoclinic 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. Mo ...


References

{{Reflist *
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


External links


Homoclinic orbits in Henon map
with Java applets and comments Dynamical systems