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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471701.png" /> that belongs to the domain of definition of the Hamilton function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471702.png" /> of the [[Hamiltonian system|Hamiltonian system]]
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{{TEX|done}}
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A point $(p=p^*,q=q^*)$ that belongs to the domain of definition of the Hamilton function $H=H(p,q)$ of the [[Hamiltonian system|Hamiltonian system]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471703.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\begin{equation}\dot p=-\frac{\partial H}{\partial q},\quad\dot q=\frac{\partial H}{\partial p},\quad p=(p_1,p_2),\quad q=(q_1,q_2),\label{*}\end{equation}
  
such that the solution of the system (*) passing through this point asymptotically approaches some periodic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471704.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471705.png" />, and asymptotically approaches another periodic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471706.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471707.png" />. The solution itself, which passes through the heteroclinic point, is called a heteroclinic solution.
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such that the solution of the system \eqref{*} passing through this point asymptotically approaches some periodic solution $T_1$ as $t\to\infty$, and asymptotically approaches another periodic solution $T_1'$ as $t\to-\infty$. The solution itself, which passes through the heteroclinic point, is called a heteroclinic solution.
  
There is a connection between the heteroclinic solutions of the system (*) and the two-dimensional invariant surfaces of this system. If a two-dimensional invariant surface separates the periodic solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h0471709.png" />, there is no heteroclinic solution joining these periodic solutions. In many cases the converse is true. In the non-degenerate case, in a neighbourhood of a homoclinic solution (cf. [[Homoclinic point|Homoclinic point]]) there exists an infinite sequence of periodic solutions any two of which may be joined by a heteroclinic solution. A neighbourhood of a contour consisting of a finite number of periodic and heteroclinic solutions of the system (*) (a so-called homoclinic cycle) has a structure that in many respects resembles that of a homoclinic solution.
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There is a connection between the heteroclinic solutions of the system \eqref{*} and the two-dimensional invariant surfaces of this system. If a two-dimensional invariant surface separates the periodic solutions $T_1$ and $T_1'$, there is no heteroclinic solution joining these periodic solutions. In many cases the converse is true. In the non-degenerate case, in a neighbourhood of a homoclinic solution (cf. [[Homoclinic point|Homoclinic point]]) there exists an infinite sequence of periodic solutions any two of which may be joined by a heteroclinic solution. A neighbourhood of a contour consisting of a finite number of periodic and heteroclinic solutions of the system \eqref{*} (a so-called homoclinic cycle) has a structure that in many respects resembles that of a homoclinic solution.
  
The above definition of a heteroclinic point may be applied practically unchanged to the case of a Hamiltonian system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717010.png" /> degrees of freedom if the periodic solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717012.png" /> are replaced by invariant tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717014.png" /> whose respective dimensions are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717017.png" />. Heteroclinic solutions play an important part in the study of instability in Hamiltonian systems with number of degrees of freedom higher than two and in the theory of structurally-stable dynamical systems (cf. [[Rough system|Rough system]]).
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The above definition of a heteroclinic point may be applied practically unchanged to the case of a Hamiltonian system with $n>2$ degrees of freedom if the periodic solutions $T_1$ and $T_1'$ are replaced by invariant tori $T_k$ and $T'_{k'}$ whose respective dimensions are $k$ and $k'$, $0<k,k'<n$. Heteroclinic solutions play an important part in the study of instability in Hamiltonian systems with number of degrees of freedom higher than two and in the theory of structurally-stable dynamical systems (cf. [[Rough system|Rough system]]).
  
 
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====Comments====
 
====Comments====
The above notion of a heteroclinic (homoclinic) point is meaningful for arbitrary continuous-time dynamical systems (not necessarily Hamiltonian). It can also be defined for dynamical systems with discrete time: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047170/h04717018.png" /> be a diffeomorphism on a manifold. Then a heteroclinic (homoclinic) point is any point which is in the intersection of the stable manifold of one invariant point and the unstable manifold of another (the same) invariant point.
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The above notion of a heteroclinic (homoclinic) point is meaningful for arbitrary continuous-time dynamical systems (not necessarily Hamiltonian). It can also be defined for dynamical systems with discrete time: Let $f$ be a diffeomorphism on a manifold. Then a heteroclinic (homoclinic) point is any point which is in the intersection of the stable manifold of one invariant point and the unstable manifold of another (the same) invariant point.

Latest revision as of 21:17, 23 November 2018

A point $(p=p^*,q=q^*)$ that belongs to the domain of definition of the Hamilton function $H=H(p,q)$ of the Hamiltonian system

\begin{equation}\dot p=-\frac{\partial H}{\partial q},\quad\dot q=\frac{\partial H}{\partial p},\quad p=(p_1,p_2),\quad q=(q_1,q_2),\label{*}\end{equation}

such that the solution of the system \eqref{*} passing through this point asymptotically approaches some periodic solution $T_1$ as $t\to\infty$, and asymptotically approaches another periodic solution $T_1'$ as $t\to-\infty$. The solution itself, which passes through the heteroclinic point, is called a heteroclinic solution.

There is a connection between the heteroclinic solutions of the system \eqref{*} and the two-dimensional invariant surfaces of this system. If a two-dimensional invariant surface separates the periodic solutions $T_1$ and $T_1'$, there is no heteroclinic solution joining these periodic solutions. In many cases the converse is true. In the non-degenerate case, in a neighbourhood of a homoclinic solution (cf. Homoclinic point) there exists an infinite sequence of periodic solutions any two of which may be joined by a heteroclinic solution. A neighbourhood of a contour consisting of a finite number of periodic and heteroclinic solutions of the system \eqref{*} (a so-called homoclinic cycle) has a structure that in many respects resembles that of a homoclinic solution.

The above definition of a heteroclinic point may be applied practically unchanged to the case of a Hamiltonian system with $n>2$ degrees of freedom if the periodic solutions $T_1$ and $T_1'$ are replaced by invariant tori $T_k$ and $T'_{k'}$ whose respective dimensions are $k$ and $k'$, $0<k,k'<n$. Heteroclinic solutions play an important part in the study of instability in Hamiltonian systems with number of degrees of freedom higher than two and in the theory of structurally-stable dynamical systems (cf. Rough system).

References

[1] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 1–3 , Gauthier-Villars (1892–1899)
[2] E. Zehnder, "Homoclinic points near elliptic fixed points" Comm. Pure Appl. Math. , 26 (1973) pp. 131–182
[3] V.K. Mel'nikov, "On the stability of the centre for time-periodic perturbations" Trans. Moscow Math. Soc , 12 (1963) pp. 1–56 Trudy Moskov. Mat. Obshch. , 12 (1963) pp. 3–52
[4a] S. Smale, "Dynamical systems and the topological conjugacy problem for diffeomorphisms" , Proc. Internat. Congress mathematicians (Stockholm, 1962) , Inst. Mittag-Leffler (1963) pp. 490–496
[4b] S. Smale, "Diffeomorphisms with many periodic points" S.S. Cairns (ed.) , Differential and Combinatorial Topol. (Symp. in honor of M. Morse) , Princeton Univ. Press (1965) pp. 63–80
[5] L.P. Shil'nikov, "On a Poincaré–Birkhoff problem" Math. USSR-Sb. , 3 : 3 (1967) pp. 353–371 Mat. Sb. , 74 (116) : 3 (1967) pp. 378–397
[6a] V.M. Alekseev, "Quasirandom dynamical systems I" Math. USSR-Sb. , 5 : 1 (1968) pp. 73–128 Mat. Sb. , 76 (118) : 1 (1968) pp. 72–134
[6b] V.M. Alekseev, "Quasirandom dynamical systems II" Math. USSR-Sb. , 6 : 4 (1968) pp. 505–560 Mat. Sb. , 77 (119) : 4 (1968) pp. 545–601
[6c] V.M. Alekseev, "Quasirandom dynamical systems III" Math. USSR-Sb. , 7 : 1 (1969) pp. 1–43 Mat. Sb. , 78 (120) : 1 (1969) pp. 3–50


Comments

The above notion of a heteroclinic (homoclinic) point is meaningful for arbitrary continuous-time dynamical systems (not necessarily Hamiltonian). It can also be defined for dynamical systems with discrete time: Let $f$ be a diffeomorphism on a manifold. Then a heteroclinic (homoclinic) point is any point which is in the intersection of the stable manifold of one invariant point and the unstable manifold of another (the same) invariant point.

How to Cite This Entry:
Heteroclinic point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Heteroclinic_point&oldid=12950
This article was adapted from an original article by V.K. Mel'nikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article