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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Henon_heiles_potential.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Henon_heiles_potential.svg/220px-Henon_heiles_potential.svg.png" decoding="async" width="220" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Henon_heiles_potential.svg/330px-Henon_heiles_potential.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Henon_heiles_potential.svg/440px-Henon_heiles_potential.svg.png 2x" data-file-width="518" data-file-height="444" /></a><figcaption>Contour plot of the Hénon–Heiles potential</figcaption></figure> <p>While at <a href="/wiki/Princeton_University" title="Princeton University">Princeton University</a> in 1962, <a href="/wiki/Michel_H%C3%A9non" title="Michel Hénon">Michel Hénon</a> and <a href="/wiki/Carl_Heiles" class="mw-redirect" title="Carl Heiles">Carl Heiles</a> worked on the non-linear motion of a star around a galactic center with the motion restricted to a plane. In 1964 they published an article titled "The applicability of the third integral of motion: Some numerical experiments".<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Their original idea was to find a third <a href="/wiki/Integral_of_motion" class="mw-redirect" title="Integral of motion">integral of motion</a> in a galactic dynamics. For that purpose they took a simplified two-dimensional nonlinear rotational symmetric potential and found that the third integral existed only for a limited number of initial conditions. In the modern perspective the initial conditions that do not have the third integral of motion are called chaotic orbits. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=H%C3%A9non%E2%80%93Heiles_system&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Hénon–Heiles potential</b> can be expressed as<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x,y)={\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(x,y)={\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ba9320bbd648750fc22dceb8836fb42eba58045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.93ex; height:6.343ex;" alt="{\displaystyle V(x,y)={\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}"></span></dd></dl> <p>The Hénon–Heiles <a href="/wiki/Hamiltonian_function" class="mw-redirect" title="Hamiltonian function">Hamiltonian</a> can be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H={\frac {1}{2}}(p_{x}^{2}+p_{y}^{2})+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H={\frac {1}{2}}(p_{x}^{2}+p_{y}^{2})+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8263a40acf958ff570f702a879cd417914d0320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.932ex; height:6.343ex;" alt="{\displaystyle H={\frac {1}{2}}(p_{x}^{2}+p_{y}^{2})+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {y^{3}}{3}}\right).}"></span></dd></dl> <p>The Hénon–Heiles system (HHS) is defined by the following four equations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}=p_{x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}=p_{x},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff86917a51e8f9b268b458991c1ea4f8ce4c2ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.417ex; height:2.509ex;" alt="{\displaystyle {\dot {x}}=p_{x},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {p_{x}}}=-x-2\lambda xy,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>λ</mi> <mi>x</mi> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {p_{x}}}=-x-2\lambda xy,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc113119560af71dbd6440b21a4883c6a6f95d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.158ex; height:2.509ex;" alt="{\displaystyle {\dot {p_{x}}}=-x-2\lambda xy,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {y}}=p_{y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {y}}=p_{y},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1155af44c65bfa83f7f1cf1c5bc70f29a82d085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.266ex; height:2.843ex;" alt="{\displaystyle {\dot {y}}=p_{y},}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {p_{y}}}=-y-\lambda (x^{2}-y^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {p_{y}}}=-y-\lambda (x^{2}-y^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dd00bd76f83d44afd3947dc3c3899828a1996f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:22.461ex; height:3.343ex;" alt="{\displaystyle {\dot {p_{y}}}=-y-\lambda (x^{2}-y^{2}).}"></span></dd></dl> <p>In the classical chaos community, the value of the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is usually taken as unity. Since HHS is specified in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>, we need a Hamiltonian with 2 degrees of freedom to model it. It can be solved for some cases using <a href="/wiki/Painlev%C3%A9_transcendents" title="Painlevé transcendents">Painlevé analysis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_Hénon–Heiles_Hamiltonian"><span id="Quantum_H.C3.A9non.E2.80.93Heiles_Hamiltonian"></span>Quantum Hénon–Heiles Hamiltonian</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=H%C3%A9non%E2%80%93Heiles_system&action=edit&section=2" title="Edit section: Quantum Hénon–Heiles Hamiltonian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the quantum case the Hénon–Heiles <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> can be written as a two-dimensional <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. </p><p>The corresponding two-dimensional Schrödinger equation is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,y)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {1}{3}}y^{3}\right)\right]\Psi (x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,y)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {1}{3}}y^{3}\right)\right]\Psi (x,y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eafb4878553ad33c4b296611437b1e304c6557bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.658ex; height:6.343ex;" alt="{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,y)=\left[{\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+{\frac {1}{2}}(x^{2}+y^{2})+\lambda \left(x^{2}y-{\frac {1}{3}}y^{3}\right)\right]\Psi (x,y).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Wada_property_of_the_exit_basins">Wada property of the exit basins</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=H%C3%A9non%E2%80%93Heiles_system&action=edit&section=3" title="Edit section: Wada property of the exit basins"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hénon–Heiles system shows rich dynamical behavior. Usually the <a href="/wiki/Wada_property" class="mw-redirect" title="Wada property">Wada property</a> cannot be seen in the <a href="/wiki/Hamiltonian_system" title="Hamiltonian system">Hamiltonian system</a>, but Hénon–Heiles exit basin shows an interesting Wada property. It can be seen that when the energy is greater than the critical energy, the Hénon–Heiles system has three exit basins. In 2001 <a href="/wiki/Miguel_Angel_Fern%C3%A1ndez_Sanju%C3%A1n" class="mw-redirect" title="Miguel Angel Fernández Sanjuán">M. A. F. Sanjuán</a> et al.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> had shown that in the Hénon–Heiles system the exit basins have the Wada property. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=H%C3%A9non%E2%80%93Heiles_system&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHénon,_M.Heiles,_C.1964" class="citation journal cs1">Hénon, M.; Heiles, C. (1964). "The applicability of the third integral of motion: Some numerical experiments". <i>The Astronomical Journal</i>. <b>69</b>: 73–79. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964AJ.....69...73H">1964AJ.....69...73H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F109234">10.1086/109234</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astronomical+Journal&rft.atitle=The+applicability+of+the+third+integral+of+motion%3A+Some+numerical+experiments&rft.volume=69&rft.pages=73-79&rft.date=1964&rft_id=info%3Adoi%2F10.1086%2F109234&rft_id=info%3Abibcode%2F1964AJ.....69...73H&rft.au=H%C3%A9non%2C+M.&rft.au=Heiles%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AH%C3%A9non%E2%80%93Heiles+system" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHénon1983" class="citation cs2">Hénon, Michel (1983), "Numerical exploration of Hamiltonian Systems", in Iooss, G. (ed.), <i>Chaotic Behaviour of Deterministic Systems</i>, Elsevier Science Ltd, pp. 53–170, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/044486542X" title="Special:BookSources/044486542X"><bdi>044486542X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Numerical+exploration+of+Hamiltonian+Systems&rft.btitle=Chaotic+Behaviour+of+Deterministic+Systems&rft.pages=53-170&rft.pub=Elsevier+Science+Ltd&rft.date=1983&rft.isbn=044486542X&rft.aulast=H%C3%A9non&rft.aufirst=Michel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AH%C3%A9non%E2%80%93Heiles+system" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAguirreVallejoSanjuán2001" class="citation journal cs1">Aguirre, Jacobo; Vallejo, Juan C.; Sanjuán, Miguel A. F. (2001-11-27). "Wada basins and chaotic invariant sets in the Hénon-Heiles system". <i>Physical Review E</i>. <b>64</b> (6). American Physical Society (APS): 066208. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreve.64.066208">10.1103/physreve.64.066208</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10261%2F342147">10261/342147</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1063-651X">1063-651X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+E&rft.atitle=Wada+basins+and+chaotic+invariant+sets+in+the+H%C3%A9non-Heiles+system&rft.volume=64&rft.issue=6&rft.pages=066208&rft.date=2001-11-27&rft_id=info%3Ahdl%2F10261%2F342147&rft.issn=1063-651X&rft_id=info%3Adoi%2F10.1103%2Fphysreve.64.066208&rft.aulast=Aguirre&rft.aufirst=Jacobo&rft.au=Vallejo%2C+Juan+C.&rft.au=Sanju%C3%A1n%2C+Miguel+A.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AH%C3%A9non%E2%80%93Heiles+system" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=H%C3%A9non%E2%80%93Heiles_system&action=edit&section=5" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/Henon-HeilesEquation.html">http://mathworld.wolfram.com/Henon-HeilesEquation.html</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Hénon–Heiles_system&oldid=1252872038">https://en.wikipedia.org/w/index.php?title=Hénon–Heiles_system&oldid=1252872038</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Stellar_astronomy" title="Category:Stellar astronomy">Stellar astronomy</a></li><li><a href="/wiki/Category:Chaotic_maps" title="Category:Chaotic maps">Chaotic maps</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 23 October 2024, at 08:31<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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