Homoclinic orbit

For an equilibrium point $p$, a homoclinic orbit is a nonconstant trajectory $x(t)$ withAt a hyperbolic equilibrium point, the trajectory leaves a neighborhood of $p$ along its unstable manifold and returns along its stable manifold. It never reaches $p$ in finite time when local solution uniqueness holds. Its closure includes the equilibrium and can form a loop-shaped separatrix.

A completely explicit example iswith Hamiltonian function $H(x,v)=v^2/2-x^2/2+x^4/4$. Its derivative is zero along every trajectory, and the origin is a saddle point with eigenvalues $\pm 1$. The curvelies on $H=0$. Differentiating gives $\dot{x}=v$ and $\ddot{x}=x-x^3$, while the hyperbolic secant tends to zero at both time infinities. Thus it is a homoclinic orbit. Changing both signs gives the second loop.

Each loop bounds periodic orbits around one potential minimum. This is an integrable Hamiltonian system, so a homoclinic loop by itself does not establish chaos. A Shilnikov bifurcation and a simple-zero Melnikov energy-balance method calculation require additional dimensional, spectral or forcing hypotheses.

For a hyperbolic equilibrium point with one unstable eigenvalue ${\lambda }_{+}>0$ and weakest stable decay rate ${\lambda }_{-}<0$, the saddle index is $\delta =-{\lambda }_{-}/{\lambda }_{+}$. For a stable complex pair, use its real part. An incoming distance $x$ from the stable manifold takes time $T={\lambda }_{+}^{-1}\log (h/|x|)$ to reach an outgoing section at unstable coordinate magnitude $h$. A stable coordinate is multiplied by $e^{{\lambda }_{-}T}=(|x|/h{)}^{\delta }$, giving a power-law local passage. A smooth nondegenerate reinjection consequently has derivative tending to zero when $\delta >1$ and an unbounded leading derivative when $\delta <1$. This determines local contraction or expansion near a homoclinic orbit; existence of a particular attractor also depends on the global return geometry.

At a planar saddle equilibrium with $m_{-}<0<m_{+}$, positive saddle quantity $m_{-}+m_{+}>0$ is equivalent to saddle index $\delta <1$. The local passage followed by the regular return has leading transverse map $s\mapsto C{s}^{\delta }$, $C>0$. Its derivative diverges as $s\downarrow 0$, so a nearby periodic orbit created by the saddle-loop splitting is repelling. This local mechanism agrees with the growth of the unstable periodic orbit born at a subcritical Hopf bifurcation in the quadratic Bogdanov–Takens unfolding.

Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

A symmetric homoclinic gluing bifurcation joins two symmetry-related periodic orbits into one orbit that traverses both lobes of a figure-eight separatrix. At the transition, the two loops are homoclinic orbits to the same saddle equilibrium. The two original cycles spend increasingly long near that saddle before the connection is reached.

A signed transverse coordinate distinguishes the lobes. Reflection can make the corresponding return map odd, so fixed points $x_{\ast }$ and $-x_{\ast }$ represent the separate cycles, while a two-cycle passing between positive and negative coordinates can represent the glued orbit. The signed gluing-map reduction describes the local power-law passage and the global reinjection; its coefficients and saddle eigenvalue ratio determine the existence and stability of these branches.

The geometry alone does not guarantee attracting cycles. For example, the exact Hamiltonian double-well homoclinic orbits form a figure-eight but are surrounded by conserved-energy families, rather than isolated attracting cycles. Dissipation, a suitable parameter unfolding and the return-map conditions must supply the gluing scenario. A damped double-well model with symmetry-related Hopf cycles is one setting in which those conditions can be investigated.

For a symmetric real saddle with one unstable direction and a strongly contracting transverse direction, composing the power-law local passage with a smooth global return gives this leading Poincaré return map, with $A>0$ and saddle index $\delta$. It is undefined at the stable-manifold crossing $x=0$. For $\delta >1$, small fixed points on each branch are attracting for $\mu <0$; for $\mu >0$ an attracting alternating-sign period-two orbit satisfies $s=\mu -A{s}^{\delta }$, giving gluing of two one-lobe flow cycles into one two-lobe cycle. For $\delta <1$ and small $\mu >0$, on $I=[-c\mu ,c\mu ]$, $0<c<1$, the inverse branchesmap into $I$ and contract. Infinite branch choices define an invariant Cantor set with full two-symbol dynamics. In the two-dimensional return, transverse contraction turns this mechanism into a Smale horseshoe. An attracting Lorenz set additionally requires appropriate trapping and global-return properties.

Consider the branch return mapwith $A>0$, $\delta >0$, and nonzero return coordinates. After one iterate, $y\in \{-1,1\}$. On these two sign sheets, define $z=-xy$. Then $|z|=|x|$ and $\mathrm{sgn}(z)=-\mathrm{sgn}(x)\mathrm{sgn}(y)$, givingThis scalar equation describes the signed gluing-map reduction.

A negative scalar fixed point has $xy>0$, so $\mathrm{sgn}(x)=y$ and the next $y$ is unchanged. The two sign choices therefore lift it to two symmetry-related fixed points of the original return map. A positive scalar fixed point instead has $\mathrm{sgn}(x)=-y$; both $x$ and $y$ change sign on each step, giving one period-two orbit.

The value zero needs separate interpretation. In the underlying homoclinic orbit model it is the separatrix limit, where the flow's return time diverges. A formal continuous extension of the scalar formula at zero does not supply an ordinary finite-period periodic orbit. All branch iterations used to model finite flow returns must remain in the valid nonzero local domain.

For $f(z)=\mu +A\mathrm{sgn}(z)|z{|}^{\delta }$, a nonzero fixed point satisfiesAssume $A>0$, $\delta >0$ and $\delta \ne 1$. A saddle-node bifurcation has multiplier $f^{\prime }(z)=A\delta |z{|}^{\delta -1}=1$, so it occurs at $z=\pm r$ withAt either nonzero point, $f^{\prime \prime }$ is nonzero and the parameter enters additively, giving the ordinary nondegenerate fold conditions.

When $0<\delta <1$, define $C=(1-\delta )r/\delta$. The scalar extension has three fixed-point branches for $|\mu |<C$: two stable outer branches and an unstable inner branch. For nonzero fixed points, their derivatives are below and above one because $A\delta |z{|}^{\delta -1}$ decreases with $|z|$. At $\mu =0$, the inner branch lies at the singular separatrix value; the signed gluing-map reduction explains why it is excluded as a finite-time flow return.

For fixed $0<A<1$ and $\delta \uparrow 1$ from below, the fold magnitude has asymptotic value $C\sim (1-\delta )A^{1/(1-\delta )}/e$. This is the half-width; the total parameter interval has width $2C$. The exponentially narrow gluing-map cusp gives the exact rescaling. For $\delta >1$ approaching one, the formal fold coordinate instead tends to infinity, outside the local return-map region.

Fix $0<A<1$ in the signed gluing-map reduction and write $\delta =1-\eta$ with $0<\eta <1$. The saddle-node curves of a signed power gluing map occur at parameter values $\mu =\pm C$, whereThus $C$ is the half-width of the three-branch interval, and $2C$ is its total width.

To see the scale, use $(1-\eta {)}^{1/\eta }\to e^{-1}$ and $(1-\eta {)}^{-1}\to 1$. ThenSince $A^{1/\eta }=\exp (\log A/\eta )$ and $\log A<0$, the width decays faster than every positive power of $\eta$. A parameter scan with an algebraic step size can therefore miss the coexistence interval entirely.

This is an essential exponential scaling, rather than the power-law fold separation in the ordinary smooth cubic cusp. The three branches counted here are scalar return-map branches. Their physical lifts differ: a negative scalar fixed point represents two symmetry-related cycles, while a positive one represents one cycle visiting both lobes. The singular zero value is a separatrix limit with infinite return time.

A Shilnikov bifurcation involves a homoclinic orbit to a saddle-focus equilibrium in three dimensions. Write its eigenvalues asand put $\delta =-{\lambda }_{-}/{\lambda }_{+}$. The Shilnikov return map combines contraction toward the equilibrium with arbitrarily many rotations before departure.

The sign of the saddle value ${\lambda }_{+}+{\lambda }_{-}={\lambda }_{+}(1-\delta )$ separates two regimes. At a nondegenerate homoclinic connection with $0<\delta <1$, the Shilnikov theorem gives countably many nearby saddle periodic orbits. When $\delta >1$, the usual generic unfolding instead creates a locally attracting periodic orbit on the appropriate side. The boundary $\delta =1$ requires additional terms.^[ref]

The spectral and nondegeneracy assumptions matter: a planar saddle loop has no rotating stable plane and does not supply this mechanism. The result is associated with Leonid Pavlovich Shilnikov; Andrey Shilnikov coauthored the cited account.

Near a saddle-focus equilibrium, linearized coordinates satisfy $\dot{z}={\lambda }_{+}z$, $\dot{r}={\lambda }_{-}r$ and $\dot{\theta }=\omega$. A trajectory entering at a small positive coordinate $z$ takesto reach a fixed outgoing section. Hence $r_{\mathrm{out}}\propto z^{\delta }$ with $\delta =-{\lambda }_{-}/{\lambda }_{+}$, while the accumulated angle is logarithmic in $z$. Composing this passage with a smooth global return gives the leading scalar formwhere $|\Omega |=|\omega |/{\lambda }_{+}$ and the phase convention absorbs constant angular shifts. The map applies only on branches whose trajectories return to the section.^[ref]

Differentiation givesFor $\delta <1$, the envelope becomes unbounded as $z\downarrow 0$, but the derivative also has infinitely many zeros. The map therefore has strongly expanding branches, rather than uniform expansion everywhere.

The full two-dimensional return has Jacobian determinant of order $z^{2\delta -1}$ in regular section coordinates: radial scaling contributes $z^{\delta -1}$ and conversion of polar coordinates contributes $z^{\delta }$. For $\delta >1/2$ this contracts area near the singular limit, but area contraction alone does not prove that a scalar reduction is valid. A one-passage periodic trajectory has period $T_g+T_{\mathrm{loc}}$, explaining the logarithmically diverging time near the homoclinic connection.

At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.

ConsiderThe circle $r=1$ is a periodic orbit. Writing $r=1+\rho$ gives $\dot{\rho }=-(1-2\sin \theta )\rho +O({\rho }^2)$, while $\theta =t+{\theta }_0$ on the circle. Its normal Floquet multiplier is therefore $\exp [-{\int }_0^{2\pi }(1-2\sin \theta )d\theta ]=e^{-2\pi }<1$.

Inside the circle, use the increasing angle as independent variable and set $w=(1-r{)}^{-1}$. Direct differentiation gives the linear ordinary differential equationLet $A(\theta )=\theta +2\cos \theta$. The solution with $w(\pi )=1$ isAt the endpoint, $w^{\prime }(\pi )=0$ and $w^{\prime \prime }(\pi )=2$, so $w=1+(\theta -\pi {)}^2+O((\theta -\pi {)}^3)>1$ immediately before $\pi$. A crossing of $w=1$ has slope $-2\sin \theta$. Thus there is no crossing on $(0,\pi )$ compatible with this terminal behavior. AlsoEvery crossing on $(-\pi ,0)$ has positive slope, giving a unique ${\theta }_b\in (-\pi ,0)$ where $w=1$. Hence $r=1-1/w>0$ for ${\theta }_b<\theta <\pi$ and tends to zero at both ends. At the incoming endpoint $r\sim (\pi -\theta {)}^2$, so $dt=d\theta /r$ has a divergent integral. At the outgoing endpoint the zero is simple and the time integral diverges logarithmically. The curve is a homoclinic orbit to the origin.

Uniqueness for the angular scalar equation prevents positive-radius trajectories from crossing this loop. Inside it they approach the origin; outside it they approach the attracting circle. The origin therefore attracts an open set, but is not Lyapunov stable: points arbitrarily close to its outgoing branch follow trajectories to a fixed positive radius before returning or escaping. Attraction and stability are distinct properties.