Homoclinic orbit

A homoclinic orbit is a nonconstant trajectory that approaches the same equilibrium point as time tends to both positive and negative infinity. Its closure includes that equilibrium. Such an orbit may form a basin boundary, although a homoclinic orbit alone does not imply a chaotic invariant set.

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.

Two symmetry-related periodic orbits can meet the two loops of a symmetric saddle's figure-eight separatrix. On the other side a single periodic orbit can traverse both lobes. The saddle value and nonlinear return-map coefficients determine stability. In a damped double-well oscillator the symmetry-related stable cycles born at the two well Hopf bifurcations provide a natural setting for this merger.

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.

For $x^{\prime }=-\mu \mathrm{sgn}(x)+A\mathrm{sgn}(y)|x{|}^{\delta }$, $y^{\prime }=\mathrm{sgn}(x)$, the invariant sign sheet has $y=\pm 1$ after one iterate. There $z=-xy$ obeys the displayed one-dimensional map. A negative fixed $z$ lifts to two symmetry-related fixed points, while a positive fixed $z$ lifts to one period-two orbit. The origin of the return map represents the separatrix, where the flow return time diverges, and is not an ordinary finite-period orbit of a group action.

The fixed point relation is $\mu =z-A\mathrm{sgn}(z)|z{|}^{\delta }$. Setting the fixed-point multiplier $A\delta |z{|}^{\delta -1}$ equal to one gives the two saddle-node bifurcations at $z=\pm r$. For fixed $0<A<1$, as $\delta \uparrow 1$ from below their parameter width tends to zero as $(1-\delta )A^{1/(1-\delta )}/e$, forming an exponentially narrow gluing-map cusp. Three scalar fixed points coexist inside this region: two stable outer branches and one unstable inner branch. For $\delta >1$ near one the formal saddle-node bifurcations recede to infinity, outside the local gluing-map domain.

For fixed $0<A<1$ and $\delta <1$, the two folds of the signed gluing-map reduction enclose three scalar fixed-point branches. Writing $\delta =1-\eta$ gives width $C=\eta (1-\eta {)}^{-1}[A(1-\eta ){]}^{1/\eta }$. Since $(1-\eta {)}^{1/\eta }\to e^{-1}$, the displayed essential asymptotic follows. The fold curves meet with exponentially small width; this is not the ordinary algebraic cusp normal form of a smooth cubic vector field. A negative fixed point lifts to two symmetry-related flow cycles, so the cusp describes distinct return-map branches rather than counting every physical cycle separately.

A Shilnikov bifurcation is a homoclinic connection to a saddle-focus in three dimensions, with one real unstable eigenvalue ${\lambda }_{+}>0$ and a stable complex pair ${\lambda }_{-}\pm i\omega$. The eigenvalue ratio $\delta =-{\lambda }_{-}/{\lambda }_{+}$ distinguishes a contracting return for $\delta >1$ from the oscillatory expanding return associated with positive saddle value when $\delta <1$. Generic local and global maps yield infinitely many saddle periodic orbits at a connection in the latter case.

A passage near a saddle-focus contracts a transverse radius by $z^{\delta }$ while rotating it by an angle proportional to $\log z$. A smooth global reinjection converts these transverse coordinates into a sinusoidal return in $z$. The map is defined only for positive passage coordinates that return on the chosen branch. The full two-dimensional map has determinant of order $z^{2\delta -1}$; for $\delta >1/2$ its thin transverse geometry supports a leading one-dimensional approximation. A one-passage orbit has period $T=T_g+{\lambda }_{+}^{-1}\log (z_0/z)$.

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.

For $\dot{r}=r(1-r)(r-2\sin \theta )$, $\dot{\theta }=r$, the circle $r=1$ has normal Floquet multiplier $e^{-2\pi }$ and is attracting. Inside it, $w=(1-r{)}^{-1}$ satisfies $dw/d\theta =(1-2\sin \theta )w-1$. The solution through $w(\pi )=1$ has $w>1$ on one interval $({\theta }_b,\pi )$ with $-\pi <{\theta }_b<0$. The corresponding radius $r=1-1/w$ is a homoclinic orbit to the origin. Near its returning endpoint $r\sim (\theta -\pi {)}^2$, so the approach takes infinite physical time. Comparison for the scalar angular equation shows that smaller radii inside this loop approach the origin, whereas all positive radii outside it approach the circle. The origin is therefore not Lyapunov stable despite attracting an open set.