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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.

A hyperbolic equilibrium point has no Jacobian eigenvalue with zero real part. Every linearized mode therefore has an exponentially growing or decaying direction, with no neutral center direction at the linear level. Stable nodes, stable foci and saddles can all be hyperbolic; hyperbolic does not mean attracting.

The absence of neutral eigenvalues makes local classification particularly useful. Small smooth nonlinear terms preserve the qualitative local structure described by the linearization under the hypotheses of the Hartman-Grobman theorem. For example, $\dot{x}=-x$, $\dot{y}=y$ is hyperbolic but unstable because of its growing direction. As a parameter moves an eigenvalue onto the imaginary axis, hyperbolicity fails and nonlinear terms can organize a bifurcation. This identifies where a familiar linear classification needs additional analysis.

A simple planar periodic orbit has Poincaré index $+1$ when the surrounding curve is traversed with the standard counterclockwise orientation. Along that circuit the nonzero vector field is tangent to the orbit and its direction makes one full positive turn.

This remains true when the actual flow runs clockwise: the field then points opposite to the chosen counterclockwise tangent, adding a constant angle $\pi$ without changing its total winding. Defining the index using traversal in the flow direction instead would change the sign for clockwise cycles, so the orientation convention must be kept explicit.

If the enclosed equilibrium points are isolated, their indices sum to one. A planar hyperbolic equilibrium point has index equal to the sign of its Jacobian determinant, giving $+1$ for a sink or source and $-1$ for a saddle equilibrium. Thus a periodic orbit cannot enclose exactly one saddle and no other equilibrium.

At an equilibrium point, $x(1-x-y)=0$ and $2-2x-y=0$. The first equation gives $x=0$ or $y=1-x$. Substituting in the second gives, respectively,These are all the equilibrium points. The Jacobian matrix of the vector field isAt $(0,2)$,The eigenvalue $-1$ has algebraic multiplicity two and eigenspace $\mathrm{span}\{(0,1{)}^T\}$. For the linearized coordinates $u=x$, $v=y-2$,Both components tend to zero. The repeated eigenvalue with one independent eigenvector makes this a stable improper degenerate node, a sink equilibrium. The term $t{e}^{-t}$ is the signature of the nontrivial Jordan block. Since it is a hyperbolic equilibrium point, the linearization stability theorem gives local asymptotic stability of the nonlinear system. Approaching trajectories are tangent to the vertical eigendirection.

At $(1,0)$,One eigenvalue is positive and one negative, so this point is a saddle equilibrium. Its stable manifold is tangent to $y=\sqrt{2}(x-1)$, and its unstable manifold is tangent to $y=-\sqrt{2}(x-1)$.

For the phase portrait, the nullclines areAbove the second line the vertical flow is downward, and below it upward. For $x>0$, horizontal flow is to the right below $y=1-x$ and to the left above it; for $x<0$ the directions reverse. The line $x=0$ is invariant, with $\dot{y}=2-y$, so it flows toward $(0,2)$. Together with the saddle's two invariant directions, these signs determine the sketch.

On the descending part of the graph, mark the two tangency points whereChoose the crossings that enclose the portion with $g^{\prime }(u)<-1/\gamma$. For fixed $\mu$ strictly between these values and sufficiently small $\epsilon$, the unique equilibrium is a repelling equilibrium by part (e). The endpoints here are leading singular limit values; at finite $\epsilon$ the boundaries instead solve $f_v-\gamma f_u=\epsilon$ and approach these tangencies when the crossings are nondegenerate.

There is a necessary qualification: such an interval exists only when $\gamma$ is large enough that the descending graph actually attains slopes below $-1/\gamma$. If $M={\max }_{u>0}[-g^{\prime }(u)]$, one needs $\gamma M>1$. The supplied graph has a very steep descending section, but an arbitrarily small positive $\gamma$ need not satisfy this condition. For $\delta =0.001$, numerical evaluation gives $M\simeq 511.49$, so the leading threshold is $\gamma \simeq 0.0019551$. A graph alone does not remove this parameter restriction.

The fast motion is especially informative. In fast time $s=\tau /\epsilon$,At leading order $z=u+\gamma v$ is constant, so fast paths are straight lines with slope $dv/du=-1/\gamma$. Along such a line, the fast eigenvalue at the nullcline isIt is negative on the two outer attracting segments and positive on the intervening repelling segment. The tangencies defining ${\mu }_{\pm }$ are consequently the fold points of a critical manifold when it is expressed in the coordinates $(z,v)$. They are not generally the maximum and minimum of $g$ itself.

For ${\mu }_{-}<\mu <{\mu }_{+}$, slow motion on an attracting segment followsOn the left attracting segment $u<{\mu }_{-}$ this moves towards increasing $u$; on the right attracting segment $u>{\mu }_{+}$ it moves towards decreasing $u$. Each segment ends at a tangency and is followed by a fast jump along a line of slope $-1/\gamma$ to the other attracting segment. This constructs the limiting relaxation oscillation. The figure uses $\gamma =0.01$, for which ${\mu }_{-}\simeq 0.62970$ and ${\mu }_{+}\simeq 1.01157$; the marked endpoints change with $\gamma$.

To justify existence of a periodic orbit of the full planar system, rather than just draw a formal cycle, first establish a compact trapping region. The positive quadrant is forward invariant: at $u=0$, $\dot{u}=\mu +\gamma \delta v/\epsilon >0$, and at $v=0$, $\dot{v}=u^2/[\epsilon (1+u^2)]\ge 0$. MoreoverChoose $V>1/\delta$ above the initial store value. Since $f(u,v)\ge -1-\delta V$ when $0\le v\le V$,Thus the rectangle $0\le u\le U$, $0\le v\le V$ is a positively invariant set if $U>\mu +\gamma (1+\delta V)/\epsilon$ and the initial point lies inside it. Every forward solution is bounded. There are no equilibria on its axes for $\mu >0$, and its sole interior equilibrium is a hyperbolic equilibrium point with both eigenvalues having positive real parts in the strict unstable range. A sufficiently small neighborhood of this repelling equilibrium has outward flow; deleting it leaves a compact trapping region with no equilibria. The Poincaré-Bendixson theorem then supplies a periodic orbit.

Hence periodic solutions exist throughout the strict unstable interval, interpreted with the finite-$\epsilon$ correction above. This argument establishes existence, without asserting uniqueness, the criticality of possible endpoint Hopf bifurcations, or absence of coexisting periodic orbits outside the interval.

A global bifurcation joins two mechanisms: a long passage near a hyperbolic equilibrium point and a finite excursion returning the unstable manifold to its neighborhood. Eigenvalue ratios control the local passage, while the geometry of the global return decides reinjection, folding and trapping. In the real symmetric case there are two returning branches and a power-law cusp. In the saddle-focus case rotation adds infinitely many oscillations to that cusp. The following maps display both mechanisms explicitly.

For the Lorenz configuration, enter on $z=h$ at $(x,y,h)$ and leave on $x=sh$, where $s=\mathrm{sgn}(x)$. Solving the linear flow gives a flight timePut $\gamma =-\hat{\lambda }/{\lambda }_{+}>\delta =-{\lambda }_{-}/{\lambda }_{+}$. The outgoing coordinates areThe smooth global map from each outgoing section to the incoming section has, to leading order,up to the choice of which unstable branch carries the sign of $\nu$. Reflection symmetry makes the offsets opposite and the leading coefficients equal. Since the stable transverse coordinate is multiplied by the higher power $|x{|}^{\gamma }$, it contributes only subleading terms to the leading scalar map. Absorbing powers of $h$ into a positive constant gives the Lorenz power return mapThe line $x=0$ lies on the stable manifold and has no finite return; the discontinuity is therefore a genuine missing return, not an arbitrary value to assign at zero.

If $\delta >1$, the derivative $A\delta |x{|}^{\delta -1}$ tends to zero. For $\mu <0$, there are two attracting fixed points, $x=\pm s$ with $s=-\mu +A{s}^{\delta }\simeq -\mu$. They describe two symmetry-related one-lobe attracting flow periodic orbits. For $\mu >0$, the orbit alternates between $s$ and $-s$, where $s=\mu -A{s}^{\delta }\simeq \mu$; its multiplier is $(A\delta s^{\delta -1}{)}^2<1$. This represents one attracting flow periodic orbit traversing both lobes. At $\mu =0$ the two homoclinic loops meet: this is the contracting gluing scenario, with diverging flow period at the connection.

If $\delta <1$, the local derivative is unbounded, and the small periodic orbits are repelling in the expanding return direction. For $\mu >0$, the small positive fixed point has $s\simeq (\mu /A{)}^{1/\delta }$, with a multiplier tending to infinity; there is a symmetry-related negative fixed point as well. More complex motion follows from the two branches, not merely from a large derivative at one fixed point. Choose $I=[-c\mu ,c\mu ]$ with $0<c<1$. The inverse branches areFor sufficiently small positive $\mu$, both lie inside $I$, their images are disjoint, and their derivatives are $O({\mu }^{1/\delta -1})\to 0$. Nested inverse images for every infinite sign sequence therefore define a Cantor invariant set. Iterating the forward map shifts the sequence. This constructs periodic itineraries of every symbolic period and nonperiodic itineraries; transverse contraction in the full return turns the expanding two-strip mechanism into a horseshoe.

This is the local chaotic Lorenz mechanism. A robust attracting Lorenz set additionally needs a global trapping region and the appropriate return properties. The scalar asymptote and $\delta <1$ alone do not prove that every nearby orbit is attracted to one chaotic set; periodic windows, escaping orbits and other global returns must be distinguished. Symmetry is crucial to obtaining the two homoclinic branches together in a single parameter variation.

A three-dimensional hyperbolic equilibrium point is a saddle-focus when its eigenvalues include a nonreal pair and a real eigenvalue with the opposite sign of real part. In the displayed orientation it has a one-dimensional unstable manifold and a two-dimensional spiralling stable manifold; reversing time interchanges the stable and unstable dimensions. The ratio $-{\lambda }_s/{\lambda }_u$ controls local transverse contraction in a Shilnikov return map.

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.