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A bound orbit approaching a separatrix can spend an increasingly long time near an unstable equilibrium. Suppose a one-dimensional potential energy has a nondegenerate maximum at $x_s$, so locallyFor energy just below $U_s$, the nearby turning point approaches $x_s$. The travel-time integral contains a contribution proportional to $\int dy/\sqrt{{\lambda }^2{y}^2-2(U_s-E)/m}$, whose growth is logarithmic as $E\to U_s$.

A simple pendulum illustrates this slowing: a release increasingly close to the upright position produces a long pause near the top of each swing. The full period of an oscillation diverges, although motion away from the unstable point remains fast. The logarithm relies on a nonzero negative curvature at the maximum; a flatter maximum can give a different divergence.

A particle constrained to a stationary smooth surface of revolution $z=f(r)$ under uniform gravity conserves $h=r^2\dot{\theta }$ and mechanical energy per unit mass. Substitute $\dot{z}=f^{\prime }(r)\dot{r}$ and $\dot{\theta }=h/r^2$ into that energy to obtainThis proves the radial reduction. Simple roots of $E-U_h$ are turning points; a stationary point $U_h^{\prime }(r_0)=0$ supports an exact circular orbit. At a local maximum of $U_h$, another trajectory at the same energy can approach the orbit asymptotically as a separatrix. Existence of roots elsewhere is insufficient for confinement: their dynamically accessible ordering relative to the launch radius matters.

For $\ddot{\varphi }={\omega }_0^2\sin \varphi$ with constant ${\omega }_0$, multiplication by $\dot{\varphi }$ gives the conserved energy $E={\dot{\varphi }}^2/2+{\omega }_0^2(1+\cos \varphi )$. The stable center is $\pi$; the separatrix has $E=2{\omega }_0^2$. Below it the resonant argument librates and above it the angle circulates. This is an invariant of the constant-coefficient pendulum approximation of a mean-motion resonance, not of arbitrary coupled orbital element evolution.

A saddle equilibrium in a smooth planar flow has one attracting and one repelling linearized direction. Its Jacobian determinant is negative, so its two eigenvalues are real and have opposite signs. This produces stable and unstable invariant directions rather than uniform attraction or uniform repulsion.

For $\dot{x}=\lambda x$, $\dot{y}=-\mu y$ with $\lambda ,\mu >0$, the $y$ axis approaches the origin in forward time and the $x$ axis moves away. A generic nearby initial state has a nonzero $x$ component and eventually leaves, even if its distance initially decreases. In a nonlinear system, the corresponding stable manifold and unstable manifold curve away from their tangent eigendirections. Such manifolds can form a separatrix between different kinds of motion, making a saddle important for global outcome selection as well as local instability.

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.

The zero-relative-speed limit of the Sine-Gordon kink-antikink scattering solution is a separatrix at twice the rest-kink energy. A Sine-Gordon Bäcklund transformation of the static kink with equal parameter gives $p_x=-p\tanh x$, $p_t=\mathrm{sech}x$ for $p=\tan (\varphi /4)$, hence the displayed solution. The two asymptotic transitions separate logarithmically in time and their speeds tend to zero; this is not a finite-period Sine-Gordon breather.

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.

A uniform circular disc of radius $a$ and mass $m$, pivoted on a perpendicular axis through its rim, has moment of inertia $I_A=3m{a}^2/2$ by the parallel axis theorem. Its hanging small oscillations have period $2\pi \sqrt{3a/(2g)}$. Measured from the upright position, the falling equation is $\ddot{\theta }=2g\sin \theta /(3a)$. Exact upright rest remains an unstable equilibrium; the energy level of that state describes a limiting separatrix after perturbation, with ${\dot{\theta }}^2=4g(1-\cos \theta )/(3a)$. The center acceleration and pivot reaction follow by resolving radial and tangential components.

In the dilute-vortex description of the two-dimensional XY model, $K={\rho }_s/(k_{B}T)$ is dimensionless stiffness and $y$ is phase vortex fugacity. A phase vortex has logarithmic energy $\pi {\rho }_s\log (L/a)$, while its placement entropy is $2k_B\log (L/a)$. Small phase vortex pairs screen the logarithmic interaction, lowering the stiffness. Near $K=2/\pi$, the rescaled variables $X=2-\pi K$, $Y=4\pi y$ obey $X^{\prime }=Y^2$, $Y^{\prime }=XY$ to leading order. The invariant $X^2-Y^2$ leads to an essential correlation-length singularity $\xi \sim a\exp (C/\sqrt{T-T_c})$, not a finite power-law exponent.

Here $\ell =\log (r/a)$ increases toward larger observation distances, and the vortex fugacity $y$ is a dimensionless weight per unit core, conventionally initialized as $y_0\simeq e^{-E_c/(k_{B}T)}$ with the microscopic measure fixed. Numerical coefficients depend on this normalization. In the displayed convention, differentiating $X=2-\pi K$ using the original flow givesOnly replacing the prefactor $(2-X{)}^2/4$ by one near $X=0$ gives the scaled leading model. Its first integral follows directly:Thus this first integral belongs to the stated scaled model, not exactly to the original $K,y$ equations. For $Y\ge 0$, the critical separatrix is $X=-Y$. The bound side has $X<-Y$ and flows to $Y=0$, $X<0$; the unbinding side ultimately runs to large vortex fugacity, where the dilute approximation ceases to be quantitative.

To derive the unusual correlation length scaling without fitting an exponent, write the invariant on a nearby unbinding trajectory as $X^2-Y^2=-c^2$. Then $X^{\prime }=X^2+c^2$ and the time to cross fixed small limits $-b$ to $b$ is $2\arctan (b/c)/c\sim \pi /c$. If a physical one-parameter path crosses the separatrix transversely, its $c^2$ is proportional to the positive reduced temperature. Since distance grows as $e^{\ell }$, this gives $\xi \sim a\exp (b_0/\sqrt{t})$, with a nonuniversal $b_0>0$. This is the dimensionless-temperature form of the singularity above.