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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 trajectory's limit set consists of its accumulation points along sequences of times tending to positive or negative infinity. The forward-time set is its omega-limit set and the backward-time set its alpha-limit set.

Let $D$ be a simply connected open subset of the plane and let $F$ be a continuously differentiable vector field. If a continuously differentiable multiplier $B$ makes $\nabla \cdot (BF)$ have one sign throughout $D$ and not vanish identically on any nonempty open subset, then $\dot{x}=F(x)$ has no nonconstant periodic orbit contained in $D$.

Indeed, suppose such an orbit bounds a region $R$. Simple connectedness puts its interior in $D$. Along the orbit the vector field is tangent, so $BF$ has zero outward normal flux. Green's theorem would then giveThe sign condition and continuity make the area integral strictly nonzero, a contradiction.

The choice of Dulac function $B$ can expose a sign that the original divergence lacks. Its regularity must hold on the entire simply connected region: a multiplier singular at an enclosed point cannot exclude a cycle enclosing that singularity.

An invariant manifold is a manifold preserved by a dynamical system: a trajectory beginning on it stays on it for as long as that trajectory is defined. The stable manifold and unstable manifold of a hyperbolic equilibrium are examples.

The part of a response that dies away after an initial condition or an input change. In a stable linear ordinary differential equation, the decaying homogeneous solution gives a transient response relative to a chosen particular solution. A homogeneous solution that grows or persists is not a decaying transient.

A differential equation whose evolution rule depends explicitly on time. A time-varying coefficient or forcing term can make a system nonautonomous. Unlike an autonomous differential equation, the state alone need not determine its future without the current time.

An autonomous differential equation has no explicit dependence on time: $\dot{x}=f(x)$. Its current complete state determines the instantaneous rate of change through the same rule at every starting time. A harmonic oscillator written as $\dot{x}=v$, $\dot{v}=-{\omega }_0^{2}x$ is autonomous even though its solution oscillates in time.

With unique solutions, shifting a solution in time gives another solution of the same equation. This allows a phase portrait to be drawn directly in state space, with one vector field assigning a direction at each point. Explicit periodic forcing instead produces a nonautonomous differential equation. One can sometimes recover an autonomous description by adding the drive phase as an extra state variable, but that enlarged state space contains information absent from the original mechanical coordinates alone.

A local transverse section to a flow is mapped back to itself by the next intersection of an orbit. Fixed points represent periodic orbits, and the derivative of this return map determines their transverse stability.

A relative equilibrium is a solution moving only through a continuous group action: its shape is stationary after the symmetry motion is removed. If a one-parameter subgroup is written $g(t)=\exp (t\xi )$, the solution has the form $x(t)=g(t)x_0$. Its image lies on a single group orbit, and quotienting by the symmetry turns it into an ordinary equilibrium point.

For example, the ordinary differential equationis equivariant under $A\mapsto e^{i\alpha }A$. At $\mu >0$, writing $A=r{e}^{i\theta }$ gives $\dot{r}=(\mu -r^2)r$ and $\dot{\theta }=\omega$. Thus $r=\sqrt{\mu }$ is stationary in the quotient, while $A(t)=\sqrt{\mu }{e}^{i(\omega t+{\theta }_0)}$ rotates in the original state space. When $\omega \ne 0$ this is a periodic orbit; when $\omega =0$ each point on the circle is an ordinary equilibrium point.

A relative equilibrium need not be periodic. A steadily translating pattern on an infinite line has a fixed profile in its moving frame but never returns to the same laboratory position. The rotating point-vortex relative equilibrium is the corresponding idea for rigidly rotating vortex configurations.

For a linear group action of $G$ on a state space, an equivariant dynamical system has a vector field satisfyingThus applying a symmetry before computing a velocity gives the same result as applying it to the computed velocity. An equivariant vector field need not be invariant as a vector: its value transforms along with the point where it is evaluated.

If the ordinary differential equation has unique solutions, its flow map obeys ${\Phi }_t(gx)=g{\Phi }_t(x)$ whenever both sides exist. Indeed, $g{\Phi }_t(x)$ has derivative $gF({\Phi }_t(x),\mu )=F(g{\Phi }_t(x),\mu )$ and starts at $gx$, so Picard-Lindelöf theorem identifies it with ${\Phi }_t(gx)$.

Consequently, every subgroup's fixed-point subspace of a group action is invariant under the flow map. If $hx=x$ for all $h$ in the subgroup, then $h{\Phi }_t(x)={\Phi }_t(hx)={\Phi }_t(x)$. This reduces some symmetry-preserving motions to a smaller state space. It also constrains normal forms: the wavevector selection rule for equivariant monomials and the cubic equivariants of the full cube symmetry group are concrete coefficient tests.

A Smale horseshoe is a compact hyperbolic invariant set of a smooth map with expanding and contracting directions and at least two strips that stretch across a common rectangle. Iterating the strips and their inverse images produces a Cantor-set product on which the dynamics are conjugate to a full shift. It supplies infinitely many saddle periodic orbits and chaotic itineraries; hyperbolicity alone does not make it an attracting invariant set. The geometric construction explains why phase-dependent reinjection near a Shilnikov bifurcation can create chaos even when the associated symmetric amplitude equations have no such mechanism.

Let $f$ be a smooth diffeomorphism and $K$ a compact invariant set. It is uniformly hyperbolic if its tangent spaces have a continuous splitting $T_{x}M=E_x^s\oplus E_x^u$ preserved by the derivative, with constants $C>0$ and $0<\rho <1$ such thatThe same constants must work for every $x\in K$. Thus the stable directions contract forward and the unstable directions contract backward, uniformly over the set. A sign observed in a finite-time numerical Lyapunov exponent calculation does not establish these uniform estimates.

For the cat map on a two-dimensional torus, the constant matrix $\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3\pm \sqrt{5})/2$. Its two eigenvectors give fixed stable and unstable directions; the smaller eigenvalue is the reciprocal of the larger. With the flat Euclidean metric, $C=1$ and $\rho =(3-\sqrt{5})/2$ give the required estimates everywhere. The whole torus is therefore hyperbolic.

A Smale horseshoe supplies a different example: its hyperbolic invariant set contains infinitely many saddle periodic orbits, yet need not attract a neighborhood. Hyperbolicity describes stretching and contraction, while being an attractor concerns nearby trajectories. For a nonsingular flow map, the analogous splitting also has the neutral direction along the vector field; it should not be omitted by copying the map definition unchanged.