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For $x\prime =f(x)$, a simple fixed point $x_{\ast }$ is locally asymptotically stable when $f\prime (x_{\ast })<0$ and unstable when $f\prime (x_{\ast })>0$.

A Floquet multiplier is an eigenvalue of a one-period linear evolution map. For a homogeneous linear differential equation with continuous coefficients of period $T>0$, this map is the monodromy matrix of a periodic linear system $N$. If $Nv=mv$ for a nonzero eigenvector $v$, the associated solution of a differential equation with $Y(0)=v$ satisfies $Y(t+T)=mY(t)$. Its exponential rate is $T^{-1}\log |m|$, the corresponding Floquet growth rate. A Floquet multiplier equal to one gives a $T$-periodic solution in that eigenvector direction; a nontrivial Jordan block at a unit-modulus Floquet multiplier can still produce polynomial growth.

For a nonlinear periodic orbit, the Floquet multipliers are eigenvalues of the one-period map of its variational equation. A nonconstant periodic orbit of a smooth autonomous differential equation has a Floquet multiplier equal to one in its tangent direction: differentiating the solution of a differential equation with respect to its starting phase gives a periodic solution of the variational equation. The derivative of a transverse Poincaré map removes this phase direction and retains the transverse Floquet multipliers. Transverse Floquet multipliers strictly inside the unit circle give local attraction, while one outside gives instability; a transverse Floquet multiplier on the unit circle requires further analysis. For a planar periodic orbit, the transverse Floquet multiplier is given by the divergence test for a planar periodic orbit.

A compact planar limit set containing no equilibrium is a periodic orbit.

A periodic orbit is a nonconstant trajectory that returns to its initial state after some least positive period.

The omega-limit set of a trajectory consists of the points approached along sequences of times tending to positive infinity. For a bounded continuous flow it is nonempty, compact, connected, and invariant.

An autonomous planar system is an ordinary differential equationwhose vector field does not depend explicitly on time.

At a hyperbolic fixed point, stable and unstable manifolds are tangent to the corresponding eigenspaces. Power-series coefficients follow from graph invariance.

$V(x_0)=0$, $V>0$ elsewhere, and $\dot{V}<0$ prove asymptotic stability; suitable sublevel sets lie in the basin.

Bifurcation theory studies qualitative changes in equilibria and invariant sets as parameters vary.

Near a nonhyperbolic equilibrium, a local invariant centre manifold is tangent to the generalized eigenspace with zero-real-part eigenvalues, and its reduced dynamics determine the local bifurcation behaviour.

A map on a metric space is chaotic in Devaney's sense when it is topologically transitive, its periodic points are dense, and it has sensitive dependence on initial conditions.

A continuous map is chaotic in Devaney's sense when it has topological transitivity, a dense set of periodic points, and sensitive dependence on initial conditions.

Forpolar coordinates giveThe trajectories rotate clockwise, while the sign of the radial equation determines stability.

A continuous interval map is chaotic in Glendinning's sense when some positive iterate has a horseshoe for an interval map.

A local center manifold is an invariant manifold tangent at a nonhyperbolic equilibrium to the generalized eigenspace whose eigenvalues have zero real part. Nearby stability reduces to the flow on this manifold when all transverse eigenvalues have negative real part.

The normal form $\dot{x}=\mu x-x^2$ has equilibria $x=0$ and $x=\mu$ that cross and exchange stability at $\mu =0$. A generic constant perturbation separates the branches or replaces the crossing by saddle-node bifurcations, so the transcritical bifurcation is not structurally stable without a constraint preserving both branches.

A one-dimensional discrete dynamical system iterates a map $x_{n+1}=F(x_n)$. A fixed point satisfies $F(x^{\ast })=x^{\ast }$, while a point of least period $k$ satisfies $F^k(x)=x$ but no corresponding equation for a smaller positive period.

A relaxation oscillation alternates slow motion along attracting branches of a critical manifold with fast jumps near its folds. It is characteristic of a fast-slow system with widely separated time scales.

For a weak perturbation of a planar Hamiltonian system, integrate the first-order change of the unperturbed Hamiltonian around each unperturbed periodic orbit. Zeros of this averaged energy drift select candidate perturbed periodic orbits, and a change from positive to negative drift indicates stability.

For $\dot{q}=p$, $\dot{p}=-V^{\prime }(q)$, the energyis constant. Local minima of $V$ give center equilibria, local maxima give saddle equilibria, and separatrices are energy contours through saddles.