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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.

Cosmic microwave background anisotropies are direction-dependent perturbations of the relic radiation temperature and polarization.

The large-scale structure of the universe is the cosmic distribution of matter on scales where galaxies and halos form a web of clusters, filaments, sheets, and voids.