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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 a dynamical system as a parameter varies: an equilibrium point can appear or disappear, lose stability, or give way to a periodic orbit. The aim is to identify the transition and its local mechanism without solving every nearby trajectory.

For example, $\dot{x}=\mu -x^2$ has two equilibria when $\mu >0$, one merged equilibrium when $\mu =0$, and none when $\mu <0$. The change cannot be described as merely a small displacement of an existing solution. It is a saddle-node bifurcation. A bifurcation diagram shows the branches against $\mu$ and marks their different stability. Near a transition, a reduced normal form can isolate the decisive nonlinear terms; comparing its predictions with a parameter sweep in an experiment tests whether that mechanism fits the observed change.

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 continuous map on a metric space has Devaney chaos when it has topological transitivity, a dense set of periodic points, and sensitive dependence on initial conditions. Transitivity allows trajectories to pass between prescribed open regions; dense periodic points place exactly repeating trajectories arbitrarily close to every state; sensitivity requires a fixed positive separation scale for some nearby initial conditions under iteration.

The two-symbol full shift illustrates all three conditions. A finite coordinate block specifies a basic open set. Concatenating two prescribed blocks with suitable spacing gives a sequence whose shift moves between the two open sets, while repeating a block gives a periodic sequence in any such neighborhood. Changing a sufficiently distant coordinate produces an initially close sequence that becomes visibly different when that coordinate shifts to the center.

These are global properties of the chosen invariant state space. A positive local stretching rate or an unstable fixed point alone does not prove all three 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 discrete dynamical system specifies a phase space $X$ and a map $F:X\to X$. An initial state $x_0$ determines its forward orbit byThe phase space may be an interval, a plane, a manifold or another suitable set; one-dimensional iteration is one special case. A two-sided orbit additionally needs backward states, which are uniquely determined when $F$ is invertible.

A fixed point obeys $F(x^{\ast })=x^{\ast }$. A point of least period $k$ obeys $F^k(x)=x$ with no smaller positive return time; its orbit is a periodic orbit. For example, $F(x)=x/2$ on the real line gives $x_n=2^{-n}{x}_0$, converging to the attracting fixed point zero. An irrational circle rotation instead gives aperiodic orbits; discrete time alone does not imply chaos.

A relaxation oscillation spends most of each cycle evolving slowly, then undergoes relatively rapid transitions between parts of its state-space path. Separated timescales, rather than a nearly sinusoidal waveform, organize the motion. In a fast-slow description, trajectories can follow attracting branches of a critical manifold until that slow tracking fails near a fold.

The Biased van der Pol relaxation period illustrates this behavior in a strongly nonlinear oscillator. Its waveform develops slow portions and fast jumps, so a single constant-frequency sinusoid does not capture its time trace even though the motion is periodic. A measured signal can be tested by comparing the durations of its slow stages with its transitions. The reduced slow motion and jump regions require different approximations; extending a slow approximation through the transition can miss the mechanism that completes the cycle.

For a weak perturbation of a planar Hamiltonian system, the energy balance method computes the net change of unperturbed energy around a closed trajectory. Ifand ${\gamma }_h$ is an unperturbed periodic orbit with energy $h$, define $M(h)={\int }_{{\gamma }_h}Gdt$. The return changes energy by $\epsilon M(h)+O({\epsilon }^2)$, away from singular or infinite-period levels.

A persistent periodic orbit must have zero net energy change. A simple zero $M(h_{\ast })=0$, $M^{\prime }(h_{\ast })\ne 0$ gives a candidate that persists under the usual smooth return-map hypotheses. For positive $\epsilon$, the first-order return derivative is $1+\epsilon M^{\prime }(h_{\ast })$, so a negative derivative gives transverse attraction and a positive one gives repulsion when the perturbation is sufficiently small.

The averaged amplitude equation expresses the same slow drift in an amplitude coordinate. Near a homoclinic orbit, the unperturbed period diverges, so the regular finite-period argument needs replacement by a connecting-orbit calculation such as the Melnikov energy-balance method.

For a one-dimensional conservative mechanical system,the chain rule gives $\dot{E}=p\dot{p}+V^{\prime }(q)\dot{q}=0$. Trajectories therefore follow contours of mechanical energy, with direction determined by $\dot{q}=p$.

A nondegenerate local minimum of the potential energy gives a center equilibrium surrounded by small closed energy contours; a nondegenerate local maximum gives a saddle equilibrium. The saddle-energy contours form separatrices between different types of motion. Degenerate stationary points require higher-order analysis.

For $V(q)=-q^2/2+q^4/4$, the minima at $q=\pm 1$ have energy $-1/4$ and the central maximum has energy zero. Levels $-1/4<E<0$ form separate periodic families around the wells, $E=0$ contains two homoclinic orbits, and $E>0$ gives periodic motion traversing both wells. These closed contours are not isolated limit cycles.

Cosmic microwave background anisotropy is the direction dependence of the relic radiation's temperature and polarization. A temperature map can be written $T(\hat{\mathbf{n}})=\bar{T}[1+\Theta (\hat{\mathbf{n}})]$, separating a mean from its fractional angular variation. Comparing directions with a Differential Microwave Radiometer measures these variations without requiring the instrument to determine the absolute mean temperature.^[ref]

Intrinsic plasma fluctuations, motions and gravitational frequency shifts all contribute. The Cosmic microwave background power spectrum organizes the signal by angular scale; the Cosmic microwave background acoustic peaks record the evolution of the photon-baryon fluid. Frequency comparisons and instrumental checks help distinguish the relic signal from galactic foregrounds and noise before the map is used to infer cosmological parameters.

The large-scale structure of the universe is the spatial organization of matter into dark matter halos, galaxy clusters, sheets, filaments and cosmic voids. Galaxy clustering traces this pattern, while the underlying matter also includes material that does not shine. Gravity amplifies initial linear cosmological density perturbations, but expansion and the matter's dynamics determine which scales grow efficiently. Comparing observed statistics with models of cosmological structure formation turns this pattern into a test of the initial conditions and subsequent evolution.