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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.

The Poincaré-Bendixson theorem strongly restricts long-term motion in a smooth autonomous planar flow. In a standard form, a nonempty compact omega-limit set containing no equilibrium point is a periodic orbit. The flow is generated by a continuously differentiable vector field on an open planar domain, and the limit set must lie within that domain.^[ref]

This gives a practical existence argument: find a compact forward-trapping region that contains no equilibria, and analyze a trajectory that remains in it. The theorem can establish periodic long-term behavior without solving the orbit explicitly. It does not by itself prove that a cycle is unique or attracting from both sides. If the limit set includes equilibria, connecting trajectories can enter the classification, while higher-dimensional flows such as the Lorenz system are outside the planar conclusion.

A periodic orbit is a nonconstant trajectory that returns to the same complete state after a positive period. Its least positive period $T$ gives $x(t+T)=x(t)$ for all times along the solution. A constant equilibrium is excluded by the nonconstant condition. For a discrete map, the analogous least period is the number of iterations required to return.

A harmonic oscillator gives a family of periodic ellipses, one for each positive mechanical energy. A limit cycle is instead an isolated periodic orbit. This distinction matters for perturbations: changing an undamped oscillator's energy selects another ellipse, while an attracting limit cycle can restore nearby trajectories toward its own orbit. Their phases need not coincide with one selected reference solution as time advances, which motivates orbital asymptotic stability rather than demanding convergence to the same timed point on the cycle.

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.