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If distinct points $x,y$ of a dynamical system on a compact metric space satisfy $d(T^{n}x,T^{n}y)\to 0$ in one compatible metric, no compatible metric can make $T$ an isometry. On a compact space, all compatible metrics give the same asymptotic-pair property by uniform continuity, whereas an isometry preserves the strictly positive distance between distinct points. In a full shift, a constant sequence and a sequence differing at just one coordinate converge to each other under forward shifts, proving this obstruction directly in the product topology.

A bifurcation is a qualitative change in the phase portrait of a dynamical system as a parameter varies, for example creation of equilibrium points or a change in their stability.

Causal state filtering estimates the current state of a dynamical system from observations up to the current time. Under quadratic loss, the conditional expectation is an optimal estimate. A Kalman filter obtains this estimate recursively for a linear Gaussian state-space model, without using future observations.

A center equilibrium of a planar dynamical system is surrounded by closed trajectories. It is stable but not asymptotically stable.

Controllability asks whether the available control inputs can move a dynamical system between every pair of states in its declared state space. Fixing a duration $T>0$ gives the precise condition ${\mathcal{R}}_T(x_0)=X$ for every $x_0\in X$, where $X$ is the state space and ${\mathcal{R}}_T(x_0)$ its reachable set. With freely chosen duration, every pair must instead be connectable in some permitted finite time, which may depend on the pair. Always specify admissible control inputs, deadlines and any constraints along the path. ^[ref]

The surprising part is that one control input can control several state variables by acting at different times. For the double integrator, acceleration changes velocity directly, while the accumulated velocity changes position. Starting from rest, accelerate during the first half of the journey and brake during the second: the final velocity can vanish although the final position has changed. The figure is an analytic mathematical example in dimensionless quantities, with no experimental measurements.

Controllability is useful before trying to design a controller: a requested endpoint outside the reachable set cannot be attained by clever choice of feedback control. The Kalman rank criterion makes this an exact linear algebra calculation for the stated linear time-invariant system model. It concerns whole states, so attaining one plant output is not enough to establish controllability. In quantum control, the declared state space may be an orbit constrained by conservation laws; see quantum controllability and unitary orbit of a density operator.

Control theory studies how to choose control inputs for a dynamical system so that its plant output follows a reference signal, despite control disturbances and imperfect knowledge of the controlled plant. A controller turns the available information into a control signal. The design must consider both the resulting response and stability theory: a rule that corrects a small mismatch can still produce growing deviations if the interconnection is badly chosen. ^[ref]

The central distinction is informational. Open-loop control executes planned commands; feedback control uses the measured output of the actual controlled plant to update those commands. A reference signal can describe a destination, while control disturbances describe influences that the controller did not command. A model can predict where the controlled plant should be; measurement tells the controller what actually happened. The authors provide a free full-book PDF through ^[ref].

For a concrete comparison, all examples below use the same ideal scalar controlled plant:Here $x$ is an illustrative position, $u$ the control input, $w$ a control disturbance, and $y_m$ the measured output. Both $x$ and $t$ are dimensionless quantities; the numbers are chosen to expose the mechanism, with no empirical interpretation. The ordinary differential equation says that the present control input and control disturbance accumulate into $x$. The measured output equals the entire state only in this ideal example; a general plant output need not reveal the entire state.

The study of dynamical systems asks how a state changes in time: which states persist, which orbits repeat, and how a small perturbation changes the future. A dynamical system combines a state space with an evolution rule. For a seasonal population, the state might be its abundance and one step one generation; for a harmonic oscillator, a state vector includes both position and velocity. Choosing a sufficient state is essential: position alone does not determine where an oscillator moves next.

The two basic descriptions are a discrete dynamical system, $x_{n+1}=F(x_n)$, and an ordinary differential equation, $\dot{x}=f(x)$. Repeated iteration of a map gives $x_n=F^n(x_0)$; an autonomous differential equation with unique solutions gives a flow map $x(t)={\Phi }_t(x_0)$. In either case, evolving for two successive durations agrees with evolving for their sum. The logistic map provides a particularly simple discrete dynamical system; the Lorenz system supplies a continuous-time example with a very different geometry.

Deterministic evolution says that the exact rule and exact initial condition specify one future. Predicting that future from measurements is a separate question. An instrument supplies a range of possible initial conditions, and a computation adds small errors. Near an asymptotically stable fixed point, nearby orbits converge and the long-term state can become easier to predict. With sensitive dependence on initial conditions, arbitrarily close starting states can eventually lead to a visible disagreement, even though each follows the same rule.

Much of the usefulness of dynamical systems comes from describing outcomes without writing every solution explicitly. Fixed points identify possible resting states; periodic orbits identify repeating behaviour; stability theory asks whether those behaviours survive small disturbances. A bifurcation diagram shows how changing a bifurcation parameter changes the possible outcomes. This is useful in population dynamics, where a stable abundance and an alternating abundance have different consequences, and in thermal convection, where a steady flow and a changing flow transport heat differently.

The beauty of the subject is that a small formula can organize a large variety of behaviour. The logistic map moves from an attracting fixed point to attracting periodic orbits and then to parameter values supporting sensitive motion. Its period-doubling cascade has a recognizable structure across many unimodal maps. Nevertheless, Devaney chaos, Glendinning chaos, and a positive Lyapunov exponent express different properties; a claim of chaos should specify its definition, its state space, and the parameter values under consideration.

Evolutionary game theory studies how a population's composition changes when the success of a pure strategy depends on which other pure strategies it encounters. The central feedback is a frequency-dependent payoff: becoming common changes the opponents a strategy meets, which can change whether it does well. In a coordination game, common behaviour can reinforce itself; in the coexistence example below, being rare is an advantage. This makes evolutionary game theory useful for asking which behaviours can persist and which initial compositions lead to different outcomes, without assuming that each participant solves an optimization problem.

In the basic model, a large, well-mixed population has finitely many inherited pure strategies; interactions sample opponents according to their population frequencies. Assume that a strategy's expected payoff determines its per capita net reproductive growth rate, with a common baseline allowed. These are modelling assumptions: reproduction must transmit the strategy, encounters must match the proposed sampling rule, and the conversion from payoff to growth must be justified for an application. The replicator equation below describes the resulting continuous-time frequencies. Taylor and Jonker's ^[ref] explicitly derive this growth model.

For learned behaviour, reproduction can instead be replaced by an assumed copying rule: an individual samples another individual uniformly and copies a better-performing pure strategy at a rate proportional to the positive payoff difference. At frequencies $x_i,x_j$, the net flow from $j$ to $i$ is $x_i{x}_j(f_i-f_j)$, because the two directional positive parts subtract to $f_i-f_j$. Summing over $j$ gives $x_i(f_i-\bar{f})$, the replicator equation. This derivation states a particular mechanism; an arbitrary rule for learning or imitation need not give the same dynamical system.

Three questions must be distinguished. A symmetric Nash equilibrium tests whether an individual can improve its expected payoff against the resident composition. An evolutionarily stable strategy tests whether each distinct rare mutant is outperformed after it is introduced. Dynamical stability tests the response of an actual dynamical system to small changes of composition, and asymptotic stability additionally requires nearby trajectories to converge. For a finite symmetric bimatrix game under the continuous-time replicator equation, every evolutionarily stable strategy is a symmetric Nash equilibrium and a locally asymptotically stable rest point; the converse from asymptotic stability to an evolutionarily stable strategy fails in general. See Ross Cressman and Yi Tao, The replicator equation and other game dynamics (2014), Theorems 1–2 and Example 1. These statements refer to stability on the full probability simplex, including perturbations introducing currently absent pure strategies.

A concrete experiment makes the frequency feedback visible. In 2009, Gore, Youk and van Oudenaarden competed two yeast strains: a producer of the enzyme invertase, which enables sucrose hydrolysis, and a mutant lacking the corresponding SUC2 gene. Released sugars can feed neighbours, so avoiding production can be advantageous. They used shaken 5 mL cultures with 5% sucrose at $30{}^{\circ }\mathrm{C}$, diluted daily after 23 hours of growth. Different labelled proteins allowed the two strains to be counted separately by flow cytometry; selective plating periodically checked the counts. The cooperator required externally supplied histidine, so restricting that amino acid imposed an adjustable growth disadvantage. See Snowdrift game dynamics and facultative cheating in yeast, pp. 253 and 255–256, Methods Summary.

Starting with rare producers and, separately, rare nonproducers tested both invasion directions. Under the reported coexistence conditions each rare strain increased; changing histidine or added glucose shifted the eventual composition. Their explanation used nonlinear growth benefits, rather than a fixed pairwise payoff matrix. This motivates testing both rarity regimes before inferring an evolutionarily stable strategy or fitting a replicator equation; it does not identify the illustrative tables below with measured payoffs. See Gore, Youk and van Oudenaarden, Figures 1–3.

An early motivation was understanding why animal contests can involve restraint even when escalation might win an individual encounter. Maynard Smith and Price's ^[ref] introduced the evolutionarily stable strategy in this setting. Taylor and Jonker's ^[ref] then studied explicit continuous and discrete dynamics. Static resistance to invasion and the paths populations follow are related questions, but they require separate definitions.

A fold-Hopf bifurcation combines a simple zero eigenvalue and a nonzero imaginary pair at an equilibrium point. In a three-dimensional centre manifold, a normal form of a dynamical system separates a real fold amplitude from a complex oscillatory amplitude. A rotationally symmetric normal form permits reduction to two real amplitude equations, while the oscillation phase continues to evolve. Secondary bifurcations of the amplitude flow can produce invariant tori in the full dynamical system.

For a continuously differentiable dynamical system $\dot{x}=f(x)$, a fixed point is hyperbolic if no eigenvalue of its Jacobian has zero real part. In a neighborhood of a hyperbolic fixed point, the nonlinear flow is locally topologically conjugate to its linearized flow. Thus the saddle, attracting or repelling character persists under the nonlinear terms. The theorem does not decide the nonlinear character of an equilibrium with purely imaginary or zero eigenvalues: a conserved Hamiltonian, Lyapunov function or higher-order analysis may be needed instead.

A plant input is a signal supplied to the controlled plant. The chosen plant input is its control input; other influences can enter as control disturbances. This use concerns a dynamical system, rather than the input of a computation.

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.

For a finite-dimensional continuous-time linear time-invariant system $\dot{x}=Ax+Bu$ on ${\mathbb{R}}^n$, with constant matrices $A,B$ over the real numbers, no state constraints, and every piecewise continuous function $u:[0,T]\to {\mathbb{R}}^m$ allowed as a control input without an amplitude bound, controllability in any fixed $T>0$ is equivalent to the controllability matrix having rank of a matrix $n$. The same equivalence holds with all square-integrable functions as inputs. A stronger useful statement iswhere $e^{AT}$ is a matrix exponential and $\mathrm{im}\mathcal{C}$ is the column space. Thus a rank of a matrix $r<n$ leaves only an $r$-dimensional affine subspace reachable at that time. ^[ref]

The Kalman rank criterion allows arbitrarily large commands as $T$ shrinks. It does not assert controllability with bounded control inputs, nor test a general nonlinear dynamical system just by counting its inputs. For the double integrator, $B=(0,1{)}^T$ and $AB=(1,0{)}^T$ give two independent directions, despite there being only one control input.

A limit cycle is an isolated periodic orbit of a dynamical system. It is attracting if nearby orbits approach it in forward time. A continuous family of closed orbits around a center equilibrium does not consist of limit cycles, because its members are not isolated.

A Lyapunov exponent measures the asymptotic exponential rate of an infinitesimal tangent displacement under a dynamical system, when the displayed limit exists for a nonzero tangent vector $v$. A positive exponent represents exponential separation along that direction. In a smooth velocity field, the tangent displacement obeys $\dot{\mathbf{l}}=(\nabla \mathbf{u})\mathbf{l}$. Finite-separation growth eventually leaves this linear regime; moments of random separation need not grow at the same rate as typical separation.

A nonempty topological dynamical system $(Y,T)$ on a compact Hausdorff space is minimal if it has no proper nonempty closed forward-invariant subset. Equivalently, every forward orbit is dense in $Y$, because an orbit closure is closed and forward invariant. Minimality implies $T(Y)=Y$.

A minimal subsystem is a nonempty closed forward-invariant subset on which the restricted dynamical system is a minimal dynamical system. Every continuous map on a nonempty compact Hausdorff space has one: intersections of chains of nonempty closed invariant sets remain nonempty by compactness, so the Zorn lemma gives a minimal member.

A real number $x$ is a normal number in base $K$ exactly when its iterates under the integer multiplication map on the circle, $T_K(x)=\{Kx\}$, form an equidistributed sequence. Braces denote the fractional part. Multiplication by $K$ shifts the positional notation one digit to the left:A length-$r$ word with base-$K$ integer value $a$ specifies the half-open interval $[a/K^r,(a+1)/K^r)$, whose length is $K^{-r}$. Visiting that interval is exactly encountering that word at the next digit position. These digit intervals form arbitrarily fine grids, so their correct limiting visiting frequencies imply equidistribution on every interval. This converts a question about digits into one about a dynamical system. Bailey and Crandall develop this connection using equidistribution of the fractional parts of $K^{n}x$. ^[ref]

In base ten the word $12$ corresponds to $[0.12,0.13)$. For Champernowne constant, counting the visits $T_{10}^n(x)$ for $0\le n\le 99998$ gives precisely the $2088$ occurrences of $12$ in its first $100000$ digits. For $x=1/3$, the base-two integer multiplication map on the circle instead alternates between $1/3$ and $2/3$; it never visits $[0,1/4)$, the interval for $00$. This two-point orbit is not an equidistributed sequence.

An orbit of a dynamical system is the set or time-ordered trajectory of states reached from one initial state under the evolution rule.

First analyze the supplied normal form as a dynamical system in its own right. Its equilibrium points are $O=(0,0)$ and, for ${\mu }_2>0$, $S_{\pm }=(\pm \sqrt{{\mu }_2},0)$. At $O$, the trace is ${\mu }_1$ and the determinant is ${\mu }_2$. At $S_{\pm }$, the determinant is $-2{\mu }_2$, so both are saddle equilibria. Thus ${\mu }_2=0$ is the pitchfork bifurcation curve, and ${\mu }_1=0,{\mu }_2>0$ is the Hopf bifurcation curve. With the allowed supercritical Hopf bifurcation, an attracting limit cycle is born on ${\mu }_1>0$ around the repelling origin.

The Hamiltonian $H=y^2/2+{\mu }_2{x}^2/2-x^4/4$ obeys $\dot{H}=({\mu }_1-x^2)y^2$. Set $s={\mu }_2>0$, $x=\sqrt{s}X$, $y=sY$, $\tau =\sqrt{s}t$, and ${\mu }_1=cs$. The limiting Hamiltonian system has two oppositely directed heteroclinic orbits with $Y=\pm (1-X^2)/\sqrt{2}$ for $-1<X<1$. The Melnikov energy-balance method givesIts unique simple zero gives a heteroclinic cycle curveThe two connections occur on the same curve because reflection interchanges them. The attracting limit cycle grows toward this heteroclinic cycle, then disappears. In the small unfolding neighborhood there is one such connection curve, rather than two independently split curves. There are no nearby homoclinic orbits replacing it: the two equal-height outer saddles are the relevant separatrix endpoints.

This calculation transfers to the original equations only if the asserted reduction is valid. The cubic calculation in part (c) shows it is not valid for the printed coefficients. Nevertheless, the claim of a single nearby global bifurcation curve for the original equations can be proved independently. Put $\eta =-(\kappa +1)>0$, $\lambda =3+{\eta }^{2}L$, $x=\varphi =\sqrt{\eta }X$, $y=\dot{\varphi }=\eta Y$, and $\tau =\sqrt{\eta }t$. Eliminating $\theta$ using $\sin q=(y+\sin 2x)/(\cos 2x-\kappa )$ producesThe order-$\eta$ correction is reversible and has no first-order separatrix energy imbalance. On the limiting positive connection, $-1/2<X<1/2$ and $Y=1/2-2X^2$. The first dissipative balance isAgain the zero is simple, and reflection makes the two heteroclinic orbits simultaneous. Hence the actual original equations have the unique local connection curveThe unstable limit cycle on the subcritical Hopf bifurcation side reaches this curve. Thus the original system still has one nearby global connection curve, but its tangency and cycle stability differ from those of the supplied assumed cubic normal form. These statements concern connections contained in the shrinking neighborhood of $P_1$; they do not exclude unrelated distant global bifurcations elsewhere on the sphere.