Dynamical systems

At a hyperbolic equilibrium, the unstable manifold is formed by nearby trajectories approaching that equilibrium as time tends to minus infinity. It is tangent to the span of eigenvectors whose eigenvalues have positive real part. Its dimension counts departing amplitudes; translation along an autonomous trajectory can remove one parameter when comparing orbit shapes. The stable manifold instead approaches the equilibrium forward in time.

A local transverse section to a flow is mapped back to itself by the next intersection of an orbit. Fixed points represent periodic orbits, and the derivative of this return map determines their transverse stability.

If a planar equilibrium with purely imaginary linear eigenvalues has radial return map $r\mapsto r+\kappa r^3+O(r^4)$, then $\kappa <0$ gives nonlinear attraction and $\kappa >0$ gives repulsion. Vanishing linear decay therefore does not imply a centre.

The Routh-Hurwitz criterion tests whether all roots of a real polynomial have negative real part, by sign conditions on determinants formed from its coefficients. Applied to a characteristic polynomial, it gives an algebraic test for strict linear asymptotic stability.

Consider the real family $s^4+2\gamma s^3+(K+{\gamma }^2)s^2+2L\gamma s+P{\gamma }^2$ with fixed $K,L,P$. The Routh-Hurwitz criterion reduces its nontrivial determinant to $4{\gamma }^2[L(K-L)+{\gamma }^2(L-P)]$. Strict decay for every finite $\gamma >0$ is equivalent to the displayed inequalities: the constant and slope must be nonnegative, with at least one positive, and coefficient positivity requires $P>0$. The stricter chain $K>L>P>0$ is sufficient but excludes valid equality cases. Neither equality case guarantees a uniform decay margin as $\gamma$ tends to zero or infinity. At $\gamma =0$ there is no asymptotic attraction.

A real $2\times 2$ Jacobian matrix has both eigenvalues in the open left half-plane exactly when its trace is negative and its determinant is positive. At zero trace the test no longer determines nonlinear stability.

A heteroclinic orbit is a trajectory tending to one invariant equilibrium or orbit in backward time and a different one in forward time. Coupled convection amplitudes can have such trajectories describing replacement of one convection roll orientation by another.

A heteroclinic cycle is a closed chain of invariant states connected by heteroclinic orbits. In convection-amplitude models it can describe successive convection roll switching. Existence and attraction require the global connections and their transverse stability; a single convection roll's positive invasion growth rate alone does not establish an attracting cycle.

Phase locking is the maintenance of a fixed phase relation (possibly an integer combination of phases) between coupled oscillations, or between a pattern and imposed forcing. The following spatial-forcing example occurs in an amplitude equation. For a real-coefficient conjugately forced amplitude equation, $A=\rho e^{i\varphi }$ obeys ${\rho }_T=(\mu +\beta \cos 2\varphi -\lambda {\rho }^2)\rho$ and ${\varphi }_T=-\beta \sin 2\varphi$. Thus forcing selects stable phases $0,\pi$ for $\beta >0$ and $\pi /2,3\pi /2$ for $\beta <0$, when a nonzero amplitude exists. At $\beta =0$ this pinning disappears and the phase is neutral.

Topological dynamics studies continuous maps and their orbits on topological spaces. For a compact metric space $X$ and a continuous map $T:X\to X$, the pair $(X,T)$ is a compact Hausdorff topological dynamical system.

In a compact metric space with a continuous map $T$, points $x,y$ are proximal ifThis property is independent of the compatible metric. If $T$ is injective and $x\ne y$, arbitrarily small proximal distances must occur at arbitrarily large times, since each finite collection of distances is strictly positive. Proximal points need not be equal or have dense orbits.

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.

For a compact metric space with a continuous map $T$, suppose $y$ is a minimal point and $x,y$ are proximal. Every neighborhood $U$ of $y$ admits arbitrarily large $n$ with $T^{n}x,T^{n}y\in U$. First shrink $U$ to an open neighborhood, then choose an open $V$ with $y\in V$ and $\overline{V}\subseteq U$. Minimality and compactness give a finite cover of the orbit closure of $y$ by $T^{-j}V$, $0\le j\le J$. Uniform continuity of these finitely many iterates turns a sufficiently close proximal encounter into a common visit to $U$ after at most $J$ more steps. If the proximal encounters occur only at bounded times, a zero distance at some time makes the two future orbits coincide, and minimality then gives arbitrarily late common visits. Thus injectivity is not needed for this general lemma.

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 point is minimal when its forward orbit closure is a minimal dynamical system. This does not require the point to be a fixed point. In a finite-alphabet full shift, minimal points are exactly the uniformly recurrent sequences.

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.

The forward orbit closure of $x$ under a continuous map $T$ is the closure of its forward orbit. It is a closed forward-invariant set. If $T$ is invertible, one may instead consider the two-sided orbit closure using $n\in \mathbb{Z}$; the convention should be specified. For a minimal point of an invertible compact system the two closures agree.

Symbolic dynamics studies dynamical systems whose points are sequences of symbols from an alphabet and whose evolution is a shift. A finite alphabet with the discrete topology gives its sequence space a product topology.

A two-sided sequence over a finite alphabet is uniformly recurrent if every finite word over an alphabet occurring in it occurs with bounded gaps. More precisely, for each such word some $M$ ensures that every length-$M$ interval contains a complete occurrence. This is equivalent to being a minimal point of the full shift: a finite cover of the orbit closure by preimages of a word's cylinder set bounds its return gaps; conversely, bounded gaps pass to all points of the orbit closure and make every forward orbit dense there. The property concerns finite words, not infinite integer intervals.

The two-sided full shift on a finite alphabet $[k]$ consists of all functions $z:\mathbb{Z}\to [k]$, with the product topology and the left shift. It is a compact metric space. A compatible metric isAgreement on increasingly large finite coordinate sets is equivalent to convergence in this topology. The metric above is compatible but is not invariant under the left shift.

On a two-sided full shift, the left shift is the homeomorphismIts inverse sends $z(j)$ to $z(j-1)$. On a one-sided sequence space, the same forward shift is generally not invertible.

A dynamical system consists of a state space together with a rule that determines how its state evolves in time.

Stability theory studies the response of a dynamical system to small changes of its state. Linear stability of a planar equilibrium describes the response near equilibria using the linearized evolution.

A model retaining an oscillation's phase while neglecting amplitude dynamics. Interactions can synchronize relative phases through phase locking; noise can cause phase slips. The Adler phase equation is a simple example for a coupled relative phase.

An event changing the unwrapped relative phase by one cycle, usually $2\pi$, compared with a phase-locked state. Thermally activated phase slips correspond to crossing neighboring barriers of a tilted washboard potential. Successive slips can produce long-term drift even while small fluctuations appear confined on shorter times.

A noise-driven barrier crossing between adjacent phase-locked wells. The Kramers escape rate is proportional to $e^{-\Delta V/T_{\mathrm{eff}}}$ when a mobility-one Langevin equation has white-noise covariance $2T_{\mathrm{eff}}\delta (t-t^{\prime })$. Both the barrier and local relaxation time determine whether an intrawell phase autocorrelation approximation is appropriate.

In a mobility-one noisy Adler phase equation, matching Kramers escape rate prefactors and a barrier difference $2\pi \omega$ give $p_{+}/p_{-}=e^{2\pi \omega /T_{\mathrm{eff}}}$. Positive mismatch favors forward winding. The ratio is a low-noise barrier-crossing result, not a claim that the particle remains in one well at arbitrarily long times.

The relative-phase drift $\dot{\theta }=\omega -ϵ\sin \theta$ describes competing frequency mismatch and phase coupling. For $0<\omega <ϵ$, the equilibrium points are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$, stable and unstable respectively. They merge in a saddle-node bifurcation at $\omega =ϵ$; greater mismatch gives running phase dynamics. Adding Gaussian white noise gives Overdamped Langevin dynamics in a tilted washboard potential.

For $0<\omega <ϵ$, set $\alpha =\arcsin (\omega /ϵ)$ and $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. The neighboring saddles of the tilted washboard potential have barriers $\Delta V_{+}=2\kappa -\omega (\pi -2\alpha )$ and $\Delta V_{-}=2\kappa +\omega (\pi +2\alpha )$ above the minimum at $\alpha$. Their difference is $2\pi \omega$; both have curvature $-\kappa$, so forward and backward Kramers escape rates have equal leading prefactors.

A phase that continues winding rather than tending to a locked value. In the noise-free Adler phase equation, $\omega >ϵ>0$ makes the drift positive at every phase, so there are no equilibrium points. The associated tilted washboard potential decreases monotonically; wells vanish at the locking threshold.

A skew product evolves a base coordinate by a dynamical system $R$ and a fibre coordinate by a map $S_x$ depending on the current base coordinate: $F(x,y)=(R(x),S_x(y))$. The base evolution is independent of the fibre, while the fibre may depend on the base. Continuity or measurability is imposed according to the setting. An irrational skew shift is a useful affine example on a torus.

For irrational $\alpha$, the irrational skew shift is the skew product $T_{\alpha }(x,y)=(x+\alpha ,y+x)$ on the torus $(\mathbb{R}/\mathbb{Z}{)}^2$. It preserves normalized Lebesgue measure and has iterates $T_{\alpha }^n(x,y)=(x+n\alpha ,y+nx+\alpha n(n-1)/2)$ modulo one. Its Koopman operator sends the Fourier basis character $e_{r,s}=e^{2\pi i(rx+sy)}$ to $e^{2\pi ir\alpha }{e}_{r+s,s}$. The resulting infinite chains of Fourier coefficients show that it is an ergodic transformation. Moreover, uniform equidistribution of an irrational skew shift proves unique ergodicity.

For an irrational skew shift, the averages of every continuous function $g$ converge uniformly in the starting point to $\int gd{m}_2$, with $m_2$ normalized Lebesgue measure. Nonconstant Fourier basis characters have either linear or quadratic phases. Linear phases are bounded geometric series; for quadratic phases the Van der Corput inequality for finite scalar sequences reduces to linear correlations of irrational frequency, uniformly in the starting point. Approximation by trigonometric polynomials proves the assertion and identifies every invariant Borel probability measure as $m_2$.

The state space of a dynamical system is the set of all states that the system may occupy. An evolution rule traces an orbit through this space from each admissible initial state.

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

An autonomous differential equation has the form $\dot{x}=f(x)$, with no explicit dependence of the vector field $f$ on time.

A state vector collects variables that determine a system's instantaneous state so that its evolution can be written as a first-order equation $\dot{x}=F(x,t)$.

Instability means that arbitrarily small perturbations can produce departures that do not remain uniformly small. Linear instability is commonly detected by an eigenmode with positive growth rate.

For an autonomous ordinary differential equation, the flow map ${\phi }_t$ sends each initial state to its state after time $t$. Uniqueness gives ${\phi }_{t+s}={\phi }_t\circ {\phi }_s$ and ${\phi }_{-t}={\phi }_t^{-1}$ whenever both sides exist.

For a smooth flow map ${\phi }_t$ generated by a vector field $F$, the chain rule and $\det (I+tDF)=1+t\mathrm{tr}(DF)+o(t)$ give ${\partial }_{t}J[{\phi }_t](x)=(\nabla \cdot F)({\phi }_t(x))J[{\phi }_t](x)$. With initial value one, its solution is the exponential of the time integral of this divergence. Thus a divergence-free generator preserves Euclidean volume and orientation. This connects the Jacobian determinant with a volume-preserving vector field.

A phase portrait depicts representative orbits of an autonomous differential equation in its state space, together with equilibria, invariant curves, and the direction of the flow.

A nullcline of a planar autonomous differential equation is a set where one component of its vector field vanishes, typically drawn as a curve. Intersections of the two component nullclines are equilibrium points. The sign of each component on either side helps determine the directions in a phase portrait.

This reflection-symmetric planar system has equilibria $(0,\pm 1)$ and, for $|a|\le 1$, $(\pm \sqrt{1-a^2},a)$. Its divergence is the constant $-2a$, excluding periodic orbits when $a\ne 0$ by the Bendixson-Dulac criterion. At $a=0$ it is Hamiltonian withand the nonzero equilibria are nonlinear centers. At $a=\pm 1$ the symmetric equilibria undergo pitchfork bifurcations.

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.

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.

Suppose $V$ is a first integral of an unperturbed planar system and $\dot{V}=\epsilon G+O({\epsilon }^2)$ after perturbation. A periodic orbit converging to an unperturbed orbit $\gamma$ can persist only ifIndeed, the net change of $V$ over every period is zero. A nonzero averaged drift therefore obstructs persistence.

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA vanishing drift is a necessary first-order selection condition for a persisting periodic orbit. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting limit cycle. In the Hamiltonian limit of three-to-one forcing, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.

Forthe perturbation gives $\dot{H}=\epsilon (1-\alpha x^2)y^2$. Along an unperturbed orbit $H=H_0$ with turning points $x_1,x_2$,

On either unperturbed homoclinic loop, $H_0=0$ and the positive-side turning points are zero and $\sqrt{2}$. The first-order persistence condition iswhich gives $\alpha =5/4$.

For the periodic energy levels $-1/4<H_0<0$ enclosing one well, the energy-balance value of $\alpha$ ranges from one at the center to $5/4$ at the homoclinic loop. Thus for small positive $\epsilon$, one expects one symmetric pair of single-well periodic orbits when $1<\alpha <5/4$.

The Lotka-Volterra predator-prey equations arewith positive parameters. Their positive equilibrium is surrounded by closed level curves of a logarithmic first integral.

For the normalized equations $\dot{x}=x(1-y)$ and $\dot{y}=ry(x-1)$ on the positive quadrant,is a first integral. It is nonnegative, strictly convex, and tends to infinity at the boundary and at infinity. Every positive regular level is therefore a compact simple closed curve traversed by one periodic orbit.

The Poincare index of a planar vector field around a simple closed curve containing no zero on the curve is the winding number of the vector-field direction along that curve.

A planar periodic orbit, oriented by the flow, has Poincare index $+1$. The index theorem equates this with the sum of the indices of the isolated equilibria enclosed by the orbit. A hyperbolic equilibrium has index $+1$ when its Jacobian matrix has positive determinant and index $-1$ when it is a saddle equilibrium.

A simply connected region containing no equilibrium, or containing equilibria whose total Poincare index is not $+1$, cannot contain a periodic orbit that encloses exactly those equilibria.

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.

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.

For a map $F:X\to X$, its iterates are defined by $F^0$ equal to the identity map and $F^{n+1}=F\circ F^n$.

A map on $(0,1)$ can satisfy $0<f(x)<x$ and still have no orbit tending to zero. Partition the domain into $I_j=(1/(j+1),1/j]\cap (0,1)$ and define $f(x)=(x+1/(j+1))/2$ on $I_j$. Every interval is invariant under the iteration of a map, and $f^n(x)=1/(j+1)+2^{-n}(x-1/(j+1))$ has a positive limit. The discontinuities at interval boundaries prevent that limit from being a fixed point. In contrast, continuity of a self-map satisfying $f(x)<x$ permits passage to a positive orbit limit and would force an impossible fixed point.

Topological entropy measures the exponential growth rate of distinguishable orbit segments of a dynamical system.

A dynamical system has positive topological entropy when $h_{\mathrm{top}}>0$; this expresses exponential orbit complexity.

A continuous interval map has positive topological entropy if and only if some positive iterate has a horseshoe for an interval map.

An interval map is a continuous function $F:I\to I$ from a real interval to itself. Its iterates form a one-dimensional discrete dynamical system.

For a real number $\beta >1$, the beta transformation is the interval mapon $[0,1)$. Each interval of continuity is expanding with slope $\beta$.

The full tent map on the unit interval isIts two affine branches expand lengths by two, and it preserves Lebesgue measure.

Given a finite measurable partition of an interval, the itinerary of $x$ records which partition element contains each iterate $T^n(x)$. Prescribing a finite initial word defines an itinerary cylinder. For the full tent map, every length-$N$ binary cylinder has Lebesgue measure $2^{-N}$ up to endpoint conventions, so its itinerary process is a fair i.i.d. Bernoulli process.

An itinerary cylinder is the set of points whose first finitely many itinerary symbols equal a prescribed finite word. Such cylinders generate the symbolic σ-algebra and pull back to finite intersections of inverse images of partition elements.

The logistic map is the one-parameter family $F_{\mu }(x)=\mu x(1-x)$ on the unit interval. As $\mu$ increases it exhibits a period-doubling cascade, periodic windows and chaotic parameter ranges.

A point $x$ is periodic for an interval map $F$ when $F^n(x)=x$ for some positive integer $n$. Its least such $n$ is its period, and the finite setis its periodic orbit.

For closed intervals $J$ and $K$, write $J⟶K$ under $F$ when $F(J)\supseteq K$. The intermediate value theorem implies that there is a closed subinterval $L\subseteq J$ with $F(L)=K$.

Given finitely many intervals $J_1,\dots ,J_r$, their directed covering graph has an directed edge $J_i\to J_j$ whenever $F(J_i)\supseteq J_j$.

Every closed walkin a directed covering graph of an interval map has a point $x\in J_0$ with $F^j(x)\in J_j$ and $F^n(x)=x$. This follows by successively pulling $J_0$ back through the covering relations and applying the fixed-point property of a closed interval. If the itinerary has least period $n$ and avoids shared endpoints, $x$ lies on an $n$-cycle.

If $A$ is the adjacency matrix of a directed covering graph, then $\mathrm{tr}(A^n)$ counts its pointed closed walks of length $n$. For a prime number $p$, subtracting the $\mathrm{tr}(A)$ constant walks and identifying the $p$ cyclic choices of starting point givesprimitive closed itineraries of length $p$.

For prescribed values $F(x_i)$ at ordered points $x_0<\cdots <x_n$, the connect-the-dots interval map is the unique piecewise-linear function obtained by linear interpolation between consecutive data points.

The Jury criterion tests whether every root of a real discrete-time characteristic polynomial lies strictly inside the unit circle. For $z^2+az+b$, this is equivalent to $|b|<1$, $1+a+b>0$, and $1-a+b>0$.

For a period-$k$ orbit $x_0,\dots ,x_{k-1}$ of a differentiable map, the multiplier isThe orbit is locally asymptotically stable when the modulus of this product is less than one.

A period-doubling bifurcation occurs when a fixed-point multiplier crosses $-1$ and a nearby period-two orbit is created. Under the generic nondegeneracy conditions, the orbit amplitude is proportional to the square root of the parameter displacement.

For $F_{\mu }(x)=\mu x+b{x}^2+a{x}^3$ near $\mu =-1$, write $\mu =-1+ϵ$. Then $F_{\mu }^2(x)-x=-2ϵx-2(a+b^2)x^3+O({ϵ}^{2}x,ϵx^2,x^4)$. The small two-cycle is attracting on $\mu <-1$ if $a+b^2>0$ and repelling on $\mu >-1$ if $a+b^2<0$. Vanishing of $a+b^2$ gives a degenerate flip requiring higher terms.

For a $C^4$ real map $f(u)=-u+c_2{u}^2+c_3{u}^3+O(u^4)$, the second iterate isIf $c_3+c_2^2>0$, sufficiently small nonzero $u$ retains its sign under $f^2$ and strictly decreases in absolute value. Its iterates therefore converge to zero, proving local asymptotic stability even though $f^{\prime }(0)=-1$. If $c_3+c_2^2<0$, the second iterate moves small positive points away from zero, proving instability. The zero-coefficient case requires higher-order terms. With strictly stable transverse directions, the centre manifold theorem for a discrete dynamical system applies this scalar test to a higher-dimensional discrete dynamical system.

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 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.

For the system in the 2024 Part II Paper 2 question at $r=1$, with
$v=(x+y)/2$ and $w=(x-y)/2$, the center manifold isIts reduced equation is

Forset $X=x-a^2$ and $Y=y-a^2$, and append $\dot{a}=0$. Near $(X,Y,a)=(0,0,0)$, the extended centre manifold and its reduced dynamics areThe reduced equation is a subcritical pitchfork with reversed parameter $\mu =-a$.

Forthe invariant center manifold near $(x,y,k)=(0,0,0)$ is $y=0$, with reduced equation $\dot{x}=x(-k-3x+x^2)$ and hence a transcritical bifurcation. Near $(x,y,k)=(1,2,0)$, put $X=x-1$ and $v=y-2-X$. The extended center manifold beginsand its reduced equation is $\dot{X}=2(X^2-k)+O(3)$, a saddle-node bifurcation.

Forthe extended centre manifold for a parameter gives the local reduced equationswhere $y_0=\pm \sqrt{2}$. Thus the first two points undergo respectively supercritical and subcritical symmetry-forced pitchfork bifurcations, while each of the last two points undergoes a saddle-node bifurcation.

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

An interval map $F:I\to I$ has a horseshoe when there are an open interval $J\subseteq I$ and disjoint open intervals $K_0,K_1\subseteq J$ such thatIterating inverse branches produces full two-symbol itinerary dynamics.

If a directed covering graph of an interval map has two distinct closed walks of the same length $n$, based at the same interval and with different first edges, successive use of the interval covering relation produces two subintervals with disjoint interiors that $F^n$ maps across the base interval. Hence $F^n$ has a horseshoe for an interval map.

Order the positive integers by odd numbers from $3$ upward, then twice the odds, then four times the odds, and so on, followed by descending powers of two and finally $1$. If a continuous interval map has a cycle of one period, it has cycles of every period later in this order. In particular, period three forces every positive period.

A period-three orbit forces an interval-covering graph with adjacency matrixIts number of closed pointed length-$n$ itineraries is $\mathrm{tr}(A^n)=L_n$. Möbius removal of lower periods and division by $n$ give four primitive cyclic classes at $n=7$ and five at $n=8$.

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

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 map $F:X\to X$ is topologically transitive when, for every pair of nonempty open sets $U,V$, some iterate satisfies $F^n(U)\cap V\ne \varnothing$.

Periodic points are dense when every nonempty open set contains a point fixed by some positive iterate.

There is a constant $\delta >0$ such that every neighbourhood of every point contains a second point whose orbit eventually separates from the first by more than $\delta$.

The doubling map on the unit circle isIt is chaotic in Devaney's sense.

The doubling map preserves normalized Lebesgue measure on the unit circle. Each point has exactly two preimages, $x/2$ and $(x+1)/2$ modulo one, and each inverse branch scales lengths by $1/2$.

If $x=0.b_1{b}_2{b}_3\dots$ in base two, thenBinary cylinders make transitivity and density of periodic points transparent: concatenate prescribed finite blocks for transitivity and repeat a finite block for a periodic point.

The fixed points of $F^n$ areThus $F^n$ has $2^n-1$ fixed points.