Dynamical systems

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.

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 $F^{n+1}=F\circ F^n$.

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.

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.

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.

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

An interval map $F$ has a horseshoe when there are closed intervals $J_0,J_1$ with disjoint interiors 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.

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.

For $n=2^k$, every proper period dividing $n$ divides $n/2$. Hence the number of points of exact period $2^k$ is

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.

Appending $\dot{\mu }=0$ turns a system parameter into a centre variable, allowing one invariant graph to describe nearby parameter values.

If a centre manifold is the graph $v=h(u)$ for $\dot{u}=f(u,v)$ and $\dot{v}=g(u,v)$, its coefficients satisfy $Dh(u)f(u,h(u))=g(u,h(u))$.

Bifurcation theory studies qualitative changes in equilibria and invariant sets as parameters vary.

A simple eigenvalue crossing the imaginary axis signals loss of hyperbolicity and a possible local bifurcation, while the remaining eigenvalues stay away from it.

A saddle-node bifurcation occurs when a stable and an unstable equilibrium coalesce at a nonhyperbolic equilibrium and disappear as a parameter crosses a critical value.

The supercritical pitchfork normal form is $\dot{x}=\mu x-x^3$: the trivial branch is stable for $\mu <0$ and two stable nonzero branches emerge for $\mu >0$.

The normal form $\dot{x}=\mu x+x^3$ has a stable trivial equilibrium for $\mu <0$ and two unstable nonzero equilibria for $\mu <0$. They collide with the trivial branch at $\mu =0$, after which that branch is unstable.

Reflection symmetry $x\mapsto -x$ forces the reduced vector field to be odd in $x$, naturally producing paired nonzero equilibrium branches.

A bifurcation diagram plots equilibrium values against a parameter and distinguishes stable, unstable, and nonhyperbolic branch segments.

The equilibria of $\dot{x}=(a^2-x)(a-y^2)$, $\dot{y}=x-y$ lie onfor every $a$, and on $x=y=\pm \sqrt{a}$ for $a\ge 0$. They meet in a subcritical pitchfork bifurcation at $a=0$; the positive square-root branch meets the $a^2$ branch and exchanges stability in a transcritical bifurcation at $a=1$.

A Hopf bifurcation occurs when a complex-conjugate pair of eigenvalues crosses the imaginary axis and a periodic orbit is created or destroyed near the equilibrium.

In the radial normal forman unstable periodic orbit exists for $\mu <0$ and shrinks into the equilibrium as $\mu \uparrow 0$. This is a subcritical Hopf bifurcation.

A stable and an unstable periodic orbit coalesce into one semistable periodic orbit and disappear at a saddle-node bifurcation of periodic orbits.

Fornonzero periodic orbits haveThey are born together at $\mu =-{\lambda }^2/4$; the inner unstable orbit then vanishes in a subcritical Hopf bifurcation at $\mu =0$, while the outer orbit is stable.

In a quadrant-invariant planar system, a stable interior equilibrium branch may emerge from one boundary equilibrium and terminate at another, transferring stability at each endpoint.

Successive symmetry-breaking bifurcations can pass stability from one boundary branch to an interior branch and then to a second boundary branch.

$V(x_0)=0$, $V>0$ elsewhere, and $\dot{V}<0$ prove asymptotic stability; suitable sublevel sets lie in the basin.

At a hyperbolic fixed point, stable and unstable manifolds are tangent to the corresponding eigenspaces. Power-series coefficients follow from graph invariance.

An autonomous planar system is an ordinary differential equationwhose vector field does not depend explicitly on time.

An equilibrium point, or fixed point, of $\dot{z}=f(z)$ is a state $z_{\ast }$ with $f(z_{\ast })=0$.

For a hyperbolic equilibrium of a smooth planar system, the eigenvalues of the Jacobian matrix determine local stability. If its determinant and trace are positive, both eigenvalues have positive real part and the equilibrium is a repeller.

A planar hyperbolic equilibrium is a saddle when the Jacobian determinant is negative. It has one stable and one unstable eigendirection and corresponding one-dimensional invariant manifolds.

A planar equilibrium is a stable node when its two Jacobian eigenvalues are real and negative. Every nearby trajectory approaches it, tangent asymptotically to an eigendirection.

A planar equilibrium is a stable spiral when its Jacobian has a complex-conjugate pair of eigenvalues with negative real part. Nearby nonstationary trajectories spiral toward it.

If $D$ is simply connected and a $C^1$ function $B$ makes $\nabla \cdot (BF)$ have one sign and not vanish identically on any open subset of $D$, then the planar system $\dot{x}=F(x)$ has no periodic orbit lying in $D$.

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.

A periodic orbit is a nonconstant trajectory that returns to its initial state after some least positive period.

A compact planar limit set containing no equilibrium is a periodic orbit.

A Floquet multiplier is an eigenvalue of the derivative of a period map and determines transverse stability of a periodic orbit.

For a planar periodic orbit $\gamma$ of period $T$, its nontrivial Floquet multiplier isThe orbit is asymptotically stable when the integral is negative and unstable when it is positive.

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 fixed point $x_{\ast }$ of $x_{n+1}=g(x_n)$ is locally asymptotically stable when $|g\prime (x_{\ast })|<1$.

Resonance is the large response produced when periodic forcing aligns with a natural mode.

An equilibrium is unstable when arbitrarily small perturbations can move trajectories away from it.

An equilibrium is a state where the autonomous vector field vanishes.

Linear stability analysis classifies a hyperbolic equilibrium from the eigenvalues of the vector field's Jacobian.

For a real $2\times 2$ Jacobian matrix $J$, both eigenvalue have negative real part exactly whenThe linearization stability theorem then makes a hyperbolic equilibrium locally asymptotically stable.

The basin of an attractor is the set of initial states whose forward trajectories converge to it.

A Lyapunov function near an equilibrium $x^{\ast }$ is continuously differentiable, satisfies $V(x^{\ast })=0$ and $V(x)>0$ away from $x^{\ast }$, and has orbital derivative

An equilibrium is Lyapunov stable if every neighbourhood contains a smaller neighbourhood whose forward trajectories remain in the original neighbourhood for all time.

A positive-definite Lyapunov function with nonpositive orbital derivative proves stability of the equilibrium.

A positive-definite Lyapunov function with strictly negative orbital derivative away from the equilibrium proves asymptotic stability.

An equilibrium is asymptotically stable when it is Lyapunov stable and every trajectory starting sufficiently nearby converges to it.

A positive-definite quadratic form defines ellipsoidal sublevel sets and often turns a polynomial vector field into an exact factored orbital derivative.

If a Lyapunov function is nonincreasing on a sublevel set, trajectories cannot cross its boundary outward.

A boundary point with $\dot{V}=0$ can still enter the sublevel set; LaSalle analysis decides whether it belongs to an invariant zero-derivative trajectory.

On a compact positively invariant set where $\dot{V}\le 0$, every trajectory approaches the largest invariant subset of $\{\dot{V}=0\}$.

For $\ddot{x}=-U^{\prime }(x)-\mu \dot{x}$ with $\mu >0$,If $U$ is radially unbounded, every trajectory is bounded, and LaSalle's principle reduces its omega-limit set to invariant points with zero velocity.

Forthe mechanical energy isThe origin is a center for $k=0$ and an asymptotically stable equilibrium for $k>0$, while $(\pm 1,0)$ are saddles. For positive damping, their stable manifolds form the boundary between the basin of the origin and the two escape regions.

For $k>0$, the trajectory beginning at $(x,y)=(1,y_0)$ with $y_0>0$ remains in $x>1$, $y>0$ and obeysIt enters every strip $0<y<\epsilon$: otherwise $\dot{E}=-k{y}^2\le -k{\epsilon }^2$ would eventually make the nonnegative energy negative.

An equilibrium is hyperbolic when its Jacobian has no eigenvalue on the imaginary axis.

If every Jacobian eigenvalue has negative real part, the equilibrium is locally asymptotically stable.

A set is positively invariant when every forward trajectory starting in it remains in it.

A trapping region is a compact region across whose boundary the vector field points inward. It is positively invariant and confines every forward trajectory that enters it.