dynamical-systems.bigb

= Dynamical systems
{wiki=Dynamical_system}

= Conservative planar phase portrait
{parent=Dynamical systems}

For $\dot q=p$, $\dot p=-V'(q)$, the energy
$$
E=\frac12p^2+V(q)
$$
is constant. Local minima of $V$ give center equilibria, local maxima give saddle equilibria, and separatrices are energy contours through saddles.

= Energy balance method
{parent=Dynamical systems}

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.

= Relaxation oscillation
{parent=Dynamical systems}
{wiki}

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.

= Discrete dynamical system
{parent=Dynamical systems}
{wiki}

A one-dimensional discrete dynamical system iterates a map $x_{n+1}=F(x_n)$. A fixed point satisfies $F(x^*)=x^*$, while a point of least period $k$ satisfies $F^k(x)=x$ but no corresponding equation for a smaller positive period.

= Iteration of a map
{title2=$F^n$}
{parent=Discrete dynamical system}
{wiki=Iterated_function}

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

= Interval map
{parent=Discrete dynamical system}

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

= Periodic point of an interval map
{title2=$F^n(x)=x$}
{parent=Interval map}
{wiki=Periodic_point}

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 set
$$
\{x,F(x),\ldots,F^{n-1}(x)\}
$$
is its periodic orbit.

= Cycle of an interval map
{synonym}

= Cycles of an interval map
{synonym}

= Interval covering relation
{title2=$J\longrightarrow K$}
{parent=Interval map}

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

= Directed covering graph of an interval map
{parent=Interval covering relation}

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

= Periodic orbit from a closed interval-covering walk
{parent=Directed covering graph of an interval map}

Every closed walk
$$
J_0\longrightarrow J_1\longrightarrow\cdots\longrightarrow J_{n-1}\longrightarrow J_0
$$
in 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.

= Counting cycles in an interval covering graph
{parent=Periodic orbit from a closed interval-covering walk}

If $A$ is the <adjacency matrix of a directed graph>[adjacency matrix] of a directed covering graph, then $\operatorname{tr}(A^n)$ counts its pointed closed walks of length $n$. For a <prime number> $p$, subtracting the $\operatorname{tr}(A)$ constant walks and identifying the $p$ cyclic choices of starting point gives
$$
\frac{\operatorname{tr}(A^p)-\operatorname{tr}(A)}p
$$
primitive closed itineraries of length $p$.

= Connect-the-dots interval map
{parent=Interval map}

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.

= Jury stability criterion
{c}
{parent=Discrete dynamical system}
{wiki}

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 $1-a+b>0$.

= Multiplier of a periodic orbit of an iteration
{parent=Discrete dynamical system}

For a period-$k$ orbit $x_0,\ldots,x_{k-1}$ of a differentiable map, the multiplier is
$$
(F^k)'(x_0)=\prod_{j=0}^{k-1}F'(x_j).
$$
The orbit is locally asymptotically stable when the modulus of this product is less than one.

= Period-doubling bifurcation
{parent=Discrete dynamical system}
{wiki}

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.

= Transcritical bifurcation
{parent=Dynamical systems}
{wiki=Transcritical_bifurcation}

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.

= Center manifold
{parent=Dynamical systems}
{wiki=Center_manifold}

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.

= Center manifold of the 2024 Cambridge cubic system
{parent=Center manifold}

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 is
$$
w=\frac{a+1}{4}v^3+O(v^5),
\qquad
z=v^2+(1-a)v^4+O(v^6).
$$
Its reduced equation is
$$
\dot v=\frac{a-1}{2}v^3+
\frac{(3a-1)(a+3)}8v^5+O(v^7).
$$

= Extended centre manifold of the 2023 Cambridge quadratic-product system
{parent=Center manifold}

For
$$
\dot x=(a^2-x)(a-y^2),
\qquad \dot y=x-y,
$$
set $X=x-a^2$ 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 are
$$
Y=X+aX+O(3),
\qquad
\dot X=-aX+X^3+O(4).
$$
The reduced equation is a subcritical pitchfork with reversed parameter $\mu=-a$.

= Bifurcations of the 2022 Cambridge quadratic-cubic system
{parent=Center manifold}

For
$$
\dot x=x(y-k-3x+x^2),
\qquad
\dot y=y(y-1-x),
$$
the 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 begins
$$
v=-k+X^2-5Xk+9k^2+O(3),
$$
and its reduced equation is $\dot X=2(X^2-k)+O(3)$, a <saddle-node bifurcation>.

= Glendinning chaos
{parent=Dynamical systems}
{c}

A continuous <interval map> is chaotic in Glendinning's sense when some positive <iteration of a map>[iterate] has a <horseshoe for an interval map>.

= Horseshoe for an interval map
{parent=Glendinning chaos}
{wiki=Horseshoe_map}

An <interval map> $F$ has a horseshoe when there are <closed intervals> $J_0,J_1$ with disjoint interiors such that
$$
F(J_i)\supseteq J_0\cup J_1
\qquad(i=0,1).
$$
Iterating inverse branches produces full two-symbol itinerary dynamics.

= Horseshoe from two closed covering walks
{parent=Horseshoe for an interval map}

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>.

= Sharkovsky theorem
{parent=Glendinning chaos}
{c}
{wiki=Sharkovsky%27s_theorem}

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.

= Fibonacci transition-graph cycle count
{parent=Sharkovsky theorem}

A period-three orbit forces an interval-covering graph with adjacency matrix
$$
A=\begin{pmatrix}1&1\\1&0\end{pmatrix}.
$$
Its number of closed pointed length-$n$ itineraries is $\operatorname{tr}(A^n)=L_n$. Möbius removal of lower periods and division by $n$ give four primitive cyclic classes at $n=7$ and five at $n=8$.

= Radially symmetric planar dynamical system
{parent=Dynamical systems}

For
$$
\dot x=y+h(r^2)x,\qquad
\dot y=-x+h(r^2)y,
\qquad r^2=x^2+y^2,
$$
polar coordinates give
$$
\dot r=h(r^2)r,\qquad \dot\theta=-1.
$$
The trajectories rotate clockwise, while the sign of the radial equation determines stability.

= Devaney chaos
{parent=Dynamical systems}
{c}
{wiki=Devaney_chaos}

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.

= Topological transitivity
{parent=Devaney chaos}
{wiki}

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

= Dense periodic points
{parent=Devaney chaos}

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

= Sensitive dependence on initial conditions
{parent=Devaney chaos}
{wiki=Butterfly_effect}

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 $\delta$.

= Doubling map
{parent=Devaney chaos}
{wiki=Dyadic_transformation}

The doubling map on the unit circle is
$$
F(x)=2x\pmod1.
$$
It is chaotic in Devaney's sense.

= Binary shift representation of the doubling map
{parent=Doubling map}

If $x=0.b_1b_2b_3\ldots$ in base two, then
$$
F(x)=0.b_2b_3b_4\ldots.
$$
Binary cylinders make transitivity and density of periodic points transparent: concatenate prescribed finite blocks for transitivity and repeat a finite block for a periodic point.

= Periodic points of the doubling map
{parent=Doubling map}

The fixed points of $F^n$ are
$$
x=\frac{j}{2^n-1},
\qquad
j=0,\ldots,2^n-2.
$$
Thus $F^n$ has $2^n-1$ fixed points.

= Exact power-of-two periods of the doubling map
{parent=Periodic points of the doubling map}

For $n=2^k$, every proper period dividing $n$ divides $n/2$. Hence the number of points of exact period $2^k$ is
$$
(2^{2^k}-1)-(2^{2^{k-1}}-1)
=2^{2^k}-2^{2^{k-1}}.
$$

= Centre manifold theorem
{parent=Dynamical systems}
{wiki=Center_manifold}

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.

= Extended centre manifold for a parameter
{parent=Centre manifold theorem}

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

= Centre-manifold invariance equation
{parent=Centre manifold theorem}

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
{parent=Dynamical systems}
{wiki=Bifurcation_theory}

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

= Eigenvalue-crossing bifurcation test
{parent=Bifurcation theory}

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

= Saddle-node bifurcation
{parent=Bifurcation theory}
{wiki}

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.

= Pitchfork bifurcation normal form
{parent=Bifurcation theory}
{wiki=Pitchfork_bifurcation}

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 $\mu>0$.

= Subcritical pitchfork bifurcation
{parent=Pitchfork bifurcation normal form}

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.

= Symmetry-forced pitchfork bifurcation
{parent=Pitchfork bifurcation normal form}

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

= Bifurcation diagram
{parent=Bifurcation theory}
{wiki}

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

= Bifurcation diagram of the 2023 Cambridge quadratic-product system
{parent=Bifurcation diagram}

The equilibria of $\dot x=(a^2-x)(a-y^2)$, $\dot y=x-y$ lie on
$$
x=y=a^2
$$
for every $a$, and on $x=y=\pm\sqrt a$ for $a\geq0$. 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 $a=1$.

= Hopf bifurcation
{parent=Bifurcation theory}
{c}
{wiki}

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.

= Subcritical Hopf bifurcation
{parent=Hopf bifurcation}
{wiki}

In the radial normal form
$$
\dot r=\mu r+r^3,\qquad \dot\theta=1,
$$
an unstable periodic orbit exists for $\mu<0$ and shrinks into the equilibrium as $\mu\uparrow0$. This is a subcritical Hopf bifurcation.

= Saddle-node bifurcation of periodic orbits
{parent=Bifurcation theory}
{wiki=Saddle-node_bifurcation}

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

= Quintic radial Hopf equation
{parent=Bifurcation theory}

For
$$
\dot r=r(\mu+\lambda r^2-r^4),\qquad \lambda>0,
$$
nonzero periodic orbits have
$$
r_\pm^2=\frac{\lambda\pm\sqrt{\lambda^2+4\mu}}2.
$$
They 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.

= Interior equilibrium branch connecting two boundary bifurcations
{parent=Bifurcation theory}

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.

= Two-stage stability exchange in a symmetric planar system
{parent=Interior equilibrium branch connecting two boundary bifurcations}

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

= Strict Lyapunov function
{parent=Dynamical systems}
{wiki=Lyapunov_function}

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

= Stable manifold
{parent=Dynamical systems}
{wiki}

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

= Autonomous planar system
{parent=Dynamical systems}

An autonomous planar system is an ordinary differential equation
$$
\dot x=f(x,y),\qquad \dot y=g(x,y)
$$
whose vector field does not depend explicitly on time.

= Equilibrium point of a dynamical system
{title2=$x^*$}
{parent=Autonomous planar system}

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

= Equilibrium points of a dynamical system
{synonym}

= Linear stability of a planar equilibrium
{parent=Equilibrium point of a dynamical system}

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.

= Saddle equilibrium
{parent=Linear stability of a planar equilibrium}
{wiki=Saddle_point}

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.

= Stable node
{parent=Linear stability of a planar equilibrium}

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.

= Stable spiral
{parent=Linear stability of a planar equilibrium}
{wiki=Stability_theory#Classification_of_equilibrium_points}

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.

= Bendixson-Dulac criterion
{parent=Dynamical systems}
{c}
{wiki=Bendixson–Dulac_theorem}

If $D$ is simply connected and a $C^1$ function $B$ makes $\nabla\mathbin{\cdot}(BF)$ 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 $D$.

= Dulac theorem
{synonym}

= Omega-limit set
{title2=$\omega(x)$}
{parent=Dynamical systems}
{wiki=Limit_set}

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.

= Periodic orbit
{parent=Dynamical systems}
{wiki}

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

= Poincare-Bendixson theorem
{c}
{parent=Dynamical systems}
{wiki=Poincaré–Bendixson_theorem}

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

= Floquet multiplier
{parent=Dynamical systems}
{wiki}

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

= Divergence test for a planar periodic orbit
{parent=Floquet multiplier}

For a planar periodic orbit $\gamma$ of period $T$, its nontrivial Floquet multiplier is
$$
\exp\left(\int_0^T\nabla\mathbin{\cdot}F(\gamma(t))\,dt\right).
$$
The orbit is asymptotically stable when the integral is negative and unstable when it is positive.

= Fixed point stability for an autonomous differential equation
{parent=Dynamical systems}

For $x\prime=f(x)$, a simple fixed point $x_*$ is locally asymptotically stable when $f\prime(x_*)<0$ and unstable when $f\prime(x_*)>0$.

= Fixed point stability for an iteration
{parent=Dynamical systems}

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

= Resonance
{parent=Dynamical systems}
{wiki=Resonance}

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

= Unstable equilibrium
{parent=Dynamical systems}
{wiki}

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

= Equilibrium of an autonomous differential equation
{parent=Dynamical systems}
{wiki=Equilibrium_point}

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

= Linear stability analysis
{parent=Equilibrium of an autonomous differential equation}
{wiki=Linear_stability}

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

= Trace-determinant stability criterion
{parent=Linear stability analysis}

For a real $2\times2$ <Jacobian matrix> $J$, both <eigenvalues>[eigenvalue] have negative <real part> exactly when
$$
\operatorname{tr}J<0
\quad\hbox{and}\quad
\det J>0.
$$
The <linearization stability theorem> then makes a hyperbolic equilibrium locally <asymptotic stability>[asymptotically stable].

= Basin of attraction
{parent=Dynamical systems}
{wiki}

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

= Lyapunov function
{title2=$V$}
{parent=Dynamical systems}
{c}
{wiki}

A Lyapunov function near an equilibrium $x^*$ is continuously differentiable, satisfies $V(x^*)=0$ and $V(x)>0$ away from $x^*$, and has orbital derivative
$$
\dot V(x)=\nabla V(x)\cdot f(x)\leq0.
$$

= Lyapunov stability
{parent=Lyapunov function}
{c}
{wiki=Lyapunov_stability}

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

= First Lyapunov theorem
{parent=Lyapunov function}
{c}

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

= Second Lyapunov theorem
{parent=Lyapunov function}
{c}

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

= Asymptotic stability
{parent=Lyapunov function}
{wiki}

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

= Ellipsoidal Lyapunov function
{parent=Lyapunov function}

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

= Invariant sublevel set
{parent=Lyapunov function}

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

= Tangency at a Lyapunov boundary
{parent=Invariant sublevel set}

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.

= LaSalle invariance principle
{parent=Lyapunov function}
{c}
{wiki=LaSalle%27s_invariance_principle}

On a compact positively invariant set where $\dot V\leq0$, every trajectory approaches the largest invariant subset of $\{\dot V=0\}$.

= Damped mechanical energy as a Lyapunov function
{parent=LaSalle invariance principle}

For $\ddot x=-U'(x)-\mu\dot x$ with $\mu>0$,
$$
E(x,\dot x)=\frac12\dot x^2+U(x),
\qquad
\dot E=-\mu\dot x^2.
$$
If $U$ is radially unbounded, every trajectory is bounded, and LaSalle's principle reduces its omega-limit set to invariant points with zero velocity.

= Damped rational double-barrier phase portrait
{parent=Damped mechanical energy as a Lyapunov function}

For
$$
\ddot x+k\dot x+
\frac{2x(1-x^2)}{(1+x^2)^3}=0,
$$
the mechanical energy is
$$
E(x,y)=\frac12y^2+\frac{x^2}{(1+x^2)^2},
\qquad
\dot E=-ky^2.
$$
The origin is a center for $k=0$ and an asymptotically stable equilibrium for $k>0$, while $(\pm1,0)$ are saddles. For positive damping, their stable manifolds form the boundary between the basin of the origin and the two escape regions.

= Outward escape from the rational potential barrier
{parent=Damped rational double-barrier phase portrait}

For $k>0$, the trajectory beginning at $(x,y)=(1,y_0)$ with $y_0>0$ remains in $x>1$, $y>0$ and obeys
$$
0<y(t)<\sqrt{y_0^2+\frac12}.
$$
It enters every strip $0<y<\varepsilon$: otherwise $\dot E=-ky^2\leq-k\varepsilon^2$ would eventually make the nonnegative energy negative.

= Hyperbolic equilibrium
{parent=Dynamical systems}
{wiki=Hyperbolic_equilibrium_point}

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

= Linearization stability theorem
{parent=Hyperbolic equilibrium}

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

= Positively invariant set
{parent=Dynamical systems}
{wiki=Invariant_set}

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

= Trapping region
{parent=Positively invariant set}
{wiki=Invariant_set}

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.