Equilibria, stability and nonlinear dynamics

An equilibrium holds the whole state fixed

Consider an autonomous differential equation z′=F(z) on an open state domain D⊆ℝⁿ. An equilibrium z_* satisfies F(z_*)=0, so z(t)=z_* is a constant solution. Assume F is continuously differentiable unless another hypothesis is stated. This supplies local existence and uniqueness, while long-time conclusions still require control of the solution’s domain and size. The geometric examples use dimensionless variables and time unless physical units are explicitly supplied.

Solving one component equation is insufficient. In a mechanical system q′=v, v′=−q−q³−cv, an equilibrium requires v=0 and q+q³=0. Thus the only real equilibrium is (0,0). At (q,v)=(1,0), position has zero instantaneous rate, but v′=−2, so this turning point is not an equilibrium. A nullcline, where a selected component rate vanishes, need not be a stationary state or an invariant curve.

Translate an equilibrium by x=z−z_*. The equation becomes x′=F(z_*+x), whose equilibrium is at zero. Translation simplifies notation without changing the physical state. In systems with different component units, such as metres and metres per second, specify reference scales before using an ordinary Euclidean norm. Changing to fixed equivalent norms or nonsingular linear coordinates preserves local stability classifications, although numerical distances and exponential constants can change.

An equilibrium, a periodic orbit and an attracting set are distinct objects. A periodic solution moves around a closed orbit; its vector field is generally nonzero. A family of equilibria can permit convergence to different points from different initial states. State exactly which object is being called stable and which perturbations are allowed. Stability is a quantified property of nearby complete initial states, not a judgement based on one attractive picture.

Staying near, approaching and approaching at a rate

The equilibrium zero is Lyapunov stable if for every sufficiently small ε>0 there is δ>0 such that ||x₀||<δ implies ||x(t;x₀)||<ε for all t≥0. Here nearby initial states must have forward solutions for all those times. Equivalently, one may initially state the bound on maximal forward intervals and then establish continuation from a compact neighbourhood inside D. The choice of δ may depend on ε but cannot depend on the particular initial state or future time.

Local attraction means that there is a neighbourhood of zero in which every initial state has a forward-complete solution tending to zero as t→∞. Local asymptotic stability combines Lyapunov stability and local attraction. Instability is the failure of stability: there is a fixed neighbourhood that arbitrarily close initial states can eventually leave. Attraction along one line, or agreement over a finite observation window, is insufficient to establish these neighbourhood statements.

Local exponential stability is stronger. There must be constants M≥1, λ>0 and δ>0 such that ||x(t;x₀)||≤M e^(−λt)||x₀|| for all t≥0 whenever ||x₀||<δ. The same M and λ work for all these states. The exponent λ has inverse-time units if time is dimensional. Asymptotic convergence can instead be algebraic, so a vanishing limit does not by itself supply an exponential rate.

Global asymptotic stability means stability together with forward completeness and convergence for every initial state in the stated domain. The basin of attraction of zero is the set of states in D with forward-complete solutions converging to zero. A proved invariant region inside that basin is a useful certificate; it need not be the whole basin. A claim of global behaviour must specify whether the domain is ℝⁿ, a constrained physical region, an angle cylinder or another state space.

Long-time conclusions need long-time solutions

A smooth field need not generate solutions for every future time. For x′=x² and x(0)=a>0, separation gives x(t)=a/(1−at), with maximal forward endpoint 1/a. The solution tends to infinity there. Describing its state as t→∞ without addressing this endpoint is meaningless. Local uniqueness and continuous dependence do not prevent finite-time escape.

A useful continuation principle is the following: if a maximal forward solution of a locally Lipschitz field remains in a compact subset K of the open domain D, then it extends for all future times. Boundedness of the field and a uniform local existence neighbourhood on K exclude a finite terminal endpoint. Bounded coordinates alone are not enough when the solution approaches an excluded boundary of D; the compact set must remain inside the valid domain.

For the nonnegative continuous Lyapunov functions on ℝⁿ used here, proper means radially unbounded: V(x)→∞ as ||x||→∞. Equivalently, every finite sublevel set {x:V(x)≤h} is compact. If V cannot increase along a solution, its initial sublevel traps that solution. For a smooth field on all of ℝⁿ this gives forward completeness. On a smaller domain, check compactness within that domain. Energy can bound speed while failing to bound an unwrapped angle, so its sublevels are not automatically compact in every coordinate representation.

A positively invariant set contains the future trajectory from each of its states. For a smooth boundary given by V=h, a strictly inward derivative V′<0 proves that trajectories cannot cross outward. Nonpositive boundary derivatives require an applicable invariance argument and uniqueness, or a limiting sublevel argument; a lone sampled boundary arrow is not a proof. Later local certificates use strict decay away from zero, while the energy example uses nonincreasing sublevels directly.

Phase lines reveal both attraction and escape

For a scalar autonomous equation x′=f(x), equilibria are zeros of f, and the sign of f on each interval gives the direction of motion. Under local uniqueness a nonconstant solution cannot cross an equilibrium in finite time and then depart. These invariant barriers organise the basin. A bounded monotone solution has a limit; when it remains in the interior of the domain, continuity forces that limit to be an equilibrium.

For x′=−x+x³=x(x²−1), the equilibria are −1,0,1. Between −1 and 0 the rate is positive, while between 0 and 1 it is negative, so the states in (−1,1) approach zero. At either endpoint the solution is constant and does not approach zero. Outside that interval the arrows point outward. In fact large escaping states blow up in finite forward time, so the basin is exactly (−1,1), not the whole real line.

The equilibrium zero in x′=−x³ is also attracting, but the derivative f′(0)=0 makes the usual strict linear test inconclusive. The exact forward solution x(t)=a/√(1+2a²t) is valid for every a and t≥0. Its magnitude never increases and tends to zero. The equilibrium is globally asymptotically stable. Its decay for nonzero a is proportional to t^(−1/2), showing why asymptotic stability must be distinguished from exponential stability.

One-sided attraction is another distinct possibility. For x′=−x², positive initial states approach zero, whereas negative initial states move away and blow down to −∞ in finite time. Thus zero is not Lyapunov stable on ℝ, even though it attracts from the right. If the physical state is restricted to an invariant half-line, state that relative domain explicitly before describing stability. The words stable and unstable otherwise refer to different perturbation sets.

Linearization decides strict spectral cases locally

Near an equilibrium of a C¹ autonomous field, write x′=Ax+R(x), where A=DF(0) and ||R(x)||/||x||→0 as x→0. If every eigenvalue of A has strictly negative real part, the nonlinear equilibrium is locally exponentially stable. If any eigenvalue has strictly positive real part, it is unstable. These are local conclusions with the stated smoothness; they do not identify the entire basin or replace the nonlinear solution by e^(tA)x₀ at arbitrary distances.

A hyperbolic equilibrium has no eigenvalue on the imaginary axis. A hyperbolic saddle has both stable and unstable directions, so it is unstable even though selected initial states on its local stable manifold converge to it. The tangent eigenspaces describe the first-order geometry; the nonlinear invariant manifolds can be curved. Avoid confusing a stable direction with a stable equilibrium.

When zero real parts remain and no positive real part settles instability, inspect nonlinear terms or another certificate. The scalar equations x′=−x³ and x′=x³ both have the zero Jacobian but opposite stability. Likewise a planar linear centre can acquire inward or outward radial cubic motion without changing its imaginary eigenvalues. A zero determinant or a pair ±i is a warning that the strict test has not answered the question, rather than evidence for a universal neutral case.

Linearization also does not require Euclidean distance to decrease at every instant. For A=[[-1,2],[0,−2]], all eigenvalues are negative, but at x=(1,1/2), the derivative of ||x||² is −1, while different nonnormal matrices with negative eigenvalues can show transient growth in that norm. A more direct illustration is B=[[-1,4],[0,−2]]: at the same state, xᵀ(B+Bᵀ)x=1>0. Stability permits a fixed overshoot factor M; a suitable weighted quadratic function can still decay monotonically.

A Lyapunov function turns level sets into a proof

A Lyapunov candidate V is a scalar C¹ function, defined near zero, with V(0)=0 and V(x)>0 for nearby nonzero x. Along x′=F(x), its derivative is V′(x)=∇V(x)·F(x). This derivative measures change along solutions, not change in every direction in state space. If V′≤0 on a neighbourhood, sufficiently small sublevels can trap nearby states and prove stability.

Here is the local argument. Choose a closed ball of radius r, centred at zero, within the domain where V is positive definite and V′≤0. The minimum m of V on its boundary sphere is positive by compactness and continuity. Choose a sublevel height 0<h<m and then δ>0 so that ||x₀||<δ implies V(x₀)<h and ||x₀||<r. Before any attempted first exit through the sphere, V cannot increase to the boundary value m. No exit is possible; the trajectory remains in a compact subset inside the ball and continues forward.

If V′<0 for every nonzero point in that neighbourhood, the same sublevel argument gives local asymptotic stability. One way to see attraction is to fix an annulus away from zero inside the trapped region: continuity makes −V′ have a positive minimum there, so a trajectory cannot stay there forever while V remains nonnegative. The standard strict Lyapunov theorem formalises this argument. When V′ merely vanishes at some nonzero states, attraction needs additional reasoning.

Quantitative comparison adds rates. If a||x||²≤V(x)≤b||x||² and V′≤−c||x||² with positive a,b,c on a positively invariant neighbourhood, then V′≤−(c/b)V. Integration gives V(t)≤V(0)e^(−ct/b), hence ||x(t)||≤√(b/a)e^(−ct/(2b))||x₀||. Include both norm bounds: decay of V alone does not identify a Euclidean exponential constant. The factor of two changes when passing from squared norm to norm.

A weighted quadratic form certifies a nonnormal system

For x′=Ax and V=xᵀPx with symmetric P, differentiation gives V′=xᵀ(AᵀP+PA)x. Positive definiteness of P makes V a valid quadratic measure. For a real 2×2 symmetric matrix, a positive first leading entry and positive determinant establish positive definiteness. The equation AᵀP+PA=−Q with positive definite Q is called a continuous-time Lyapunov equation. It checks the whole linear system, rather than a few sampled trajectories.

Take A=[[-1,2],[0,−2]] and P=[[1/2,1/3],[1/3,7/12]]. The determinant is 13/72 and the first leading entry is 1/2, so P is positive definite. Exact multiplication gives AᵀP+PA=−I. Thus V′=−||x||² everywhere, although V itself has a cross term 2x₁x₂/3 and its contours are generally tilted ellipses. A tilted ellipse is a different geometry from a Euclidean circle.

Explicit safe bounds avoid unnecessary decimal eigenvalues. The largest eigenvalue of P is at most tr P=13/12, and the smallest is at least det P/(tr P)=1/6, since the product of the two eigenvalues is det P. Therefore (1/6)||x||²≤V≤(13/12)||x||². It follows that V′≤−(12/13)V and ||x(t)||≤√(13/2)e^(−6t/13)||x₀|| for every initial state and t≥0. These are conservative proved constants, not a claim that the bound is attained.

For a nonlinear perturbation x′=Ax+R(x), the same V yields V′=−||x||²+2xᵀPR(x). If ||R(x)||≤η||x|| in a sufficiently small ball, then V′≤−(1−2||P||η)||x||². Choosing η<1/(2||P||) makes the decay strict there. This explains why a Hurwitz linearization controls a sufficiently small nonlinear neighbourhood while leaving larger states and the full basin open to further analysis. The operator norm and state scaling must be specified when numerical constants are reported.

A zero energy derivative is a set to inspect

The strict derivative condition is sufficient, not necessary. For a smooth autonomous system, a common LaSalle invariance setting is a compact positively invariant set K and a C¹ function V with V′≤0 on K. Every forward trajectory starting in K approaches, in distance, the largest invariant subset of E={x∈K:V′(x)=0}. That subset consists of states whose complete orbits can remain in E. The theorem concerns autonomous evolution and the stated compact invariant region; it is not an unrestricted rule for time-dependent fields.

Merely listing E is not enough. For q′=v, v′=−q−q³−cv with c>0, set V=v²/2+q²/2+q⁴/4. Then V′=v(−q−q³−cv)+(q+q³)v=−cv². E is the entire line v=0 within a chosen sublevel. At a point (q,0) with q≠0, v′=−q−q³≠0, so the trajectory immediately leaves that line. The only orbit staying inside E is the equilibrium zero.

Every finite sublevel of this V is compact in ℝ², positively invariant and contains a forward-complete solution from each of its states. LaSalle gives attraction to zero for every initial state. Positive definiteness and nonincrease separately give Lyapunov stability, so the equilibrium is globally asymptotically stable. The argument establishes stability and attraction for their respective reasons; convergence to an invariant set alone should not be relabelled as point stability.

Compare the undamped case c=0. Now V′=0 everywhere, and nonzero compact energy curves support continuing motion rather than decay. LaSalle only says trajectories approach the largest invariant part of the whole chosen set, which yields no attraction to zero. Likewise a derivative that is negative only at a handful of samples does not qualify a global sign condition. The set where dissipation vanishes and its actual dynamics are central to the conclusion.

Mechanical energy needs its units and its state space

For a damped pendulum with mass m, length ℓ, gravitational acceleration g and damping rate γ, take m,ℓ,g,γ>0 and write mℓ²θ″+mℓ²γθ′+mgℓ sinθ=0. The state is (θ,ω), with ω=θ′. Angle is measured in radians and ω has inverse-time units. The energy E=mℓ²ω²/2+mgℓ(1−cosθ) is in joules. Differentiation using both state equations gives E′=−mℓ²γω², in watts.

Around the downward equilibrium choose the angle chart −π<θ<π. For any 0<h<2mgℓ, the sublevel K_h={E≤h} in this chart is compact: energy bounds |ω|, and the potential excludes a neighbourhood of either chart boundary. It is positively invariant because E cannot increase. In E′=0, the condition ω=0 can persist only if sinθ=0. Within K_h the only possibility is θ=0. LaSalle plus the local positive-definite energy gives attraction and stability at the downward state.

The strict energy threshold matters. At energy 2mgℓ the upright equilibrium (π,0), represented at a chart boundary, appears. On the physical angle circle, the upright state is a distinct equilibrium; on the unwrapped real θ-axis there are infinitely many copies of both upward and downward states. Energy is periodic in θ, so it is not proper on ℝ² with an unwrapped angle. A careless claim that every real-angle state converges to (0,0) is false.

Energy monotonicity is useful even where it does not prove a particular final state. It can exclude increases in mechanical energy and bound speed, while the geometry of the state space decides whether the needed compactness and uniqueness of the limiting equilibrium hold. In a measured pendulum, friction models, driving torques and parameter ranges are additional modelling assumptions. The ideal viscous equation supports the stated theorem; an external periodic drive is a different nonautonomous system.

A certified ball can be smaller than the basin

For x′=−x+x³, y′=−2y, use V=(x²+y²)/2. Its derivative is −x²+x⁴−2y². On the closed ball x²+y²≤ρ² with 0<ρ<1, x⁴≤ρ²x², so V′≤−(1−ρ²)x²−2y²≤−(1−ρ²)(x²+y²)=−2(1−ρ²)V. The derivative is strictly inward on its nonzero boundary. Hence the ball is positively invariant, and r(t)≤r₀ e^(−(1−ρ²)t) for initial radius r₀≤ρ.

This proves an entire family of local exponential certificates, with the rate depending on the selected ball. It does not imply exponential decay with one such rate for every initial state in the full basin. Close to the unstable barriers x=±1, solutions can spend a long time before moving toward zero. Constants that are uniform on a fixed smaller ball need not be uniform near the basin boundary.

The full basin follows by separating the components. If |x₀|<1, the scalar phase line traps x between its initial value and zero and forces x→0. Meanwhile y(t)=y₀e^(−2t) for every real y₀. If |x₀|=1, x remains ±1; if |x₀|>1, x escapes. Thus the basin is the open strip |x|<1 with arbitrary y, which is much larger than any one certified ball.

The boundary lines x=±1 are invariant and approach the saddle equilibria (±1,0) as y decays. They separate convergence to zero from escape. An energy sublevel, an attraction basin, its boundary and a plotted finite collection of trajectories are different mathematical sets. A useful numerical estimate should say whether it is a proved inner approximation, a sampled conjecture or an exact result. This distinction matters in engineering safety margins and ecological recovery models.

A decreasing landscape organises several attractors

A gradient flow is x′=−∇U(x), where U is a differentiable landscape; assume enough smoothness for local uniqueness. Along a trajectory, U′=−||∇U||²≤0. The critical points ∇U=0 are the equilibria. A strict local minimum with a positive-definite Hessian gives a locally exponentially stable equilibrium, because the linearized field is minus that Hessian. Saddles and degenerate critical points need their own analysis.

Take U(x,y)=(x²−1)²/4+y²/2. Then x′=x−x³ and y′=−y. The equilibria are (−1,0),(0,0),(1,0). At x=±1, the Hessian diag(3x²−1,1) is diag(2,1), giving stable minima. At x=0 it is diag(−1,1), so the origin is a saddle for the flow. The landscape is proper, so its nonincreasing sublevels give forward completeness.

Uniqueness keeps x from crossing zero. If x₀>0, the scalar arrows drive x toward 1, including states initially larger than 1; if x₀<0 they drive x toward −1. The component y always decays. The invariant line x=0 approaches the saddle and separates the two basins. Thus two different stable equilibria coexist, with a precise boundary that no generic claim of one global minimum can replace.

For a general bounded gradient trajectory with a compact forward trapping set, the invariance principle gives approach to the critical set. It does not, under mere smoothness, automatically establish convergence to one specified critical point. Extra conditions, such as an appropriate isolation argument or stronger regularity theorem, may provide that conclusion. In the explicit double well, the scalar signs and finite listed equilibria settle the individual limits directly. A local or global minimum is also different from a statement about the speed of convergence.

An attracting orbit can keep a neutral phase direction

For a periodic orbit Γ, stability is naturally measured by dist(x,Γ), rather than by distance to one moving reference point at the same clock time. Orbital asymptotic stability means nearby states stay near the orbit and their distance to it tends to zero. Their limiting phases can differ. A periodic orbit is a set invariant under time shifts, so an autonomous system normally retains a neutral direction along the motion.

An explicit dimensionless example is x′=μx−y−(x²+y²)x, y′=x+μy−(x²+y²)y. For r=√(x²+y²)>0, differentiate r² and the angle to obtain r′=r(μ−r²), θ′=1. The polar formula is only used away from r=0; the original Cartesian field defines the origin regularly. At μ>0, the circle r=√μ is a periodic orbit of period 2π. Every initial state with r₀>0 approaches that radius.

A small radial perturbation η around r=√μ satisfies, to first order, η′=−2μη. After a period, its linear multiplier is e^(−4πμ), strictly between 0 and 1. The tangential multiplier is 1 because a phase shift stays a phase shift. These are Floquet multipliers for the linear variational motion along this explicit cycle. They concern the orbit and its transverse attraction, rather than synchronization of two arbitrary clocks.

The same family illustrates a stability change. For μ<0, r′≤μr gives exponential convergence to zero. At μ=0, r(t)=r₀/√(1+2r₀²t) gives algebraic attraction. For μ>0 the origin is unstable and the nonzero attracting circle appears. Two solutions with equal radius on that circle and different fixed phases remain a fixed positive distance apart. Their shared attraction to the same orbit does not imply convergence to each other at equal times.

Planar criteria can exclude or force periodic motion

A particularly clear Bendixson criterion is this strict-sign version: for a C¹ planar autonomous field F on a simply connected open region U, if div F is strictly positive throughout U or strictly negative throughout U, no nonconstant periodic orbit can lie entirely in U. Simple connectivity matters because the interior of a closed orbit must lie in the region where the argument applies. Less strict sign variants need their own hypotheses; the strict version suffices here.

The proof connects to the divergence theorem from the preceding multivariable lessons. Uniqueness makes a regular planar periodic orbit a simple closed curve after choosing its minimal period. The field is tangent to that curve, so its outward flux is zero. Green’s flux identity equates this to the integral of div F over the enclosed area. A continuous strictly signed divergence makes that area integral nonzero, a contradiction. A visually spiralling trajectory is not itself this proof.

For q′=v, v′=−q−q³−cv with c>0, the planar divergence is −c everywhere. Thus there are no nonconstant periodic orbits anywhere in ℝ², consistently with energy dissipation. For c=0 the divergence is zero, so the strict criterion says nothing; it does not exclude the closed energy curves. Failure of a sufficient criterion is not evidence that a cycle exists.

In the other direction, a useful Poincaré–Bendixson form says that a nonempty compact omega-limit set of a forward planar C¹ autonomous trajectory, if it contains no equilibrium, is a periodic orbit. A forward trajectory trapped in a compact region inside the domain has a nonempty compact omega-limit set. Excluding all equilibria from that trapping region therefore supplies a route to a cycle. If equilibria remain in the limit set, connecting orbits and equilibria are possible; boundedness alone does not force a cycle.

For the radial system with μ=1, the annulus 1/2≤r≤2 is positively invariant: r′>0 on the inner circle and r′<0 on the outer circle. There is no equilibrium in the annulus since θ′=1. The planar theorem supplies periodic motion, and the direct radial equation identifies the exact cycle r=1 and its attraction. This is a proved trapping-region argument. Planar autonomous hypotheses cannot be carried over unchanged to three-dimensional or periodically forced systems.

A parameter can change equilibria and their stability

A bifurcation is a qualitative change in the local or global dynamics as a parameter varies. Equilibrium locations, their stability and periodic orbits can change. In x′=μx−x³, zero exists for every μ. For μ<0 it is stable; at μ=0 it is still asymptotically stable with algebraic decay. When μ>0, zero becomes unstable and the two stable equilibria ±√μ appear. This explicit symmetric family has a supercritical pitchfork diagram. Breaking its symmetry can change the diagram.

For x′=μ−x², no equilibrium exists when μ<0. At μ=0, zero is one-sided attracting and is unstable under unrestricted real perturbations. For μ>0, +√μ is stable and −√μ unstable, because f′(x)=−2x. This is a saddle-node family. Time direction and sign conventions matter: switching the sign of the whole field reverses those stability assignments.

For x′=μx−x², the branches x=0 and x=μ exchange stability at μ=0. Their derivatives are μ and −μ, respectively. In an unrestricted real state space this is a transcritical family; in a nonnegative population domain a negative equilibrium branch is not physically admissible. A physical boundary can alter which states and perturbations are meaningful even though the algebraic branches remain available on paper.

The planar radial family just studied is an explicit supercritical Hopf normal form: its Jacobian eigenvalues are μ±i, its cubic radial term controls the nonlinear result, and a stable cycle appears at μ>0. A general Hopf theorem needs smoothness, a simple isolated imaginary pair, a nonzero crossing rate and a nondegeneracy condition on nonlinear coefficients; its applicable hypotheses must be checked for the particular theorem used. Merely finding imaginary eigenvalues does not prove a cycle is born or determine its stability.

Parameter diagrams are statements about a specified model and parameter range. Near a critical value, slow transients can resemble an equilibrium or an unobserved instability during a short experiment. Include the zero-parameter case explicitly: strict-sign classifications for μ≠0 do not settle that boundary. At the same time, local bifurcation information does not automatically exclude distant equilibria, remote cycles or finite-time escape elsewhere in the state space.

Perturbations and numerical evidence need honest bounds

For a stable model subject to forcing, the original equilibrium need not remain an exact solution. In z′=−kz+d(t), assume k>0, z∈ℝⁿ and a bounded measurable disturbance with ||d(t)||≤ε almost everywhere. The solution is an absolutely continuous function satisfying the equation almost everywhere. Variation of constants and the triangle inequality give ||z(t)||≤e^(−kt)||z₀||+(ε/k)(1−e^(−kt)). Thus limsup ||z(t)||≤ε/k. Persistent forcing permits an ultimate neighbourhood; it does not promise convergence to the old zero equilibrium.

If all components have the same state unit, k has inverse-time units, d and ε have state-per-time units, and ε/k has state units. Different component units require scaling or an appropriate weighted norm before using this bound. With constant d=d₀, the actual equilibrium is d₀/k, which makes the distinction concrete. More general nonlinear robustness or input-to-state stability theorems require their own hypotheses and cannot be inferred from the unforced eigenvalues alone.

Numerical trajectories can test calculations and reveal mistakes, but a finite collection of time steps does not prove an infinite-time neighbourhood claim. For the scalar stable equation x′=−kx, explicit Euler gives x_{n+1}=(1−kh)x_n. The numerical origin attracts only when |1−kh|<1, that is 0<kh<2. At kh=2 it alternates without decay, and above 2 it grows, even though the exact solution decays for every k>0. Mark exact, approximate and sampled layers separately, with the step and time horizon visible.

For a proposed certificate, verify its derivative symbolically or by direct algebra on its entire stated region, establish compact trapping and continuation where needed, and identify the correct limiting invariant set. Computational residuals and interval or exact-arithmetic checks support those steps. A successful sample grid cannot certify an unexamined region, and an unexplained solver warning must not be discarded as a cosmetic issue.

These ideas connect phase portraits to mechanics through energy and power, to linear algebra through quadratic forms and spectral bounds, to multivariable calculus through gradients and flux, and to feedback design through disturbance rejection and invariant safety regions. Ecology and chemistry use basins and equilibria too, but nonnegative domains, conservation constraints and fitted parameters must be retained. The useful conclusion is the strongest one justified by the model, assumptions and evidence: local or global, point or orbit, exact or approximate, with an explicitly stated rate when one has actually been proved.

Worked examples

Try it yourself

1. For q′=v,v′=−3q−v, is (q,v)=(2,0) an equilibrium?

   Show the answer and reasoning

   No. The only equilibrium is (0,0).

   At (2,0), q′=0 but v′=−6. Both state rates must vanish simultaneously, so a turning point is not enough.

2. In the stability definition, may δ depend on the chosen future observation time?

   Show the answer and reasoning

   No. For each ε, one δ must work for every future time.

   A different tolerance for each finite window would not provide the all-time neighbourhood bound required by Lyapunov stability.

3. For z′=Jz, J=[[0,−1],[1,0]], is the origin stable, attracting or exponentially stable?

   Show the answer and reasoning

   Stable, but not attracting and therefore not exponentially stable.

   The exact flow preserves ||z||. Nearby initial states stay equally near forever, while each nonzero radius remains nonzero.

4. For x′=x²,x(0)=2, what is the maximal forward interval?

   Show the answer and reasoning

   0≤t<1/2.

   The solution 2/(1−2t) blows up as t approaches 1/2 from below. A smooth polynomial field gives local existence without guaranteeing forward completeness.

5. Why does the common Jacobian 0 not classify both x′=−x³ and x′=x³?

   Show the answer and reasoning

   The strict linearization test is inconclusive; the first is attracting and stable, the second unstable.

   The nonlinear signs point inward or outward. For positive initial a, the second solution a/√(1−2a²t) also escapes in finite time.

6. If 2||z||²≤V≤8||z||² and V′≤−4||z||² on an invariant neighbourhood, give a Euclidean exponential estimate.

   Show the answer and reasoning

   ||z(t)||≤2e^(−t/4)||z₀||.

   V′≤−V/2 gives V(t)≤V₀e^(−t/2). The norm-squared ratio is at most 8/2=4, and taking a square root halves the exponent.

7. For A=diag(−1,−3) and P=diag(1/2,1/6), compute AᵀP+PA.

   Show the answer and reasoning

   −I.

   Each diagonal entry is twice the product of the corresponding entries of A and P: 2(−1)(1/2)=−1 and 2(−3)(1/6)=−1. The positive P gives V′=−||z||².

8. For q′=v,v′=−q−v with V=(q²+v²)/2, identify E={V′=0} and its largest invariant subset.

   Show the answer and reasoning

   E is v=0; its largest invariant subset is the origin.

   V′=−v², but at (q,0) with q≠0, v′=−q leaves the line. Thus a point in the zero-derivative set need not stay there.

9. In E′=−mℓ²γω², what are the SI units of the right side?

   Show the answer and reasoning

   Watts, kg·m²·s⁻³.

   γ has s⁻¹ and ω² has s⁻², so their product with mℓ² has energy-per-time units. E itself is measured in joules.

10. Why use h<2mgℓ for a compact local pendulum sublevel around the downward state?

    Show the answer and reasoning

    It excludes the upright energy and keeps the sublevel away from the ±π chart boundaries.

    At those boundaries the potential is 2mgℓ. Strict inequality gives an interior compact trapping region with only the downward invariant zero-dissipation state.

11. For x′=−x+x³,y′=−2y, use the radiusρ=3/4 certificate to give a radius decay rate.

    Show the answer and reasoning

    r(t)≤r₀e^(−7t/16), for r₀≤3/4.

    The general estimate is r(t)≤r₀e^(−(1−ρ²)t). Here 1−ρ²=1−9/16=7/16. The selected ball is a sufficient invariant region, rather than the full basin.

12. For the double-well gradient flow, which limit follows from initial state (−2,5), and what happens on x=0?

    Show the answer and reasoning

    (−2,5) approaches (−1,0); every state on x=0 approaches the saddle (0,0).

    The sign of x is preserved; the scalar arrows direct negative states toward −1. On the invariant boundary x=0, only y decays. One converging boundary does not stabilize the saddle.

13. For the radial family at μ=4, find the cycle radius, period and radial Floquet multiplier.

    Show the answer and reasoning

    Radius 2, period 2π, radial multiplier e^(−16π).

    The radius is√μ. Linear radial decay rate is 2μ=8 and one revolution takes 2π, so the multiplier is e^(−8·2π). The tangential multiplier remains 1.

14. Can two different phases on one attracting autonomous cycle remain separated at equal times?

    Show the answer and reasoning

    Yes. Orbital attraction does not require equal-time phase synchronization.

    On the radial example θ′=1, the phase difference is constant. Distance to the orbit can be zero for both states while their distance to one another stays positive.

15. Does divergence zero prove there are no cycles? May strict signed divergence on an annulus automatically use the simply connected criterion?

    Show the answer and reasoning

    No to both.

    Zero divergence fails the strict sign hypothesis. An annulus is not simply connected, so a closed orbit may enclose a region outside the domain where the proposed flux argument was assumed.

16. For x′=μ−x² at μ=9, classify both equilibria. What happens at μ=0 on ℝ?

    Show the answer and reasoning

    +3 is stable, −3 unstable; at μ=0 the origin is one-sided attracting but unstable on ℝ.

    f′=−2x gives derivatives −6 and 6. At the critical value, negative initial states move away and can blow down in finite time, so unrestricted stability fails.

17. For z′=−5z+d(t), ||d||≤0.2 state-units/s, what ultimate bound follows? Does it promise convergence to zero?

    Show the answer and reasoning

    limsup ||z(t)||≤0.04 state-units; zero convergence is not guaranteed.

    The disturbance bound divided by the decay rate gives 0.2/5=0.04 with state units. A constant disturbance can sustain a nonzero shifted equilibrium.

18. For explicit Euler on x′=−3x, state the strict step interval for numerical attraction. What happens at its upper endpoint?

    Show the answer and reasoning

    0<h<2/3; at h=2/3 the multiplier is −1, so nonzero states alternate without decay.

    The recurrence multiplier is 1−3h. Requiring its absolute value below 1 gives the open interval. The exact differential equation is stable for every positive time, so the numerical boundary is a method restriction.

What to carry forward

• Specify the complete state, domain, norm scaling and object whose stability is being claimed.
• Stability controls all future departures from a neighbourhood; attraction and exponential rates are additional claims.
• Forward completeness must be established before making an infinite-time conclusion.
• Phase-line barriers can determine an exact basin, including states that escape or remain on a boundary.
• Strict spectral linearization tests are local; zero or imaginary real parts require further analysis.
• A positive-definite Lyapunov function with nonpositive derivative can prove stability through compact local trapping.
• Strict decay gives attraction; explicit upper and lower norm comparisons are required for a reported Euclidean exponential rate.
• Quadratic certificates use the symmetric matrix residual AᵀP+PA and can certify nonnormal evolution.
• LaSalle concerns the largest invariant part of the zero-derivative set in a compact positively invariant autonomous setting.
• Mechanical energy and its derivative have different units, and the angle state space changes compactness and global claims.
• A proved sublevel or ball can be a conservative inner subset of a much larger attraction basin.
• Gradient landscapes can have several stable equilibria; convergence to a critical set is not automatically convergence to one point.
• Periodic-orbit attraction is measured relative to the orbit and permits a neutral phase direction.
• Planar divergence and limit-set criteria require their actual sign, topology, compactness and equilibrium hypotheses.
• The critical parameter case and physical state constraints must be checked separately in bifurcation diagrams.
• Persistent disturbance bounds give ultimate neighbourhoods, while numerical step restrictions can create artificial instability.
• Algebra, computations and visual trajectories support the reasoning without replacing its quantified assumptions.