Dynamical Systems — Foundations¶
Job: Describe long-time behavior of continuous- and discrete-time systems; linearize at equilibria; use Lyapunov theory, invariant manifolds, and topological conjugacy (Hartman–Grobman).
Core¶
Flows and maps¶
Continuous-time: \(\dot x=f(x)\) on a manifold \(M\) (here open \(M\subset\mathbb{R}^n\)). Flow \(\varphi_t\) of the autonomous ODE (local if incomplete). Discrete-time: \(x_{n+1}=F(x_n)\), iterates \(F^{\circ n}\).
Equilibrium \(x_*\): \(f(x_*)=0\) (resp. \(F(x_*)=x_*\)). Orbit: \(\{\varphi_t(x)\}\). \(\omega\)-limit: \(\omega(x)=\bigcap_{T\ge0}\overline{\{\varphi_t(x):t\ge T\}}\). \(\alpha\)-limit as \(t\to-\infty\). Invariant sets: \(\varphi_t(A)=A\).
Linearization and hyperbolicity¶
\(A=Df(x_*)\). Spectrum \(\sigma(A)\):
| Type | Spectral condition |
|---|---|
| Hyperbolic equilibrium | \(\operatorname{Re}\lambda\neq0\) all \(\lambda\in\sigma(A)\) |
| Asymptotically stable (linear) | \(\operatorname{Re}\lambda<0\) all \(\lambda\) |
| Unstable (linear) | some \(\operatorname{Re}\lambda>0\) |
| Center directions | some \(\operatorname{Re}\lambda=0\) |
For maps: hyperbolicity means \(\sigma(DF(x_*))\) off the unit circle; attractivity when all \(|\mu|<1\).
Hartman–Grobman. If \(x_*\) is hyperbolic, a neighborhood homeomorphism conjugates the nonlinear flow to \(e^{tA}\) near \(0\). (\(C^1\) linearization needs nonresonance / Sternberg hypotheses — harder.)
Lyapunov stability¶
\(x_*\) stable if \(\forall\varepsilon\,\exists\delta\): \(|x(0)-x_*|<\delta\Rightarrow|x(t)-x_*|<\varepsilon\) for \(t\ge0\). Asymptotically stable if stable and attractive. Exponentially stable if \(\|x(t)-x_*\|\le Ce^{-\alpha t}\|x(0)-x_*\|\).
Lyapunov function: \(V\) \(C^1\), positive definite near \(x_*\), \(\dot V=\nabla V\cdot f\le0\) \(\Rightarrow\) stable; \(\dot V<0\) definite \(\Rightarrow\) asymptotically stable. LaSalle invariance: trajectories approach the largest invariant set in \(\{\dot V=0\}\). Chetaev: instability certificates via \(V\) increasing in a cone.
Invariant manifolds and bifurcations¶
Stable/unstable manifolds \(W^{s/u}(x_*)\) tangent to \(E^{s/u}\) of \(A\) (Hadamard–Perron). Center manifold when \(\operatorname{Re}\lambda=0\) eigenvalues exist; reduced dynamics on \(W^c\) determine local bifurcations.
| Bifurcation | Normal-form sketch |
|---|---|
| Saddle-node | \(\dot x=r\pm x^2\) |
| Transcritical | \(\dot x=rx\pm x^2\) |
| Pitchfork | \(\dot x=rx\pm x^3\) |
| Hopf | complex pair crosses \(i\mathbb{R}\); limit cycle born |
Planar and global tools¶
Nullclines; Dulac / Bendixson exclude periodic orbits (div\((\rho f)\) sign). Poincaré–Bendixson: compact \(\omega\)-limit in \(\mathbb{R}^2\) with finitely many equilibria is equilibrium, cycle, or graphic. Higher dimension: chaos possible (Lorenz, Smale horseshoe).
Structural stability (pointer): hyperbolic chain-recurrent set; Peixoto in 2D; higher-D open (Smale).
Discrete-time notes¶
Hyperbolic fixed points of \(F\); multipliers \(\sigma(DF)\) off unit circle. Lyapunov for maps: \(V(F(x))\le V(x)\). Logistic map \(x\mapsto rx(1-x)\): period-doubling cascade to chaos (Feigenbaum). Symbolic dynamics / shift spaces encode horseshoes.
Ergodic one-liners (pointer)¶
Invariant probability \(\mu\); ergodicity: invariant sets have measure \(0\) or \(1\); Birkhoff: time averages \(\to\) space averages \(\mu\)-a.e. Lyapunov exponents measure average expansion (Oseledets).
See it¶
\(\dot x=x-x^3\) (supercritical pitchfork normal form): equilibria \(0,\pm1\). \(f'(0)=1>0\) unstable; \(f'(\pm1)=-2<0\) asymptotically stable. Gradient structure \(\dot x=-\nabla V\) with double-well \(V\) explains basins.
Damped pendulum \(\ddot\theta+\dot\theta+\sin\theta=0\): energy \(E=\frac12\dot\theta^2+1-\cos\theta\), \(\dot E=-\dot\theta^2\le0\); LaSalle \(\Rightarrow\) approach equilibria.
Check¶
Why does Hartman–Grobman fail at a nonhyperbolic equilibrium?
Center directions: linearization has eigenvalues on \(i\mathbb{R}\) (or unit circle for maps); topological type need not match \(e^{tA}\) (e.g. \(\dot x=-x^3\) vs \(\dot x=0\) at \(0\): both \(A=0\), different stability).
Sketch LaSalle for the damped pendulum.
\(E\) as above, \(\dot E=-\dot\theta^2\le0\). Largest invariant set in \(\dot\theta=0\) forces \(\sin\theta=0\) along trajectories \(\Rightarrow\) approach equilibria (attracting at \(2\pi\mathbb{Z}\) under damping).
State a Bendixson criterion excluding cycles in \(\mathbb{R}^2\).
If \(\operatorname{div} f\) is continuous and sign-definite on a simply connected region \(D\), then no periodic orbit lies entirely in \(D\).
Pitfalls¶
- Linear stability \(\operatorname{Re}\lambda\le0\) with pure imaginary eigenvalues does not decide nonlinear stability (center manifold / normal forms).
- Structural stability requires hyperbolicity of the chain-recurrent set — rare in high dimension.
- Numerical integration can create spurious attractors; trust structural results + validated bounds.
- Finite-time blowup: incomplete flows; maximal existence interval may be finite.
- “Chaos” needs a definition (sensitive dependence, positive entropy, horseshoe) — not just “looks messy.”
Refs¶
Perko; Guckenheimer–Holmes; Katok–Hasselblatt; Hirsch–Smale–Devaney; Arnold, Geometrical Methods; Wiggins.
Next¶
DE foundations · ODE second-order · Game theory · PDE · Stochastic processes · DS index · Hyperbolic Dynamics · Hamiltonian & KAM · Bifurcations & Chaos · Ergodic & Physical Measures