Dynamical Systems
Year 3 · 5 sections · 52 concepts · 101 practice questions
What's covered
free1 · Flows and invariant sets11 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Flows
- Proposition 1.2 · The group property
- Definition 1.3 · Invariant and trapping sets
- Definition 1.4 · Omega- and alpha-limit sets
- Proposition 1.5 · Properties of omega-limit sets
- Theorem 1.6 · Lyapunov's stability theorem
- Theorem 1.7 · LaSalle's invariance principle
- Example 1.8 · The damped pendulum
- Corollary 1.9 · Estimating basins of attraction
- Check 1.10 · Check: proving stability without linearisation
2 · Planar dynamics12 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · The index of a closed curve
- Proposition 2.2 · Properties of the index
- Theorem 2.3 · The index sum formula
- Theorem 2.4 · Periodic orbits enclose equilibria
- Theorem 2.5 · The Poincaré–Bendixson theorem
- Corollary 2.6 · Trapping annuli contain periodic orbits
- Definition 2.7 · Hamiltonian systems
- Proposition 2.8 · Equilibria of Hamiltonian systems
- Proposition 2.9 · Gradient systems
- Theorem 2.10 · Liénard's theorem
- Check 2.11 · Check: periodic orbits in the plane
3 · Bifurcations9 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Bifurcations
- Theorem 3.2 · The saddle-node bifurcation
- Example 3.3 · The transcritical bifurcation
- Example 3.4 · Pitchfork bifurcations
- Example 3.5 · Bifurcation diagrams and hysteresis
- Theorem 3.6 · The Hopf bifurcation
- Example 3.7 · The Hopf normal form
- Check 3.8 · Check: classifying a bifurcation
4 · Discrete dynamics10 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Orbits, fixed and periodic points
- Theorem 4.2 · Stability of fixed points of maps
- Example 4.3 · Cobweb diagrams
- Example 4.4 · The logistic map
- Proposition 4.5 · The period-doubling bifurcation
- Definition 4.6 · Topological conjugacy
- Example 4.7 · The logistic map at r = 4 and the tent map
- Theorem 4.8 · Period three implies all periods
- Check 4.9 · Check: analysing a one-dimensional map
5 · Chaos10 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Sensitive dependence and Devaney chaos
- Proposition 5.2 · The doubling map is chaotic
- Definition 5.3 · Lyapunov exponents
- Proposition 5.4 · The shift map
- Theorem 5.5 · Symbolic dynamics for the logistic map
- Example 5.6 · The Lorenz system
- Proposition 5.7 · Volume contraction
- Definition 5.8 · Attractors and strange attractors
- Check 5.9 · Check: is it chaotic?
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Proposition 1.2 · The group property
; orbits are points, loops, or non-self-intersecting curves.
Proof
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