Standard syllabus
Dynamical systems · Graduate · Math
Learning objectives from the Dynamical systems syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Dynamical Systems — scope drawn from open graduate dynamical systems notes and typical US graduate dynamical systems syllabi.
Flows and invariant sets
- Existence — Existence
- Uniqueness for ODEs — Uniqueness for ODEs
- Flows, orbits, and invariant sets — Flows, orbits, and invariant sets
- Limit sets, ω-limit sets, and attractors — Limit sets, ω-limit sets, and attractors
- Lyapunov stability — Lyapunov stability
- Lyapunov functions — Lyapunov functions
- Stable and unstable manifolds (local theory) — Stable and unstable manifolds (local theory)
Bifurcation theory
- Parameter-dependent equilibria — Parameter-dependent equilibria; structural stability (introduction)
- Saddle-node and transcritical bifurcations — Saddle-node and transcritical bifurcations
- Pitchfork bifurcations — Pitchfork bifurcations
- Hopf bifurcation — Hopf bifurcation
- Limit cycle creation — Limit cycle creation
- Normal forms — Normal forms
- Center manifold reduction (introduction) — Center manifold reduction (introduction)
- Codimension-two bifurcations (overview) — Codimension-two bifurcations (overview)
Discrete and symbolic dynamics
- Diffeomorphisms and return maps — Diffeomorphisms and return maps
- Smale horseshoe and symbolic dynamics — Smale horseshoe and symbolic dynamics
- Topological entropy (introduction) — Topological entropy (introduction)
- Period doubling cascade and universality — Period doubling cascade and universality
- Hamiltonian dynamics — Hamiltonian dynamics
- KAM theory (overview) — KAM theory (overview)
Applied dynamical systems
- Pattern formation in PDEs (reaction–diffusion, introduction) — Pattern formation in PDEs (reaction–diffusion, introduction)
- Celestial mechanics — Celestial mechanics
- N-body problems (overview) — N-body problems (overview)
- Neuronal dynamics and bursting models — Neuronal dynamics and bursting models
- Epidemiological — Epidemiological
- Ecological models with bifurcations — Ecological models with bifurcations
- Control of nonlinear systems near equilibria — Control of nonlinear systems near equilibria
Computation and analysis tools
- Numerical continuation — Numerical continuation
- Bifurcation software (AUTO, MatCont overview) — Bifurcation software (AUTO, MatCont overview)
- Lyapunov exponent computation — Lyapunov exponent computation
- Delay differential equations (introduction) — Delay differential equations (introduction)
- Random dynamical systems (introduction) — Random dynamical systems (introduction)
- Data-driven reconstruction of attractors (overview) — Data-driven reconstruction of attractors (overview)
Learning objectives
Click an objective for study materials.
Flows and invariant sets
- Existence — Existence
- Uniqueness for ODEs — Uniqueness for ODEs
- Flows, orbits, and invariant sets — Flows, orbits, and invariant sets
- Limit sets, ω-limit sets, and attractors — Limit sets, ω-limit sets, and attractors
- Lyapunov stability — Lyapunov stability
- Lyapunov functions — Lyapunov functions
- Stable and unstable manifolds (local theory) — Stable and unstable manifolds (local theory)
Bifurcation theory
- Parameter-dependent equilibria — Parameter-dependent equilibria; structural stability (introduction)
- Saddle-node and transcritical bifurcations — Saddle-node and transcritical bifurcations
- Pitchfork bifurcations — Pitchfork bifurcations
- Hopf bifurcation — Hopf bifurcation
- Limit cycle creation — Limit cycle creation
- Normal forms — Normal forms
- Center manifold reduction (introduction) — Center manifold reduction (introduction)
- Codimension-two bifurcations (overview) — Codimension-two bifurcations (overview)
Discrete and symbolic dynamics
- Diffeomorphisms and return maps — Diffeomorphisms and return maps
- Smale horseshoe and symbolic dynamics — Smale horseshoe and symbolic dynamics
- Topological entropy (introduction) — Topological entropy (introduction)
- Period doubling cascade and universality — Period doubling cascade and universality
- Hamiltonian dynamics — Hamiltonian dynamics
- KAM theory (overview) — KAM theory (overview)
Applied dynamical systems
- Pattern formation in PDEs (reaction–diffusion, introduction) — Pattern formation in PDEs (reaction–diffusion, introduction)
- Celestial mechanics — Celestial mechanics
- N-body problems (overview) — N-body problems (overview)
- Neuronal dynamics and bursting models — Neuronal dynamics and bursting models
- Epidemiological — Epidemiological
- Ecological models with bifurcations — Ecological models with bifurcations
- Control of nonlinear systems near equilibria — Control of nonlinear systems near equilibria
Computation and analysis tools
- Numerical continuation — Numerical continuation
- Bifurcation software (AUTO, MatCont overview) — Bifurcation software (AUTO, MatCont overview)
- Lyapunov exponent computation — Lyapunov exponent computation
- Delay differential equations (introduction) — Delay differential equations (introduction)
- Random dynamical systems (introduction) — Random dynamical systems (introduction)
- Data-driven reconstruction of attractors (overview) — Data-driven reconstruction of attractors (overview)
Definitions and structure
- Existence — Existence
- Uniqueness for ODEs — Uniqueness for ODEs
- Flows, orbits, and invariant sets — Flows, orbits, and invariant sets
- Limit sets, ω-limit sets, and attractors — Limit sets, ω-limit sets, and attractors
- Lyapunov stability — Lyapunov stability
Proofs and reasoning
- Lyapunov functions — Lyapunov functions
- Stable and unstable manifolds (local theory) — Stable and unstable manifolds (local theory)
- Parameter-dependent equilibria — Parameter-dependent equilibria; structural stability (introduction)
- Saddle-node and transcritical bifurcations — Saddle-node and transcritical bifurcations
- Pitchfork bifurcations — Pitchfork bifurcations
Abstraction and generalization
- Hopf bifurcation — Hopf bifurcation
- Limit cycle creation — Limit cycle creation
- Normal forms — Normal forms
- Center manifold reduction (introduction) — Center manifold reduction (introduction)
- Codimension-two bifurcations (overview) — Codimension-two bifurcations (overview)
Modeling and computation
- Apply existence in engineering contexts — Apply existence in engineering contexts
- Apply uniqueness for odes in engineering contexts — Apply uniqueness for odes in engineering contexts
- Apply flows, orbits, and invariant sets in engineering contexts — Apply flows, orbits, and invariant sets in engineering contexts
- Apply limit sets, ω-limit sets, and attractors in engineering contexts — Apply limit sets, ω-limit sets, and attractors in engineering contexts
- Apply lyapunov stability in engineering contexts — Apply lyapunov stability in engineering contexts
Data and technology
- Use software to explore dynamical systems problems numerically — Use software to explore dynamical systems problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Lyapunov functions — Lyapunov functions
- Stable and unstable manifolds (local theory) — Stable and unstable manifolds (local theory)
- Parameter-dependent equilibria — Parameter-dependent equilibria; structural stability (introduction)
- Saddle-node and transcritical bifurcations — Saddle-node and transcritical bifurcations
- Pitchfork bifurcations — Pitchfork bifurcations
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Flows and invariant sets
Existence
Coming soon - Bifurcation theory
Parameter-dependent equilibria; structural stability (introduction)
Coming soon - Discrete and symbolic dynamics
Diffeomorphisms and return maps
Coming soon - Applied dynamical systems
Pattern formation in PDEs (reaction–diffusion, introduction)
Coming soon - Computation and analysis tools
Numerical continuation
Coming soon
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$1,162 · Dynamical systems · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.