Dynamical systems
Undergraduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
First-order and planar systems
- Autonomous ODEs; equilibrium points and phase lines
- Bifurcations in one-dimensional systems (saddle-node, transcritical)
- Planar linear systems: classification via eigenvalues
- Phase plane analysis and nullclines
- Liapunov functions (introduction)
Nonlinear dynamics
- Linearization and Hartman–Grobman theorem (statement)
- Stable and unstable manifolds (introduction)
- Limit cycles and Poincaré–Bendixson theorem (2D)
- Hamiltonian and gradient systems (introduction)
- Index theory for closed orbits (optional)
Discrete dynamical systems
- Iterated maps and cobweb diagrams
- Fixed points and stability for discrete systems
- Period doubling and route to chaos (logistic map)
- Symbolic dynamics and shift maps (introduction)
- Fractals and sensitive dependence (overview)
STEM / applied
Applications in science and engineering
- Population dynamics: logistic growth and harvesting models
- Epidemic models (SIR) and compartmental ODEs
- Mechanical systems: pendulum and coupled oscillators
- Chemical kinetics and reaction–diffusion (introduction)
- Control of equilibria and feedback linearization (preview)
Computation and analysis
- Numerical phase portrait generation
- Poincaré sections and return maps
- Parameter continuation and bifurcation diagrams
- Time series analysis and Lyapunov exponents (introduction)
- Chaos in physical and biological systems (case studies)
Notes
Topics reflect common dynamical systems syllabi at US colleges and universities. Prerequisites typically include differential equations and linear algebra.