Nonlinear Dynamics and Chaos

Flows, stability, oscillation and chaos, seen through their trajectories

About this course

Flows, stability, bifurcations, oscillations, iterated maps and chaos, from one-dimensional flows through the phase plane to the Lorenz system, ending with a runnable laboratory capstone.

What you will be able to do

  • Find fixed points and classify their stability and bifurcations in one-dimensional flows
  • Sketch phase portraits of planar systems using linearisation and nullclines
  • Use energy and Lyapunov functions to establish stability
  • Identify limit cycles and Hopf bifurcations in planar oscillators
  • Analyse iterated maps, period doubling and chaos using Lyapunov exponents and Poincare maps

What this course covers

  • Phase portraits and the matrix exponential solution
  • Real eigenvalues: nodes, saddles, and invariant directions
  • Complex eigenvalues: spirals and linear centers
  • Trace-determinant classification map
  • Repeated eigenvalues, transient growth, and linearization limits
  • Iterated maps and their basic setup
  • Cobweb diagrams as geometric iteration
  • Fixed-point stability and the derivative multiplier
  • Global contraction on an invariant closed interval
  • Periodic orbits, cycle stability, and the logistic map
  • Limit Cycles vs. Periodic Orbits and Centers
  • Circular Limit Cycle: Exact Radial Analysis
  • Poincaré–Bendixson Theorem and Divergence Criterion
  • Van der Pol Oscillator and Rigorous Limit-Cycle Reasoning
  • State space and autonomous differential equations
  • Logistic model and dimensionless normalisation
  • Phase-line analysis, equilibria, and stability
  • Linearisation near equilibria and uniqueness of trajectories
  • Designing a precise, measurable experiment question
  • Damped double-well oscillator and energy balance check
  • Hopf normal form, logistic map, and Lorenz sensitivity experiments
  • Stroboscopic map, experimental controls, and capstone reporting
  • Lyapunov stability, attraction, and asymptotic stability
  • Phase-line sign analysis and the derivative shortcut
  • Basins of attraction and gradient-flow potential
  • Non-hyperbolic fixed points and the inconclusive linear test
  • Nullclines: definition, properties, and pitfalls
  • Sign-region analysis for the damped double-well system
  • Jacobian classification and basin boundary via saddle manifold
  • Energy function as global consistency check
  • Linear signature of a Hopf bifurcation
  • Supercritical Hopf bifurcation and radial normal form
  • Subcritical Hopf bifurcation, bistability, and hysteresis
  • Genericity conditions and numerical investigation of Hopf bifurcations
  • Logistic map fixed points and their stability
  • Period-doubling bifurcation and the two-cycle
  • Bifurcation diagram construction and numerical limitations
  • Feigenbaum scaling, periodic windows, and universality
  • Stretching and folding as the mechanism for bounded complexity
  • Lyapunov Exponent Definition and Scalar Map Formula
  • Interpreting Exponent Sign and Its Limits
  • Checking and Computing Logistic Exponents
  • Prediction Horizon and Finite-Separation Saturation
  • Extension to Continuous Flows and Interpretation Limits
  • Pendulum state space and equations of motion
  • Equilibria and their linear stability
  • Conserved energy and phase portrait structure
  • Libration period, elliptic integral, and amplitude dependence
  • Bifurcation diagrams: structure and reading
  • Saddle-node (fold) bifurcation and critical slowing
  • Transcritical bifurcation and stability exchange
  • Supercritical and subcritical pitchfork bifurcations
  • Comparing normal forms and limits of static diagrams
  • Lyapunov Stability: Definitions and Geometric Setup
  • Stability vs. Asymptotic Stability: Non-Positive vs. Strictly Negative Derivative
  • LaSalle's Invariance Principle
  • Quadratic Lyapunov Functions and Global vs. Local Certificates
  • Lorenz equations: structure and state space
  • Equilibria: location and local stability
  • Phase-volume contraction and global boundedness
  • Sensitive dependence and responsible numerical evidence
  • Circle as State Space for Phase Dynamics
  • The Adler Phase Equation and Phase Locking
  • Phase Slipping and the Slip-Period Formula
  • Saddle-Node on an Invariant Circle (SNIC) Bifurcation
  • Mutual Coupling and the Factor-of-Two Effective Coupling
  • Poincaré Return Map Construction
  • Fixed Points, Multipliers, and Transverse Stability
  • Stroboscopic Map for Periodically Forced Oscillators
  • Interpreting Strobe Plots and the Dimension Argument

Syllabus

  1. One-Dimensional Flows

    1. L1 · States, Flows and the Geometry of Change
    2. L2 · Fixed Points and Stability on the Line
    3. L3 · Bifurcations in One Dimension
    4. L4 · Flows on a Circle and Synchronization

  2. The Phase Plane

    1. L5 · Linear Systems and the Phase Plane
    2. L6 · Nullclines and Nonlinear Phase Portraits
    3. L7 · The Pendulum and Conservative Dynamics
    4. L8 · Dissipation, Energy and Lyapunov Functions

  3. Oscillations and Hopf Bifurcations

    1. L9 · Limit Cycles and Planar Oscillations
    2. L10 · Hopf Bifurcations and the Birth of Oscillation

  4. Maps and Chaos

    1. L11 · Iterated Maps and Cobweb Dynamics
    2. L12 · Period Doubling and the Logistic Map
    3. L13 · Sensitivity and Lyapunov Exponents
    4. L14 · The Lorenz System and Deterministic Chaos
    5. L15 · Poincare Maps and Driven Oscillators

  5. Laboratory Capstone

    1. L16 · A Nonlinear Dynamics Laboratory and Capstone