Unit 9 of 13

Nonlinear Dynamics and Stability

  • 7 lessons
  • Videos 49 to 55

Nonlinear systems are studied through the geometry of their solutions. This unit uses phase planes, linearization and Lyapunov functions, then bifurcations, limit cycles and an introduction to chaos.

Nullclines, equilibria and the direction of flow organize a phase plane. The Jacobian decides local stability at a hyperbolic equilibrium, while a zero or purely imaginary eigenvalue needs another argument, such as a Lyapunov function. As a parameter changes, equilibria appear, exchange stability or split in saddle-node, transcritical and pitchfork bifurcations. Limit cycles are found with the Poincaré-Bendixson theorem or ruled out by the Bendixson-Dulac criterion, and in a Hopf bifurcation a small cycle branches from an equilibrium. The unit ends with chaos in the Lorenz system.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Phase Planes and Stability

  1. Nonlinear Phase Planes and Nullclines

    How to read a two-state autonomous system in its phase plane, with the vector field, nullclines, equilibrium points and trajectories, and how a conserved quantity reveals the cycles of a predator-prey model.

    Video 49

  2. Linearization and Local Stability

    How the Jacobian matrix gives the linearization of a nonlinear system at an equilibrium, when a hyperbolic equilibrium's eigenvalues decide local asymptotic stability, and why nonhyperbolic and global questions need more.

    Video 50

  3. Lyapunov Functions, Energy, and Invariance

    How a positive definite Lyapunov function with a nonpositive orbital derivative proves Lyapunov stability, and how LaSalle's invariance principle and invariant sets give convergence when that derivative is only semidefinite.

    Video 51

Bifurcations and Limit Cycles

  1. One-Dimensional Bifurcations

    How a parameter changes the equilibria of a scalar equation, read from a bifurcation diagram, with saddle-node, transcritical and pitchfork bifurcations, exchange of stability and reflection symmetry.

    Video 52

  2. Limit Cycles and Planar Limit-Set Theorems

    How to recognize periodic orbits and isolated limit cycles in the plane, and how trapping regions, omega-limit sets, the Poincaré-Bendixson theorem and the Bendixson-Dulac criterion prove or rule out closed orbits.

    Video 53

  3. Hopf Bifurcation and Oscillation Onset

    How a small periodic orbit is born from an equilibrium as a complex pair of eigenvalues crosses the imaginary axis, with the polar normal form, transversality, nondegeneracy and the first Lyapunov coefficient.

    Video 54

Sensitive Dependence and Chaos

  1. Chaos and Sensitive Dependence

    What chaos means and what it takes to establish it, with sensitive dependence on initial conditions, the Lorenz system and its volume contraction, attractors, the largest Lyapunov exponent and the forced Duffing equation.

    Video 55