Nonlinear Differential Equation Course

Lesson 1 • Classification and Standard Forms

Categorises nonlinear ODEs by order, type, and structure. Provides a taxonomy students apply when selecting solution methods.

Lesson 2 • Existence and Uniqueness Theorems

Presents Picard-Lindelöf and Peano theorems for nonlinear equations. Students verify conditions before attempting analytical or numerical solutions.

Lesson 3 • Geometric Interpretation of Solutions

Introduces direction fields and integral curves as visual tools. Connects algebraic structure to solution behaviour in the phase plane.

Lesson 1 • Bernoulli and Riccati Equations

Transforms Bernoulli and Riccati equations into linear forms via substitution. Demonstrates how substitution reduces complexity.

Lesson 2 • Implicit Differentiation and Parametric Solutions

Handles equations solvable only in parametric or implicit form. Prepares students for singular solutions and envelope theory.

Lesson 3 • Substitution and Transformation Methods

Covers homogeneous substitution, Clairaut equations, and other transformations. Expands the toolkit for equations resisting standard methods.

Lesson 4 • Separation of Variables

Applies separation to first-order nonlinear equations and evaluates implicit solutions. Reinforces integration skills central to later chapters.

Lesson 1 • Equilibrium Points and Linearisation

Locates equilibria and applies Jacobian linearisation to classify them. Links local linear behaviour to nonlinear dynamics near fixed points.

Lesson 2 • Lyapunov's Direct Method

Constructs Lyapunov functions to prove stability without explicit solutions. Covers energy-based and quadratic candidate functions.

Lesson 3 • Autonomous Systems and Phase Portraits

Converts scalar equations to first-order systems and plots trajectories. Establishes the phase plane as the primary analysis tool.

Lesson 4 • Stability Definitions and Criteria

Defines Lyapunov, asymptotic, and global stability rigorously. Students apply definitions to determine long-term behaviour of solutions.

Lesson 1 • Hopf Bifurcation

Derives conditions for the birth of limit cycles from equilibria in planar systems. Students apply the Hopf criterion to applied models.

Lesson 2 • Saddle-Node and Transcritical Bifurcations

Analyses creation and exchange of equilibria in one-dimensional systems. Students derive normal forms and sketch bifurcation diagrams.

Lesson 3 • Pitchfork Bifurcations

Covers supercritical and subcritical pitchfork bifurcations and symmetry breaking. Connects to buckling and symmetry-breaking phenomena.

Lesson 4 • Global Bifurcations

Introduces homoclinic and heteroclinic bifurcations that cannot be detected locally. Connects global orbit structure to parameter changes.

Lesson 1 • Regular Perturbation Expansions

Constructs power-series solutions in a small parameter for weakly nonlinear equations. Establishes the framework for all perturbation methods.

Lesson 2 • Singular Perturbation and Boundary Layers

Handles equations where the small parameter multiplies the highest derivative. Students construct inner, outer, and composite expansions.

Lesson 3 • Averaging Method

Applies Krylov-Bogoliubov averaging to slowly varying oscillators. Reduces nonautonomous systems to autonomous averaged equations.

Lesson 4 • Multiple Scales Method

Introduces slow and fast time scales to remove secular terms in oscillatory systems. Applies to Duffing and van der Pol oscillators.

Lesson 1 • Stiffness and Implicit Methods

Identifies stiff nonlinear systems and applies implicit schemes to maintain stability. Covers backward Euler and trapezoidal methods.

Lesson 2 • Multistep and Predictor-Corrector Methods

Introduces Adams-Bashforth and Adams-Moulton schemes for efficiency. Analyses zero-stability and convergence of multistep methods.

Lesson 3 • Boundary Value Problem Solvers

Applies shooting, finite difference, and collocation methods to nonlinear BVPs. Compares convergence and implementation complexity.

Lesson 4 • Error Analysis and Convergence

Quantifies global error, consistency, and convergence for nonlinear solvers. Students verify numerical solutions against known analytical benchmarks.

Lesson 1 • Routes to Chaos

Surveys period-doubling, quasiperiodicity, and intermittency as routes to chaos. Students identify which route applies to a given system.

Lesson 2 • Lyapunov Exponents

Computes Lyapunov exponents to quantify chaos and attractor geometry. Connects positive exponents to chaotic behaviour.

Lesson 3 • Lorenz System and Strange Attractors

Analyses the Lorenz equations as a prototype chaotic system. Students identify the strange attractor and its fractal structure.

Lesson 4 • Poincaré Maps and Return Maps

Reduces continuous dynamics to discrete maps via Poincaré sections. Connects fixed points of maps to periodic orbits of flows.

Lesson 1 • Travelling Wave Solutions

Reduces nonlinear PDEs to ODEs via travelling wave ansatz. Applies phase plane analysis to classify wave profiles.

Lesson 2 • Solitons and Integrable Systems

Introduces the KdV equation and soliton solutions via inverse scattering. Highlights conservation laws and integrability conditions.

Lesson 3 • Hamiltonian and Gradient Systems

Identifies Hamiltonian structure and gradient flow in nonlinear systems. Uses conserved quantities to simplify analysis.

Lesson 4 • Reaction-Diffusion Systems

Analyses Turing instability and pattern formation in coupled reaction-diffusion PDEs. Connects to biological and chemical applications.