Engineering Math - Differential Equations and Dynamical Systems

Steve Brunton via YouTube

Start learning Write review

Explore differential equations and dynamical systems through this comprehensive 22-hour video series that bridges mathematical theory with real-world applications. Learn to model and analyze systems that change over time, from fluid dynamics and weather patterns to epidemiology and space mission design. Begin with calculus fundamentals including derivatives, power law, and chain rule before progressing through ordinary differential equations, matrix systems, and eigenvalue problems. Master solution techniques for first-order, second-order, and higher-order ODEs using exponential solutions, power series, and analytical methods. Dive into linear algebra concepts including eigenvalues and eigenvectors to solve matrix systems of differential equations. Analyze stability through phase portraits, fixed points, and linearization of nonlinear systems near equilibrium. Study forced systems using undetermined coefficients and variation of parameters methods, with applications in control theory and convolution. Develop numerical skills through finite difference methods, integration schemes including Forward Euler, Backward Euler, and Runge-Kutta methods, with hands-on coding examples in both Python and MATLAB. Investigate chaotic dynamics, uncertainty propagation, and error analysis in numerical simulations. Gain practical experience modeling physical systems like spring-mass-dampers, harmonic oscillators, and particles in potential wells while understanding the mathematical foundations that govern dynamic behavior across engineering and scientific disciplines.

Syllabus

Differential Equations and Dynamical Systems: Overview
Calculus Review: The Derivative (and the Power Law and Chain Rule)
Gentle Introduction to Modeling with Matrices and Vectors: A Probabilistic Weather Model
The Simplest Ordinary Differential Equation (ODE) and Its Exponential Solution
Solving Differential Equations with Power Series: A Simple Example
Taylor Series and Power Series Made Easy (with Pictures): Review of Calculus
Taylor Series of the Exponential Function and Euler's Formula!
Second-Order Ordinary Differential Equations: Solving the Harmonic Oscillator Four Ways
Example Second-Order ODE: Spring-Mass-Damper
More Examples of Second Order Differential Equations
High-Order Ordinary Differential Equations with More Derivatives (from Physics)
Solving General High-Order, Linear Ordinary Differential Equations (ODEs)
Matrix Systems of Differential Equations
Motivating Eigenvalues and Eigenvectors with Differential Equations
Eigenvalues and Eigenvectors
Solving Systems of Differential Equations with Eigenvalues and Eigenvectors
2x2 Systems of ODEs: Sources and Sinks
2x2 Systems of ODEs: Saddle Points and Instability
2x2 Systems of ODEs: Imaginary Eigenvalues and Center Fixed Points
Stability and Eigenvalues: What does it mean to be a "stable" eigenvalue?
What is a "Linear" Differential Equation?
Linearizing Nonlinear Differential Equations Near a Fixed Point
A Particle in a Potential Well: Nonlinear Dynamics
Drawing Phase Portraits for Nonlinear Systems
Phase Portrait for Double Well Potential
The Hartman-Grobman Theorem, Structural Stability of Linearization, and Stable/Unstable Manifolds
Non-Normal Linear Systems and Transient Energy Growth: Bypass Transition to Turbulence
Repeated Eigenvalues and Secular Terms: Transient Growth in Non-Normal Systems
Systems of Differential Equations: Diagonalization and Jordan Canonical Form
Differential Equations with Forcing: Method of Undetermined Coefficients
Differential Equations with Forcing: Method of Variation of Parameters
Systems of Differential Equations with Forcing: Example in Control Theory
Linear Systems of Differential Equations with Forcing: Convolution and the Dirac Delta Function
Forced Systems of Differential Equations in Matlab and Python
Numerical Differentiation with Finite Difference Derivatives
Numerical Differentiation: Second Derivatives and Differentiating Data
Why we can't take "dt" to 0 in a computer: Sources of error in numerical differentiation
Numerical Integration: Discrete Riemann Integrals and Trapezoid Rule
Numerical Simulation of Ordinary Differential Equations: Integrating ODEs
Deriving Forward Euler and Backward/Implicit Euler Integration Schemes for Differential Equations
Numerical Integration of ODEs with Forward Euler and Backward Euler in Python and Matlab
Error Analysis of Euler Integration Scheme for Differential Equations Using Taylor Series
Stability of Forward Euler and Backward Euler Integration Schemes for Differential Equations
Runge-Kutta Integrator Overview: All Purpose Numerical Integration of Differential Equations
Coding a Fourth-Order Runge-Kutta Integrator in Python and Matlab
Numerical Integration of Chaotic Dynamics: Uncertainty Propagation & Vectorized Integration
Chaotic Dynamical Systems
Engineering Math Pre-Req: Quick and Dirty Introduction to Python
Engineering Math Pre-Req: Quick and Dirty Introduction to Matlab