Hyperbolic vs Non-Hyperbolic Fixed Points - Computing Their Invariant Manifolds via Taylor Series
Explore hyperbolic and non-hyperbolic fixed points, their invariant manifolds, and computational methods using Taylor series expansion in dynamical systems and nonlinear dynamics.
Stability of Periodic Orbits - Floquet Theory - Stable and Unstable Invariant Manifolds
Explore stability of periodic orbits using Floquet theory. Learn about monodromy matrices, Floquet multipliers, and invariant manifolds in Hamiltonian systems. Understand the foundations of chaos in dynamical systems.
Periodic Systems and Periodic Motion - Parametric Resonance Tongues of Instability, Mathieu Equation
Explore periodic systems, parametric resonance, and the Mathieu equation. Learn about resonance tongues, stability analysis, and the fascinating Kapitza pendulum phenomenon in nonlinear dynamics.
Hamiltonian Normal Forms, Lie Series Method for Transformations - Infinite-Dimensional Hamiltonians
Explore Hamiltonian normal forms, Lie series transformations, and infinite-dimensional systems. Learn to simplify complex dynamics and analyze PDEs like shallow water equations.
Poisson Brackets, Non-Canonical Hamiltonian Systems and Euler's Rigid Body Equations
Explore Poisson brackets, non-canonical Hamiltonian systems, and Euler's rigid body equations in this advanced dynamics lecture. Gain insights into abstract algebra, Lie algebra, and their applications in physics and mechanics.
Integrable and Non-Integrable Hamiltonian Systems - KAM Tori - Poincare Sections - Poisson Brackets
Explore integrable and non-integrable Hamiltonian systems, KAM theory, Poincaré sections, and Poisson brackets in this advanced dynamics lecture, offering insights into complex nonlinear systems and their behavior.
Lagrangian Coherent Structures Introduction
Explore Lagrangian coherent structures in fluid flows, revealing particle transport patterns. Learn theoretical foundations and applications in fluid mechanics, with insights on reduced order modeling.
Hamilton-Jacobi Theory - Finding the Best Canonical Transformation + Examples
Explore Hamilton-Jacobi theory for optimal canonical transformations in Hamiltonian systems. Learn to solve the Hamilton-Jacobi equation and apply it to oscillators and central force problems.
Generating Function of a Canonical Transformation - Examples and the Big Picture
Explore generating functions for canonical transformations in Hamiltonian systems. Learn about harmonic oscillator examples, near-identity transformations, and connections to quantum mechanics and gauge invariance.
Principle of Least Action and Lagrange's Equations of Mechanics - Basics of Calculus of Variations
Explore variational principles in mechanics, including the Principle of Least Action and Lagrange's equations. Learn calculus of variations techniques and their applications in physics and engineering problems.
Hamiltonian Systems Introduction - Why Study Them?
Explore Hamiltonian systems and their advantages in mechanics. Learn about generalized momentum, canonical equations, and the transformation from Lagrangian to Hamiltonian formalism.
Bifurcations in 2D, Part 4 - Global Bifurcations, Limit Cycle Creation - Homoclinic Bifurcation
Explores global bifurcations in 2D systems, focusing on limit cycle creation through saddle-node, SNIPER, and homoclinic bifurcations. Examines universal behaviors and applications to biological systems.
Limit Cycles - Poincare-Bendixson Worked Examples: Glycolysis
Explore Poincaré-Bendixson theorem applications in limit cycle analysis, including a glycolysis model. Learn to construct trapping regions using nullclines and understand parameter spaces for stable limit cycles.
Insect Outbreak Model - A Cusp Catastrophe Population Dynamics Example
Explore insect outbreak modeling using cusp catastrophe theory, analyzing population dynamics, fixed points, and bifurcations in a budworm-fir tree ecosystem.
The Problem of Traffic: A Mathematical Modeling Journey
Explore mathematical modeling of traffic flow, from individual car behavior to stop-and-go patterns, using differential-delay equations. Learn about model building, assumptions, and solving complex real-world problems.
Basic Excel Business Analytics - Intro to Linear Programming Using Algebra, Pencil & Paper
Introduction to linear programming using algebra and graphing methods to solve manufacturing problems and maximize contribution margin, with focus on mathematical modeling and feasible region analysis.
Koopman Analysis
Explore advanced techniques in dynamical systems analysis, including Koopman theory, spectral analysis, and applications to control and chaos in various scientific domains.
Dynamical Systems
Explore advanced techniques in dynamical systems analysis, including SINDy, Koopman theory, and deep learning applications for modeling complex nonlinear systems.
Sparsity and Compression
Explore sparsity and compression techniques, from compressed sensing to robust regression, with practical applications in image processing, signal analysis, and machine learning.
Euler’s Method - How to Use It
Comprehensive guide to Euler's method for approximating solutions to differential equations, covering its application, methodology, limitations, accuracy, and error analysis.
Julia for Biologists - A Perspective on Speed, Abstraction, and Metaprogramming
Explore Julia's potential in biology: speed, abstraction, and metaprogramming for data analysis, modeling, and research. Learn practical examples and discover the Julia ecosystem for biologists.