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Introduction to nonlinear dynamics
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Maps and difference equations
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Transients and attractors
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Parameters and bifurcations
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Field trip: Boulder Creek
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Maps I: Unit test
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Logistic Map App
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Return maps
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Constructing the bifurcation diagram
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Exploring the bifurcation diagram
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Feigenbaum and universality
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Field trip: The standard map (with Jim Meiss)
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Maps II: Unit test
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What is a flow?
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State variables and state space
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Introduction to ordinary differential equations (ODEs)
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Nonlinearity and nonintegrability
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Field trip: ODEs and the human insulin system (with Sriram Sankaranarayanan)
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Flows I: Unit test
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Pendulum App
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Fixed points and stability
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Saddle points and eigenvectors
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Stable and unstable manifolds
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Attractors, strange and otherwise
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Field trip: Using stable and unstable manifolds to design spacecraft trajectories (with Jeff Parker)
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Flows II: Unit test
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ODEs, vector fields, and dynamical landscapes
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Introduction to ODE solvers
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Two simple ODE solvers: forward and backward Euler
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Solving the simple harmonic oscillator ODEs
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Field trip: Systems that can't be modeled with ODEs (with Jean Hertzberg)
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Flows III: Unit test
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ODE solvers, round II: Error and adaptation
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Production ODE solvers
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Numerical dynamics and due diligence
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Shadowing and chaos
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Field trip: Solving partial differential equations (with Christine Hrenya)
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Flows IV: Unit test
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Dynamics and state-space deformation
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Lyapunov exponents
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Sections and projections
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Unstable periodic orbits
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Fractals and chaos
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Field trip: Diffusion-limited aggregation, fractals, and snowflakes (with Dave Feldman)
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Flows V: Unit test
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Time-series analysis and the observer problem
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Delay-coordinate embedding
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Topology, diffeomorphisms, and reconstruction of dynamics
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Estimating embedding parameters
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Caveats and extensions
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Field trip: Detecting extreme events (with Holger Kantz)
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Nonlinear time-series analysis I: Unit test
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Computing fractal dimensions
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Computing Lyapunov exponents
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Noise and filtering
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Field trip: Chaotic mixing and marine invertebrate reproduction (with John Crimaldi)
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Nonlinear time-series analysis II: Unit test
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Prediction
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Control of chaos
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Classical mechanics
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Music and dance (with a coda on the difference between chaos and complexity)
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Applications: Unit test
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Nonlinear Dynamics: Mathematical and Computational Approaches (2024)
Lead instructor: Liz Bradley