Nonlinear Dynamics and Chaos with Steven Strogatz

Video Lectures

Displaying all 25 video lectures.
Lecture 1
Course Introduction and Overview
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Course Introduction and Overview
Historical and logical overview of nonlinear dynamics. The structure of the course: work our way up from one to two to three-dimensional systems. Simple examples of linear vs. nonlinear systems. 1-D systems. Why pictures are more powerful than formulas for analyzing nonlinear systems. Fixed points. Stable and unstable fixed points. Example: Logistic equation in population biology.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Chapter 1 and Section 2.0--2.3.
Lecture 2
One Dimensional Systems
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One Dimensional Systems
Linearization for 1-D systems. Existence and uniqueness of solutions. Bifurcations. Saddle-node bifurcation. Bifurcation diagrams.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 3.0--3.2, 3.4.
Lecture 3
Overdamped Bead on a Rotating Hoop
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Overdamped Bead on a Rotating Hoop
Equations of motion. When can we neglect the second derivative? Dimensional analysis and scaling. A singular limit.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Section 3.5.
Lecture 4
Model of an Insect Outbreak
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Model of an Insect Outbreak
Model of spruce budworm outbreaks in the forests of northeastern Canada and United States. Nondimensionalization. Saddle-node bifurcations. Jump phenomena. Hysteresis. Cusp catastrophe.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Section 3.7.
Lecture 5
Two Dimensional Linear Systems
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Two Dimensional Linear Systems
Phase plane analysis. Eigenvectors and eigenvalues. Classification of 2-D linear systems. Saddle points. Stable and unstable nodes and spirals. Centers. Non-isolated fixed points.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Chapter 5 and Sections 6.0--6.2.
Lecture 6
Two Dimensional Nonlinear Systems: Fixed Points
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Two Dimensional Nonlinear Systems: Fixed Points
Linearization. Jacobian matrix. Borderline cases. Example: Centers are delicate. Polar coordinates. Example of phase plane analysis: rabbits versus sheep (Lotka-Volterra model of competition in population biology). Stable manifold of a saddle point.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 6.3, 6.4.
Lecture 7
Conservative Systems
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Conservative Systems
Mechanical systems with one degree of freedom. Particle in a double well. Symmetry. Homoclinic orbits. Energy surface. Theorem about nonlinear centers. Pendulum. Cylindrical phase space.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 6.5, 6.7.
Lecture 8
Index Theory and Introduction to Limit Cycles
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Index Theory and Introduction to Limit Cycles
Index of a curve (with respect to a given vector field). Properties of the index. Index of a point. Using index theory to rule out closed trajectories. Some strange things: Index theory in biology. Hairy ball theorem. Combing a torus and connection to tokamaks and fusion. Index theory on compact orientable 2-manifolds.
Limit cycles. Examples in science and engineering. Definitions, and why linear systems can't have limit cycles.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 6.8, 7.0, 7.1.
Lecture 9
Testing for Closed Orbits
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Testing for Closed Orbits
Techniques for ruling out closed orbits: index theory and Dulac's criterion. Techniques for proving closed orbits exist: Poincaré-Bendixson theorem. Trapping regions. Example in polar coordinates. Simple model of oscillatory glycolysis in metabolism. Nullclines.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 7.2, 7.3.
Lecture 10
Van der Pol Oscillator
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Van der Pol Oscillator
Origins of the van der Pol oscillator in radio engineering. Strongly nonlinear limit. Liénard transformation. Relaxation oscillations. Weakly nonlinear limit. Energy method for estimating the amplitude of the limit cycle.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 7.4--7.6.
Lecture 11
Averaging Theory for Weakly Nonlinear Oscillators
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Averaging Theory for Weakly Nonlinear Oscillators
Derivation of averaged equations for slowly-varying amplitude and phase. Explicit solution of amplitude equation for weakly nonlinear van der Pol oscillator. Duffing equation.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Section 7.6.
Lecture 12
Bifurcations in Two Dimensional Systems
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Bifurcations in Two Dimensional Systems
Bifurcations of fixed points: saddle-node, transcritical, pitchfork. Hopf bifurcations. Other bifurcations of periodic orbits.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.0--8.2.
Lecture 13
Hopf Bifurcations in Aeroelastic Instabilities and Chemical Oscillators
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Hopf Bifurcations in Aeroelastic Instabilities and Chemical Oscillators
Supercritical vs subcritical Hopf. Airplane wing vibrations. Flutter. Chemical oscillations. Computer simulations.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.2, 8.3.
Lecture 14
Global Bifurcations of Cycles
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Global Bifurcations of Cycles
Hopf, saddle-node bifurcation of cycles, SNIPER, and homoclinic bifurcation. Coupled oscillators. Knotted cycles. Quasiperiodicity.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.4, 8.6.
Lecture 15
Chaotic Waterwheel
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Chaotic Waterwheel
Schematic diagram of the waterwheel. Video of waterwheel in action. Derivation of governing equations for the waterwheel. Contuinuity equation and torque balance. Amplitude equations. Using orthogonality of the Fourier modes. A miracle: Exact decoupling of a three-dimensional subsystem from the rest of the modes.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 9.0, 9.1.
Lecture 16
Waterwheel Equations and Lorenz Equations
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Waterwheel Equations and Lorenz Equations
Analysis of the waterwheel equations. Lorenz equations. Simple properties of the Lorenz system. Volume contraction. Fixed points and their linear stability.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 9.1, 9.2.
Lecture 17
Chaos in the Lorenz Equations
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Chaos in the Lorenz Equations
Global stability for the origin for r is less than 1. Liapunov function. Boundedness. Hopf bifurcations. No quasiperiodicity. Simulations of the Lorenz system. Stories of how Lorenz made his discovery. Strange attractor and butterfly effect. Exponential divergence of nearby trajectories. Liapunov exponent. Predictability horizon for the weather and solar system. Formula for the predictability horizon (Liapunov time), based on rate of separation of nearby trajectories. What is meant by saying that chaotic systems are unpredictable?

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 9.2, 9.3.
Lecture 18
Strange Attractor for the Lorenz Equations
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Strange Attractor for the Lorenz Equations
Defining attractor, chaos, and strange attractor. Transient chaos in games of chance. Dynamics on the Lorenz attractor. Reduction to a 1-D map: the Lorenz map. Ruling out stable limit cycles for the Lorenz system when r = 28. Cobweb diagrams.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 9.3, 9.4.
Lecture 19
One Dimensional Maps
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One Dimensional Maps
Logistic map: a simple mathematical model with very complicated dynamics. Influential article by Robert May. Numerical results: Fixed points. Cycles of period 2, 4, 8, 16, .... The period-doubling route to chaos. An icon of chaos: The orbit diagram. Chaos intermingled with periodic windows. Period-3 window. Analytical results: Fixed points and their stability. Flip bifurcation (eigenvalue = --1) at period doubling. Period-2 points and their stability.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 10.0-10.2.
Lecture 20
Universal Aspects of Period Doubling
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Universal Aspects of Period Doubling
Exploring the logistic map and period doubling with online applets. Interactive cobweb diagrams. Interactive orbit diagram. Zooming in to see the periodic windows. Self-similar fractal structure: each periodic window contains miniature copies of the whole orbit diagram. Smooth curves running the orbit diagram: supertracks. How are periodic windows born? Example: Birth of period three. Tangent bifurcation. The mechanism underlying the fractal structure.
Introduction to Mitchell Feigenbaum's work on universality, what he found, and why it matters. Testable predictions about periodic doubling in physical and chemical systems. Sine map vs. logistic map. The universal scaling constants alpha and delta.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 10.2¬--10.4, 10.6.
Lecture 21
Feigenbaum's Renormalization Analysis of Period Doubling
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Feigenbaum's Renormalization Analysis of Period Doubling
Superstable fixed points and cycles. Intuition behind renormalization, based on self-similarity. Renormalization transformation. Defining a family of universal functions. Explaining geometrically where the universal aspects of period doubling come from. Functional equation for alpha and the universal function g.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 10.7.
Lecture 22
Renormalization: Function Space and a Hands-on Calculation
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Renormalization: Function Space and a Hands-on Calculation
The concept of an infinite-dimensional space of functions. Each point represents a function. Renormalization transformation T as a mapping. Sheets of functions with the same superstability type. The universal function g as a saddle point of T. The scaling factor delta is the unstable eigenvalue of DT. Concrete renormalization, using only high school algebra, gives alpha and delta to within about 10 percent.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 10.7.
Lecture 23
Fractals and the Geometry of Strange Attractors
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Fractals and the Geometry of Strange Attractors
Analogy to making pastry. The geometry underlying chaos: Stretching, folding, and reinjection of phase space. The same process generates the fractal microstructure of strange attractors. Rössler attractor. Visualizing a strange attractor as an "infinite complex of surfaces" (in the words of Edward Lorenz). The Cantor set as a model for the cross-section of strange attractors. Dimension of Cantor set and other self-similar fractals.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 11.0--11.3, 12.0, 12.1, 12.3.
Lecture 24
Hénon Map
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Hénon Map
The Hénon map: a two-dimensional map that sheds light on the fractal structure of strange attractors. Deriving the Hénon map. Analyzing the map. Simulations.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Section 12.2.
Lecture 25
Using Chaos to Send Secret Messages
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Using Chaos to Send Secret Messages
Lou Pecora and Tom Carroll's work on synchronized chaos. Proof of synchronization by He and Vaidya, using a Liapunov function. Kevin Cuomo and Alan Oppenheim's approach to sending secret messages with chaos. Secure versus private communications. Anecdotes about Princess Diana, the early days of cell phones, Francis Ford Coppola's movie "The Conversation." Parameter modulation and signal masking. Demonstration of Cuomo's method for synchronizing Lorenz circuits and using them to send and receive private messages.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Section 9.6.