# Nonlinear Dynamics and Chaos with Steven Strogatz

## Video Lectures

Displaying all 25 video lectures.

Lecture 1Play Video |
Course Introduction and OverviewHistorical 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 2Play Video |
One Dimensional SystemsLinearization 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 3Play Video |
Overdamped Bead on a Rotating HoopEquations 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 4Play Video |
Model of an Insect OutbreakModel 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 5Play Video |
Two Dimensional Linear SystemsPhase 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 6Play Video |
Two Dimensional Nonlinear Systems: Fixed PointsLinearization. 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 7Play Video |
Conservative SystemsMechanical 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 8Play Video |
Index Theory and Introduction to Limit CyclesIndex 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 9Play Video |
Testing for Closed OrbitsTechniques 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 10Play Video |
Van der Pol OscillatorOrigins 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 11Play Video |
Averaging Theory for Weakly Nonlinear OscillatorsDerivation 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 12Play Video |
Bifurcations in Two Dimensional SystemsBifurcations 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 13Play Video |
Hopf Bifurcations in Aeroelastic Instabilities and Chemical OscillatorsSupercritical vs subcritical Hopf. Airplane wing vibrations. Flutter. Chemical oscillations. Computer simulations. Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 8.2, 8.3. |

Lecture 14Play Video |
Global Bifurcations of CyclesHopf, 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 15Play Video |
Chaotic WaterwheelSchematic 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 16Play Video |
Waterwheel Equations and Lorenz EquationsAnalysis 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 17Play Video |
Chaos in the Lorenz EquationsGlobal 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 18Play Video |
Strange Attractor for the Lorenz EquationsDefining 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 19Play Video |
One Dimensional MapsLogistic 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 20Play Video |
Universal Aspects of Period DoublingExploring 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 21Play Video |
Feigenbaum's Renormalization Analysis of Period DoublingSuperstable 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 22Play Video |
Renormalization: Function Space and a Hands-on CalculationThe 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 23Play Video |
Fractals and the Geometry of Strange AttractorsAnalogy 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 24Play Video |
Hénon MapThe 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 25Play Video |
Using Chaos to Send Secret MessagesLou 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. |