Exploring Chaos Theory: A Journey from Kepler to Fractals

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Delve into the fascinating world of chaos theory, tracing its evolution from the seminal works of Kepler and Newton to the complexities of Poincare's findings and modern chaos concepts. Discover the dynamics of chaotic systems like the simple pendulum and understand the implications of sensitive dependence on initial conditions. Explore how chaos theory has revolutionized our understanding of nonlinear systems and their behavior.


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  1. Introduction to Chaos Clint Sprott Department of Physics University of Wisconsin - Madison Presented to Physics 311 at University of Wisconsin in Madison, WI on October 31, 2014

  2. Abbreviated History n Kepler (1605) n Newton (1687) n Poincare (1890) n Lorenz (1963)

  3. Kepler (1605) n Tycho Brahe n 3 laws of planetary motion n Elliptical orbits

  4. Newton (1687) n Invented calculus n Derived 3 laws of motion F = ma n Proposed law of gravity F = Gm1m2/r2 n Explained Kepler s laws n Got headaches (3 body problem)

  5. Poincare (1890) n 200 years later! n King Oscar (Sweden, 1887) n Prize won 200 pages n No analytic solution exists! n Sensitive dependence on initial conditions (Lyapunov exponent) n Chaos! (Li & Yorke, 1975)

  6. 3-Body Problem

  7. Chaos n Sensitive dependence on initial conditions (positive Lyapunov exp) n Aperiodic (never repeats) n Topologically mixing n Dense periodic orbits

  8. Simple Pendulum F = ma -mg sin x = md2x/dt2 dx/dt = v dv/dt = -g sin x dv/dt = -x (for g = 1, x << 1) Dynamical system Flow in 2-D phase space

  9. Phase Space Plot for Pendulum

  10. Features of Pendulum Flow n Stable (O) & unstable (X) equilibria n Linear and nonlinear regions n Conservative / time-reversible n Trajectories cannot intersect

  11. Pendulum with Friction dx/dt = v dv/dt = -sin x bv

  12. Features of Pendulum Flow n Dissipative (cf: conservative) n Attractors (cf: repellors) n Poincare-Bendixson theorem n No chaos in 2-D autonomous system

  13. Damped Driven Pendulum dx/dt = v dv/dt = -sin x bv + sin t 2-D nonautonomous dx/dt = v dv/dt = -sin x bv + sin z dz/dt = 3-D autonomous

  14. New Features in 3-D Flows n More complicated trajectories n Limit cycles (2-D attractors) n Strange attractors (fractals) n Chaos!

  15. Stretching and Folding

  16. Chaotic Circuit

  17. Equations for Chaotic Circuit dx/dt = y dy/dt = z dz/dt = az by + c(sgn x x) Jerk system Period doubling route to chaos

  18. Bifurcation Diagram for Chaotic Circuit

  19. Invitation I sometimes work on publishable research with bright undergraduates who are crack computer programmers with an interest in chaos. If interested, contact me.

  20. References n http://sprott.physics.wisc.edu/ lectures/phys311.pptx (this talk) n http://sprott.physics.wisc.edu/chaost sa/ (my chaos textbook) n sprott@physics.wisc.edu (contact me)

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