1 / 51

San Diego State University Nonlinear Dynamical Systems Group Department of Mathematics

Order-Chaos transition and the Magnetic Pendulum. Ricardo Carretero Nonlinear Dynamical Systems (NLDS) Group Department of Mathematics and Statistics San Diego State University Web: http://nlds.sdsu.edu. San Diego State University Nonlinear Dynamical Systems Group Department of Mathematics

treval
Download Presentation

San Diego State University Nonlinear Dynamical Systems Group Department of Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Order-Chaos transition and the Magnetic Pendulum Ricardo CarreteroNonlinear Dynamical Systems (NLDS) GroupDepartment of Mathematics and Statistics San Diego State UniversityWeb:http://nlds.sdsu.edu San Diego State University Nonlinear Dynamical Systems Group Department of Mathematics San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720

  2. Chaos in Mechanical Oscillations: Magnetic pendulum In class demonstration • Chaotic oscillations • Sensitive dependence to Initial Conditions • Fractal basins of attraction • Friction controls degree of “fractality” → Fractal dimension (more later…) Fractal basin of attraction

  3. High energy Low energy Chaos in Mechanical Oscillations: Sensitive dependence to initial conditions • Sensitive dependence to initial conditions • Small error in IC → amplified exponentially • Unpredictability →Chaos • Chaos vs Random

  4. Fractals in Nature I

  5. Fractals in Nature (crystal growth)II

  6. Fractals in Nature (with help from Physics) III Lichtenberg figure:High voltage dielectric breakdown in plexiglass block creates a beautiful fractal pattern called a Lichtenberg figure. The branching discharges ultimately become hair-like, but are thought to extend down to the molecular level Dendritic structures:Air displacing a vacuum formed by pulling two glue-covered acrylic sheets apart. The air travels in the direction the horns are pointing, creating dendritic structures into the glue in an effort to equalize the pressures in the room and between the sheets. Sometimes LIGHTENINGS create Lichtenberg figures: Ex. golfcourse

  7. Fractals in Nature (anatomy) IV

  8. Mathematical Theory of fractals: attractors (1D) • Iterated Maps: • Orbit with IC: 0 < x(0) < 1: 0.1 → 0.01 → 0.0001 → 0.00000001, … → … 0 • Orbit with IC: x(0) = 1: 1 → 1 → 1 → … → … 1 • Orbit with IC:x(0) > 1: 2 → 4 → 16 → 256 → 65,536 → 4,294,967,296, … → … ∞ • In general: which ICs go to zero and which ones go to infinity ? -1 0 1 - ∞ 0 ∞

  9. 0 Mathematical Theory of fractals: attractors (2D) • Iterated Maps: • Complex Numbers: • Mandelbrot Set: ?

  10. Fractals: The Mandelbrot set : self-similarity

  11. 1

  12. 0. 1

  13. 0.0 1

  14. 0.00 1

  15. 0.000 1

  16. 0.000 01

  17. 0.000 00 1

  18. 0.000 000 1

  19. 0.000 000 0 1

  20. 0.000 000 00 1

  21. 0.000 000 000 1

  22. 0.000 000 000 0 1

  23. 0.000 000 000 001

  24. 0.000 000 000 000 1

  25. 0.000 000 000 000 01

  26. 0.000 000 000 000 001

  27. Sensitive dependence on ICs: Fractals ↔ Chaos An IC in this zone might end up in any of the magnets. Any perturbation, however small, might kick our system to a completely different configuration. Sensitive dependence to changes on the initial condition. Examples: 1) Roulette. 2) The butterfly effect.

  28. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  29. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  30. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  31. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  32. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  33. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  34. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  35. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  36. Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079Friction:Γ=3.0000Γ=1.0000Γ=0.5000Γ=0.2500Γ=0.1259Γ=0.1018Γ=0.0787Γ=0.0433Γ=0.0079

  37. Self similarityof the basins of attraction

  38. Self similarityof the basins of attraction (cont.)

  39. Sensitive dependence on ICs: Fractals ↔ Chaos

  40. The magnetic pendulum: Eqs of motion

  41. Fixed points and Stability: Bifurcation diagram

  42. Eqns of Motion →Potential Energy Surface Eqs of motion: Potential:

  43. = = The Gaussian curvature criterion Eqs of motion: Gaussian curvature: Gaussian curvature criterion: If K(x,y) < 0 (and β>0) then ALL eigenvalues of J are purely imaginaryand thus CHAOS IS NOT POSSIBLE

  44. Transition order → chaos as energy increases K > 0(no chaos) K < 0

  45. Transition order → chaos as energy increases Lyapunov Exponent vs Energy

  46. Back to order as D is increased D=1 D=0.64

  47. Chaos, fractals, predictability, determinism, randomness and all that • The world around us is deterministic: • We need infinite precision in initial configuration and an infinitely precise computer to be able to predict for long times • Does randomness really exist? • Perfect knowledge of past should determine future • Problem: cannot have exact measure of present configuration • If system is sensitive dependent to ICs (ie chaotic) then, in practice, we won’t be able to make long term predictions

  48. THE END Blue Oyster Fractal

  49. SDSU – Research Group Nonlinear Dynamical Systems Web:http://nlds.sdsu.edu/[Members] • Core Members: • Dr. Peter Blomgren (blomgren@terminus.sdsu.edu) • Numerical Analysis, Reconstruction and image processing • Wave propagation in random media • Dr. Ricardo Carretero (carreter@math.sdsu.edu) • Pattern Formation, Chaos and fractals • Nonlinear waves and lattices • Dr. Joseph Mahaffy (mahaffy@math.sdsu.edu) • Mathematical Biology, Aged-structured population models • Delay differential equations • Dr. Antonio Palacios (palacios@euler.sdsu.edu) • Symmetry and bifurcations, Chains of coupled oscillators • Radars and sensors • Graduate Programs: • MS in Applied Mathematics • MS in Dynamical Systems • PhD in Computational Sciences • PhD in Dynamical Systems

  50. Graduate Programs in Dynamical Systems Web:http://nlds.sdsu.edu/[Graduate Programs] • Required Courses: • M538 Introduction to Nonlinear Dynamics • M638 Advanced Nonlinear Dynamics • M636 Mathematical Modeling • M537 Ordinary Differential Equations • M531 Partial Differential Equations • Recommended electives: • Pattern Formation (M635) • Fractal Geometry • Nonlinear Waves • Bifurcation Theory • Neural Modeling/Math Biology • Nonlinear Time-Series Analysis • Numerical Methods for Dynamical Systems Blue oyster fractal

More Related