1 / 20

Complex Behavior of Simple Systems

Complex Behavior of Simple Systems. J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to International Conference on Complex Systems in Nashua, NH on May 23, 2000. Lorenz Equations (1963). d x /d t = s y - s x d y /d t = - xz + rx - y d z /d t = xy - bz

pearl
Download Presentation

Complex Behavior of Simple Systems

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. Complex Behavior of Simple Systems J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to International Conference on Complex Systems in Nashua, NH on May 23, 2000

  2. Lorenz Equations (1963) dx/dt = sy - sx dy/dt = -xz + rx - y dz/dt = xy - bz 7 terms, 2 quadratic nonlinearities, 3 parameters

  3. Rössler Equations (1976) dx/dt = -y - z dy/dt = x + ay dz/dt = b + xz - cz 7 terms, 1 quadratic nonlinearity, 3 parameters

  4. Lorenz Quote (1993) “One other study left me with mixed feelings. Otto Roessler of the University of Tübingen had formulated a system of three differential equations as a model of a chemical reaction. By this time a number of systems of differential equations with chaotic solutions had been discovered, but I felt I still had the distinction of having found the simplest. Roessler changed things by coming along with an even simpler one. His record still stands.”

  5. Rössler Toroidal Model (1979) “Probably the simplest strange attractor of a 3-D ODE” (1998) dx/dt = -y - z dy/dt = x dz/dt = ay - ay2 - bz 6 terms, 1 quadratic nonlinearity, 2 parameters

  6. Sprott (1994) • 14 more examples with 6 terms and 1 quadratic nonlinearity • 5 examples with 5 terms and 2 quadratic nonlinearities

  7. Gottlieb (1996) What is the simplest jerk function that gives chaos? Displacement: x Velocity: = dx/dt Acceleration: = d2x/dt2 Jerk: = d3x/dt3

  8. Sprott (1997) “Simplest Dissipative Chaotic Flow” dx/dt = y dy/dt = z dz/dt = -az + y2 - x 5 terms, 1 quadratic nonlinearity, 1 parameter

  9. Bifurcation Diagram

  10. Return Map

  11. Fu and Heidel (1997) Dissipative quadratic systems with less than 5 terms cannot be chaotic. They would have no adjustable parameters.

  12. Weaker Nonlinearity dx/dt = y dy/dt = z dz/dt = -az + |y|b - x Seek path in a-b space that gives chaos as b1.

  13. Regions of Chaos

  14. Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = -az - y + |x| - 1 6 terms, 1 abs nonlinearity, 2 parameters (but one =1)

  15. General Form dx/dt = y dy/dt = z dz/dt = -az - y + G(x) G(x) =±(b|x| - c) G(x) =-bmax(x,0) + c G(x) =±(bx - csgn(x)) etc….

  16. First Circuit

  17. Bifurcation Diagram for First Circuit

  18. Second Circuit

  19. Third Circuit

  20. Chaos Circuit

More Related