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Second Lecture : pp. 59-82 George G. Lendaris, May 7, 2002

Support slides for second lecture on Chapter 3 in Analysis of Dynamic Psychological Systems v. 1 , Plenum, 1992: “Basic Principles of Dynamical Systems” F.D. Abraham, R.H. Abraham, & C.D. Shaw. Second Lecture : pp. 59-82 George G. Lendaris, May 7, 2002

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Second Lecture : pp. 59-82 George G. Lendaris, May 7, 2002

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  1. Support slides for second lecture on Chapter 3 in Analysis of Dynamic Psychological Systems v. 1, Plenum, 1992:“Basic Principles of Dynamical Systems” F.D. Abraham, R.H. Abraham, & C.D. Shaw Second Lecture: pp. 59-82 George G. Lendaris, May 7, 2002 SySc 610: Systems Approach to Research in Applied Psychology Portland State University

  2. TOPICS (May 7, 2002) • INTERACTING BIOLOGICAL POPULATIONS • Lotka-Volterra Model • SUSTAINED OSCILLATORS • FORCED COUPLED OSCILLATORS • Periodically Driven Damped Oscillators • Forced Linear Spring and Response Diagram • Forced Hard Springs and the Cup Catastrophe • Periodically Driven Self-Sustaining Oscillators • Entrainment and Braids • Response Diagram for Frequency Changes

  3. INTERACTIVE BIOLOGICAL POPULATIONS LOTKA-VOLTERRA MODEL The Lotka-Volterra model describes interactions between two species in an ecosystem: predator and prey. The Lotka-Volterra model is by definition a “Dynamical Model” in which the rate of change of the two populations is a function of both populations’ densities

  4. (Fox) (Hare) INTERACTIVE BIOLOGICAL POPULATIONS, cont. LOTKA-VOLTERRA MODEL (PREDATOR-PREY)

  5. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PREDATOR–PREY: STELLA MODEL

  6. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PREDATOR–PREY: STELLA MODEL OUTPUT

  7. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PREDATOR-PREY: RATES OF CHANGE Equations used to model the dynamics look like the following: rate of change of the prey population (variable x) rate of change of the predator p population (variable y)

  8. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PREDATOR-PREY: SAMPLE VECTOR FIELD Figure 16

  9. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PREDATOR-PREY: SAMPLE VECTOR FIELD Figure 17

  10. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PHASE PORTRAIT - PERIODIC ORBITS (no friction) • The populations follow one and only one of the trajectories in the phase portrait, depending on: • Their initial sizes • Rates of change of their respective sizes. Rabbit Fox Source: http://two.ucdavis.edu/~aking/mam99/phspln.htm

  11. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PHASE PORTRAIT - PERIODIC ORBITS (with friction) B) Ecological friction induces ecological static equilibrium (to attractor state) With intraspecific competition, the dynamics have only a single, attracting equilibrium, with damped oscillations relaxing to it. Fixed Point, or Rabbit Fox Source: http://two.ucdavis.edu/~aking/mam99/phspln.htm

  12. INTERACTIVE BIOLOGICAL POPULATIONS, cont. PHASE PORTRAIT - PERIODIC ORBITS (with friction), cont. C) Limit cycle attractor. Rabbit Fox Source: http://two.ucdavis.edu/~aking/mam99/phspln.htm

  13. INTERACTIVE BIOLOGICAL POPULATIONS, cont. VECTOR FIELD/PHASE PORTRAIT Source: http://two.ucdavis.edu/~aking/mam99/phspln.htm

  14. SUSTAINED OSCILLATORS

  15. SUSTAINED OSCILLATORS MUSICAL INSTRUMENTS Lord Rayleigh (1877) used the damped oscillator to describe percussive musical instruments, then generalized to sustained instruments (reeds, strings, etc.). These systems exhibit internal friction (damping) and restoring forces (which oppose/assist the externally applied force sustaining the oscillation) As before, dimensions of the state space are defined to represent the displacement and velocity of the vibrating object.

  16. SUSTAINED OSCILLATORS MUSICAL INSTRUMENTS, cont. In basic model: Restoring Force is a negative linear function of the displacement Friction is a cubic function of the velocity (for small motions near the origin, friction is a positive function of velocity, thus assisting motion [~ negative friction]). Figure 18A

  17. SUSTAINED OSCILLATORS MUSICAL INSTRUMENTS, cont. GENERAL MODELFOR REED WOODWIND INSTRUMENT Left insert: Force is a negative linear function of reed displacement Right insert: Force is a cubic function of reed velocity Phase Portrait: Point repellor at origin -- from which trajectories spiral outward to a limit cycle. Larger motions - regular friction: damps trajectories in toward the limit cycle. The SIGN CHANGE of friction forces creates SUSTAINED OSCILLATION. Figure 18A Reed displacement: X Reed velocity: V

  18. SUSTAINED OSCILLATORS ELECTRONIC OSCILLAOR (Helmholtz, Van der Pol) Phase Portrait: Repelling Equilibrium Point at origin. Periodic Attractor around the origin. Amperage, i Average Voltage, V Figure 18B Vacuum Tube Radio Transmitter

  19. SUSTAINED OSCILLATORS RELAXATION OSCILLAOR Figure 18C This type of model was employed to model heartbeat. Contributed to start of electronic experimental dynamics

  20. FORCED COUPLED OSCILLATORS • Periodically Driven Damped Oscillators • Periodically Driven Self-Sustaining Oscillators ----------------------------------------------------------------- Classic example: Effect of mechanical vibration on pendulum or spring (e.g., via Duffing) Biological example: Effect of climatic seasons on predator-prey model.

  21. FORCEDCOUPLEDOSCILLATORS: DAMPED Actual devices studied by Raleigh, Duffing, and Ludeke. The driven oscillator only approximately sustained; the driven oscillator is damped Figure 19A

  22. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont. Model equivalent to Duffing’s. A connecting rod drives a spring connected to the sliding wieght. There is a frequency control on the driving turntable and a strobe light is switched on at fixed point in the driving cycle. Figure 19B

  23. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont. a) Driving cycle bent into ring, with isochronous harmonic attractor b) Trajectory approaching the attractor, strobe plane as phase zero Figure 20A Figure 20B

  24. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont c) Strobe section (Poincaré section), phase zero, showing several attractive points and several trajectories approaching one of them (P4) d) Response diagram: response amplitude as a function of driving frequency, ω, ωo is the resonant frequency of the driven spring Figure 20C

  25. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont : HARD SPRING Figure 21A Figure 21B a). Force is an inverse cubic function of displacement b). Response diagram: Amplitude of the attractor as the control parameter, the driving frequency, ω, is increased and decreased, showing the hysteresis loop of Duffing (double fold catastrophe)

  26. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont.: HARD SPRING Figure 21C Figure 21D c). Large and small amplitude attractors at an intrahysteresis frequency d). Strobe plane of C, showing the basins for each of the attractors, and the saddle cycle and separatrix between the two cyclic attractors

  27. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont : HARD SPRING Figure 21 The Ring Model completed for the Forced Hard Spring

  28. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont : HARD SPRING Figure 22

  29. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont : 2-D TORUS MODEL We consider forcing an oscillator with a periodic attractor rather than a point attractor. The state space may be a toroidal surface. Figure 23A

  30. FORCEDCOUPLEDOSCILLATORS: DAMPED, cont : 2-D TORUS MODEL Figure 23B Coupled system considered as perturbation of uncoupled system. Alternate closed closed trajectories are cyclic attractors and repellors.

  31. FORCEDOSCILLATORS: SELF-SUSTAINING 3-D RING MODEL The 1D vertical ring of the driven oscillator is replaced by 1 2D V/X plane Figure 24A

  32. FORCEDOSCILLATORS: SELF-SUSTAINING UNCOUPLED OSCILLATOR Figure 24B The torus is an invariant manifold. It is attractive, but not an attractor.

  33. FORCEDOSCILLATORS: SELF-SUSTAINING UNCOUPLED OSCILLATOR, cont. In the Poincaré section or strobe plane we would see a trajectory (discrete) approaching a point as a series of points as the trajectory successively crossed the plane while spiraling closer to the limit cycle. Figure 25A Oscillators coupled showing isochronous trajectory. In phase case; the periodic trajectory is an attractor.

  34. FORCEDOSCILLATORS: SELF-SUSTAINING UNCOUPLED OSCILLATOR, cont. Figure 25B Periodic out-of-phase saddle, attracts amplitudes but repels phases as ribbons arrows show

  35. FORCEDOSCILLATORS: SELF-SUSTAINING UNCOUPLED OSCILLATOR, cont. Figure 26 Coupled oscillators showing isochronous trajectory. Out-of-phase case; this periodic trajectory is a repellor shown winding around its locating torus.

  36. FORCEDOSCILLATORS: SELF-SUSTAINING UNCOUPLED OSCILLATOR, cont. Figure 27 Composite of braided periodic attractor and saddle on invariant torus with central repellor.

  37. FORCEDOSCILLATORS: SELF-SUSTAINING RESPONSE DIAGRAM Figure 28 Van der Pol System. Superimposition of Frequency Response Diagram for Three Different Strength Springs. Two parameters: Driving frequency and Spring strength.

  38. NEXT CLASS • CLASS 12 • Dynamical Systems 3 • “Basic Principles of Dynamical Systems” F.D. Abraham, R.H. Abraham, & C.D. Shaw • Pages 82-113

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