second lecture : pp. 59-82 george g. lendaris, may 7, 2002

Systems Approach to Research in Applied Psychology, Spring 2002, G. Lendaris 1/38

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. 59-82George G. Lendaris, May 7, 2002

SySc 610: Systems Approach to Research in Applied PsychologyPortland State University


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


INTERACTIVE BIOLOGICAL POPULATIONSLOTKA-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


INTERACTIVE BIOLOGICAL POPULATIONS, cont.

LOTKA-VOLTERRA MODEL(PREDATOR-PREY)

(Hare)(Fox)



PREDATOR–PREY: STELLA MODEL



PREDATOR–PREY: STELLA MODEL OUTPUT



PREDATOR-PREY: RATES OF CHANGE

Equations used to model the dynamics look like the following:

dycxydtdy

bxyaxdtdx

rate of change of the prey population (variable x)

rate of change of the predator p population (variable y)



PREDATOR-PREY: SAMPLE VECTOR FIELD

Figure 16



PREDATOR-PREY: SAMPLE VECTOR FIELD

Figure 17



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.

Source: http://two.ucdavis.edu/~aking/mam99/phspln.htm

Rab

bit

Fox



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 Source: http://two.ucdavis.edu/~aking/mam99/phspln.htm

Rab

bit

Fox


INTERACTIVE BIOLOGICAL POPULATIONS, cont.PHASE PORTRAIT - PERIODIC ORBITS (with friction), cont.

C) Limit cycle attractor.


Rab

bit

Fox


INTERACTIVE BIOLOGICAL POPULATIONS, cont.VECTOR FIELD/PHASE PORTRAIT



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.



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


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.

SUSTAINED OSCILLATORSMUSICAL INSTRUMENTS, cont.

GENERAL MODELFOR REED WOODWIND INSTRUMENT

Reed displacement: X

Reed velocity: V

Figure 18A


Phase Portrait:

Repelling Equilibrium Point at origin.

Periodic Attractor around the origin.

Amperage, i

Average Voltage, V


ELECTRONIC OSCILLAOR (Helmholtz, Van der Pol)

Vacuum Tube Radio TransmitterFigure 18B



RELAXATION OSCILLAOR

This type of model was employed to model heartbeat.Contributed to start of electronic experimental dynamics

Figure 18C


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.


FORCED COUPLED OSCILLATORS: DAMPED

Actual devices studied by Raleigh, Duffing, and Ludeke. The driven oscillator only approximately sustained; the driven oscillator is damped

Figure 19A


FORCED COUPLED OSCILLATORS: 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


a) Driving cycle bent into ring, with isochronous harmonic attractor b) Trajectory approaching the attractor, strobe plane as

phase zero

FORCED COUPLED OSCILLATORS: DAMPED, cont.

Figure 20A

Figure 20B


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

FORCED COUPLED OSCILLATORS: DAMPED, cont

Figure 20C


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)

FORCED COUPLED OSCILLATORS: DAMPED, cont : HARD SPRING

Figure 21A Figure 21B


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

FORCED COUPLED OSCILLATORS: DAMPED, cont.:HARD SPRING

Figure 21C Figure 21D


The Ring Model completed for the Forced Hard Spring


Figure 21



Figure 22


We consider forcing an oscillator with a periodic attractor rather than a point attractor.

The state space may be a toroidal surface.

FORCED COUPLED OSCILLATORS: DAMPED, cont : 2-D TORUS MODEL

Figure 23A


Coupled system considered as perturbation of uncoupled system. Alternate closed closed trajectories are cyclic attractors and repellors.

FORCED COUPLED OSCILLATORS: DAMPED, cont : 2-D TORUS MODEL

Figure 23B


The 1D vertical ring of the driven oscillator is replaced by 1 2D V/X plane

FORCED OSCILLATORS: SELF-SUSTAINING

3-D RING MODEL

Figure 24A


FORCED OSCILLATORS: SELF-SUSTAININGUNCOUPLED OSCILLATOR

The torus is an invariant manifold. It is attractive, but not an attractor.

Figure 24B


FORCED OSCILLATORS: SELF-SUSTAINING UNCOUPLED OSCILLATOR, cont.

Oscillators coupled showing isochronous trajectory. In phase case; the periodic trajectory is an attractor.

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


FORCED OSCILLATORS: SELF-SUSTAININGUNCOUPLED OSCILLATOR, cont.

Periodic out-of-phase saddle, attracts amplitudes but repels phases as ribbons arrows show

Figure 25B


FORCED OSCILLATORS: SELF-SUSTAINING

UNCOUPLED OSCILLATOR, cont.

Coupled oscillators showing isochronous trajectory. Out-of-phase case; this periodic trajectory is a repellor shown winding around its locating torus.

Figure 26


FORCED OSCILLATORS: SELF-SUSTAININGUNCOUPLED OSCILLATOR, cont.

Composite of braided periodic attractor and saddle on invariant torus with central repellor.

Figure 27


FORCED OSCILLATORS: SELF-SUSTAININGRESPONSE DIAGRAM

Van der Pol System. Superimposition of Frequency Response Diagram for Three Different Strength Springs. Two parameters: Driving frequency and Spring strength.

Figure 28


NEXT CLASS

CLASS 12

Dynamical Systems 3

“Basic Principles of Dynamical Systems” F.D. Abraham, R.H. Abraham, & C.D. Shaw

Pages 82-113

second lecture : pp. 59-82 george g. lendaris, may 7, 2002

Documents