Nonlinear systems, chaos and control in Engineering

Transcription

1 Nonlinear systems, chaos and control in Engineering Module 1 block 3 One-dimensional nonlinear systems Cristina Masoller Cristina.masoller@upc.edu

2 Schedule Flows on the line (Strogatz ch.1 & 2) (5 hs) Introduction Fixed points and linear stability Examples Solving equations with computer Bifurcations (Strogatz ch. 3) (3 hs) Introduction Saddle-node Transcritical Pitchfork Examples Flows on the circle (Strogatz ch. 4) (2 hs) Introduction to phase oscillators Nonlinear oscillator Fireflies and entrainment

3 Outline Introduction to phase oscillators Nonlinear oscillator (Adler equation) Example: overdamped pendulum Entrainment and locking Example: fireflies

4 Phase oscillator Flow on a circle: a function that assigns a unique velocity vector to each point on the circle. Example:

5 Linear oscillator: Two non-interacting oscillators periodically go in and out of phase Beat frequency = 1/T lap

6 Example of a phase oscillator

7 Interacting oscillators

8 In phase vs out of phase oscillation

9 Example: Integrate (accumulate) and fire oscillator

10 Other examples: heart beats, neuronal spikes

11 Nonlinear oscillator (Adler equation) Simple model for many nonlinear phase oscillators (neurons, circadian rhythms, overdamped driven pendulum), etc. Robert Adler ( ) is best known as the co-inventor of the television remote control using ultrasonic waves. But in the 1940s, he and others at Zenith Corporation were interested in reducing the number of vacuum tubes in an FM radio. The possibility that a locked oscillator might offer a solution inspired his 1946 paper A Study of Locking Phenomena in Oscillators.

12 For <1: simple model of an excitable system: With a small perturbation: fast return to the stable state But if the perturbation in larger than a threshold, then, long excursion before returning to the stable state.

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14 Fixed points when a > Linear stability: The FP with cos *>0 is the stable one.

15 Oscillation period when a < a=0: uniform oscillator T when a :

16 Bottleneck Period grows to infinite as critical slowing down : early warning signal of a critical transition ahead. Generic feature at a saddle-node bifurcation

17 Example: overdamped pendulum driven by a constant torque b very large: Dimension-less equation:

18 Gravity helps the external torque Gravity opposes the external torque

19 Strogatz video Synchronous rhythmic flashing of fireflies

20 Model (Ermentrout and Rinzel 1984) There is a external periodic stimulus with frequency : Response of a firefly ( ) to the stimulus ( ): if the stimulus is ahead on the firefly cycle, the firefly tries to speed up to synchronize; But if the firefly is flashing too early, then slows down If is ahead [0 < - < ] then sin( - )>0 and the firefly speeds up [d /dt > ] The parameter A measures the capacity of the firefly to adapt its flashing frequency.

21 Analysis Dimensionless model: detuning parameter (Adler equation)

22 The firefly and the stimulus flash simultaneously The firefly and the stimulus are phase locked (entrainment): there is a stable and constant phase difference The firefly and the stimulus are unlocked: phase drift

23 Entrainment is possible only if the frequency of the external stimulus,, is close to the firefly frequency, A

24 Potential interpretation V( ) cos Small detuning Large detuning

25 Arnold tongues If the external frequency is not close to the firefly frequency,, then, a different type of synchronization is possible: the firefly can fire m pulses each n pulses of the external signal. A

26 Summary A vector field on a circle is a rule that assigns a unique velocity vector to each point on the circle simple model to describe phaselocking of a nonlinear oscillator to an external periodic signal. In the phase-locked state, the oscillator maintains a constant phase difference relative to the signal. An oscillator can be entrained to an external periodic signal if the frequencies are similar.

27 x Period T Class / Home work Solve Adler s equation with =1, A=0.99 and (0)= /2 With =1, calculate the average oscillation period and compare with the analytical expression With =1/sqrt(2), calculate the trajectory for an arbitrary initial condition t A (t) Time

28 Bibliography Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology, chemistry and engineering (Addison-Wesley Pub. Co., 1994). Ch. 4 A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization, a universal concept in nonlinear science (Cambridge University Press 2001). Chapters 1-3