nonlinear systems, chaos and control in engineeringcris/teaching/masoller_part3_2018.pdf · steven...
TRANSCRIPT
![Page 1: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/1.jpg)
Nonlinear systems, chaos
and control in Engineering
Phase oscillators, entrainment and
locking
Cristina Masoller
http://www.fisica.edu.uy/~cris/
![Page 2: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/2.jpg)
Introduction to phase oscillators
Nonlinear oscillator: the Adler equation
Entrainment and locking
Outline
![Page 3: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/3.jpg)
Example:
Phase oscillator
Flow on a circle: a function that assigns a unique
velocity vector to each point on the circle.
![Page 4: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/4.jpg)
Linear oscillator:
Two non-interacting oscillators
periodically go in and out of phase
Beat frequency = 1/Tlap
![Page 5: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/5.jpg)
Example of a phase oscillator
![Page 6: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/6.jpg)
Interacting oscillators
![Page 7: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/7.jpg)
In phase vs out of phase
oscillation
![Page 8: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/8.jpg)
Example: Integrate (accumulate)
and fire oscillator
![Page 9: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/9.jpg)
Other examples: heart beats, neuronal spikes
![Page 10: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/10.jpg)
Introduction to phase oscillators
Nonlinear oscillator: the Adler equation
Entrainment and locking
Outline
![Page 11: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/11.jpg)
An overdamped pendulum driven
by a constant torque
b very large:
Dimension-less equation:
![Page 12: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/12.jpg)
(Adler equation) Simple model for
many nonlinear phase oscillators
(neurons, circadian rhythms, over-
damped driven pendulum), etc.
Nonlinear oscillator
Robert Adler (1913–2007) 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.”
![Page 13: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/13.jpg)
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.
![Page 14: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/14.jpg)
![Page 15: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/15.jpg)
Fixed points when a >
Linear stability:
The FP with cos *>0 is the stable one.
![Page 16: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/16.jpg)
Oscillation period when a <
a=0: uniform
oscillator T when a:
![Page 17: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/17.jpg)
Period grows to infinite as
Bottleneck
Generic feature at a saddle-node bifurcation
“critical slowing down”: early warning signal of a
critical transition ahead.
![Page 18: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/18.jpg)
Introduction to phase oscillators
Nonlinear oscillator: the Adler equation
Entrainment and locking
Outline
![Page 19: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/19.jpg)
Synchronous rhythmic flashing
of fireflies
Strogatz
video
![Page 20: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/20.jpg)
There is a external periodic stimulus with frequency :
Model
(Ermentrout and Rinzel 1984)
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.
Response of a firefly () to the stimulus (): if the
stimulus is ahead on the firefly cycle, the firefly tries
to “speed up” to synchronize;
![Page 21: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/21.jpg)
Analysis
Dimensionless model:
detuning parameter
(Adler equation)
![Page 22: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/22.jpg)
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
![Page 23: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/23.jpg)
A
Entrainment is possible only if the frequency of the
external stimulus, , is close to the firefly frequency,
![Page 24: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/24.jpg)
Small detuning
Potential interpretation
Large detuning
cos)( V
![Page 25: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/25.jpg)
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
![Page 26: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/26.jpg)
Experimental observation of
entrainment
Time
2:1
3:1
T 2 T mod
4:1
Time
![Page 27: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/27.jpg)
A vector field on a circle is a rule that assigns a
unique velocity vector to each point on the circle
Summary
An oscillator can be entrained to an external
periodic signal if the frequencies are similar.
simple model to describe phase-
locking 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.
![Page 28: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/28.jpg)
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.
Class / Home work
0 50 100 1500
5
10
15
20
25
t
x
0 0.2 0.4 0.6 0.8 15
10
15
20
25
30
35
40
45
50
A
Peri
od T
0 5 10 150
1
2
3
4
5
6
7
8
Time
(t
)
![Page 29: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part3_2018.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c47eacd8c7612337c339c/html5/thumbnails/29.jpg)
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
Bibliography