nonlinear systems, chaos and control in engineeringcris/teaching/masoller_part2_2019.pdf · steven...
TRANSCRIPT
![Page 1: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/1.jpg)
Nonlinear systems, chaos
and control in Engineering
Bifurcations: saddle-node, transcritical
and pitchfork
Cristina Masoller
http://www.fisica.edu.uy/~cris/
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Flows on the line = first-order ordinary differential
equations.
dx/dt = f(x)
Fixed point solutions: f(x*) =0
• stable if f´(x*) <0
• unstable if f´(x*) >0
• neutral (bifurcation point) if f´(x*) = 0
There are no periodic solutions; the approach to fixed point
solutions is monotonic (sigmoidal or exponential).
Reminder Summary Part 1
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Introduction to bifurcations
Saddle-node, transcritical and pitchfork bifurcations
Examples
Imperfect bifurcations & catastrophes
Outline
![Page 4: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/4.jpg)
A qualitative change (in the structure of the phase
space) when a control parameter is varied:
• Fixed points can be created or destroyed
• The stability of a fixed point can change
There are many examples in physical systems,
biological systems, etc.
What is a bifurcation?
![Page 5: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/5.jpg)
Example
Control parameter increases in time
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Bifurcation and potential
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Bifurcation but no change of
behavior
Change of behavior but no
bifurcation
Bifurcations are not equivalent to
qualitative change of behavior
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Introduction to bifurcations
Saddle-node, transcritical and pitchfork
bifurcations
Examples
Imperfect bifurcations & catastrophes
Outline
![Page 9: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/9.jpg)
Saddle-node bifurcation
rx *
rrf 2)('
rrf 2)('
At the bifurcation
point r*=0: 0*)(' xf
Stable if r<0
unstable
Basic mechanism for the creation or the destruction of fixed points
![Page 10: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/10.jpg)
Calculate the fixed points and their stability
as a function of the control parameter r
Example
rx *
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Are representative of all saddle-node bifurcations.
Close to the saddle-node bifurcation the dynamics
can be approximated by
Normal forms
or
Example:
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Near a saddle-node bifurcation
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“
A pair of fixed points appear (or disappear) out of the “clear
blue sky” (“blue sky” bifurcation, Abraham and Shaw 1988).
Bifurcation diagram
Two fixed points → one fixed point → 0 fixed point
![Page 14: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/14.jpg)
Transcritical bifurcation
rx
x
*
* 0are the fixed points for all r
Transcritical bifurcation: general mechanism for changing
the stability of fixed points.
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Exchange of stability at r = 0.
Bifurcation diagram
xrxf 2)('
rf )0('
rrf )('
Exercise:
show that a transcritical bifurcation occurs near x=1
(hint: consider u = x-1 small)
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Pitchfork bifurcation
Symmetry x → -x
One fixed point → 3 fixed points
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The governing equation is symmetric: x -x
but for r > 0: symmetry broken solutions.
Bifurcation diagram
Bistability
0* x
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Potential
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Pitchfork bifurcations
Supercritical:
x3 is stabilizing
Subcritical:
x3 is destabilizing
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Exercise: find the fixed points
and compute their stability
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Subcritical bifurcation:
Hysteresis
Critical or dangerous transition! A lot of effort in trying to
find “early warning signals” (more latter)
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Hysteresis: sudden changes in
visual perception
Fischer
(1967):
experiment
with 57
students.
“When do you
notice an
abrupt change
in perception?”
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Bifurcation condition: change in the stability of a fixed
point
f ´(x*) = 0
In first-order ODEs: three possible bifurcations
• Saddle node
• Pitchfork
• Trans-critical
The normal form describes the behavior near the
bifurcation.
Summary
![Page 24: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/24.jpg)
Introduction to bifurcations
Saddle-node, transcritical and pitchfork
bifurcations
Examples
Imperfect bifurcations & catastrophes
Outline
![Page 25: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/25.jpg)
Example: neuron model
![Page 26: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/26.jpg)
Saddle-node Bifurcation
![Page 27: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/27.jpg)
Near the bifurcation point: slow
dynamics
This slow transition is an “early warning signal” of a
critical or dangerous transition ahead (more latter)
![Page 28: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/28.jpg)
If the control parameter now
decreases
![Page 29: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/29.jpg)
Class/home work
0 10 20 30 40 50-80
-60
-40
-20
0
20
40
t
V(t
)
I=20
I=10
0 5 10 15 20-60
-40
-20
0
20
40
t
V(t
)
I=10
Simulate the neuron model with different values of the
control parameter I and/or different initial conditions.
![Page 30: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/30.jpg)
Example: laser threshold
![Page 31: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/31.jpg)
Transcritical Bifurcation
![Page 32: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/32.jpg)
“imperfect” bifurcation due to
noise
![Page 33: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/33.jpg)
Laser turn on
![Page 34: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/34.jpg)
Laser turn-on delay
vtrtr 0)( 0*0 rr
Linear increase of
control parameter
Start before the
bifurcation point
r
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Comparison with
experimental observations
Dynamical hysteresis
Quasi-static very slow
variation of the control
parameter
![Page 36: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/36.jpg)
Class/home work with matlab
-1 0 1 2 3 40
1
2
3
4
r
x
v=0.01
v=0.1
01.0
)(
0
0
x
vtrtr
-3 -2 -1 0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
3.5
4
r
x
-3 -2 -1 0 1-5
0
5
10x 10
-3
r
x
r0=-1
r0=-2
r0=-3
01.0
1.0
0
x
v
Simulate the equation with r increasing linearly in time.
Consider different variation rate (v) and/or different initial
value of the parameter (r0).
![Page 37: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/37.jpg)
-1 -0.5 0 0.5 1 1.5 20
0.5
1
1.5
2
r
x
0 100 200 300 400 500 600-1
-0.5
0
0.5
1
1.5
2
t
x (
bla
ck),
r (
red)
01.0v
Now consider that the control parameter r increases and
then decreases linearly in time.
Plot x and r vs time and plot x vs r.
![Page 38: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/38.jpg)
0 5 10 15 200
0.5
1
1.5
2
2.5
3
t
x
r=1
r=2
r=3
01.0
)(
0
x
rtr
Calculate the “turn on”
when r is constant, r>r*=0.
Calculate the bifurcation
diagram by plotting x(t=50) vs r.
-1 -0.5 0 0.5 1-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r
x
h=0.01
h=0
![Page 39: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/39.jpg)
A particle moves along a
wire hoop that rotates at
constant angular velocity
Example: particle in a
rotating wire hoop
![Page 40: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/40.jpg)
Neglect the second derivative (more latter)
Fixed points from: sin = 0
Fixed points from: cos -1 = 0
stable unstable
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Pitchfork Bifurcation:
When is this “first-order” description valid?
When is ok to neglect the second derivative d2x/dt2 ?
Dimensional analysis and scaling
![Page 42: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/42.jpg)
Dimensionless time
(T = characteristic time-scale)
Dimensionless equation
We want the lhs very small, we define T such that
and
Define:
![Page 43: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/43.jpg)
The dimension less equation suggests that the first-
order equation is valid in the over damped limit: 0
Over damped limit
Problem: second-order equation has two independent
initial conditions: (0) and d/d(0)
But the first-order equation has only one initial condition
(0), d/d(0) is calculated from
Paradox: how can the first-order equation represent
the second-order equation?
cossinsin
d
d
![Page 44: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/44.jpg)
First order system:
Trajectories in phase space
Second order system:
cossinsin)( f
d
d
Second order system:
![Page 45: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/45.jpg)
Introduction to bifurcations
Saddle-node, transcritical and pitchfork
bifurcations
Examples
Imperfect bifurcations & catastrophes
Outline
![Page 46: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/46.jpg)
Imperfect bifurcations
![Page 47: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/47.jpg)
Parameter space (h, r)
Exercise : using these two equations
1. fixed points: f(x*) = 0
2. saddle node bifurcation: f’(x*) =0
Calculate hc(r) 33
2 rrhc
![Page 48: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/48.jpg)
Budworms population grows logistically (R>0 grow rate)
p(N): dead rate due to predation
If no budworms (N0): no predation: birds look for food
elsewhere
If N large, p(N) saturates: birds eat as much as they can.
Example: insect outbreak
![Page 49: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/49.jpg)
Dimensionless formulation
x*=0
Other FPs from the solution of
Independent
of r and k
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When the line intersects the curve tangentially
(dashed line): saddle-node bifurcation
a: Refuge level
of the budworm
population
c: Outbreak level (pest)b: threshold
Exercise : show that x*=0 is always unstable
![Page 51: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/51.jpg)
Parameter space (k, r)
![Page 52: Nonlinear systems, chaos and control in Engineeringcris/teaching/masoller_part2_2019.pdf · Steven H. Strogatz: Nonlinear dynamics and chaos, with applications to physics, biology,](https://reader034.vdocument.in/reader034/viewer/2022042917/5f5c4b2bc3741774e9615099/html5/thumbnails/52.jpg)
Steven H. Strogatz: Nonlinear dynamics and
chaos, with applications to physics, biology,
chemistry and engineering (Addison-Wesley Pub.
Co., 1994). Chapter 3
Bibliography
Thomas Erneux and Pierre Glorieux: Laser
Dynamics (Cambridge University Press 2010)
Eugene M. Izhikevich: Dynamical Systems in
Neuroscience: The Geometry of Excitability and
Bursting (MIT Press 2010)