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Naima Hammoud
Feb 22, 2017
Dynamical Systems: Lecture 2
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Recall: the logistic equation
N = rN
✓1� N
K
◆N
N
Strogatz,NonlinearDynamicsandChaos(1994)
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Recall: the logistic equation
N = rN
✓1� N
K
◆N
N
Strogatz,NonlinearDynamicsandChaos(1994)
![Page 4: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/4.jpg)
Recall: the logistic equation
N = rN
✓1� N
K
◆
fixed points
N = 0
rN
✓1� N
K
◆= 0
rN = 0
=) N = 0
✓1� N
K
◆= 0
=) N = K
N
N
Strogatz,NonlinearDynamicsandChaos(1994)
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Tumor Growth
N
N
˙N = �aN log(bN)
Strogatz,NonlinearDynamicsandChaos(1994)
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Tumor Growth
˙N = �aN log(bN)
N
N
Strogatz,NonlinearDynamicsandChaos(1994)
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Tumor Growth
˙N = �aN log(bN)
fixed points
N = 0
�aN log(bN) = 0
�aN = 0
! N = 0
log(bN) = 0
! bN = 1
! N = 1/b
N
N
Strogatz,NonlinearDynamicsandChaos(1994)
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Tumor Growth
˙N = �aN log(bN)
fixed points
N = 0
�aN log(bN) = 0
�aN = 0
! N = 0
log(bN) = 0
! bN = 1
! N = 1/b
N
N1/b
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization
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Synchronization
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Synchronization of Fireflies
• Consider a system with one firefly and one flashlight
• Start flashing the light at a constant frequency (not too far from the firefly’s natural frequency), e.g. if a firefly fires every 0.8 seconds, then maybe the light can flash every second.
• The firefly will modify its frequency by speeding up or slowing down so as to match the flashlight
• If the stimulus is too slow or too fast, entrainment won’t occurStrogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
Model
✓(t)
✓ = 0
phase of the firefly’s flashing rhythm
the instant when a flash is emitted by the firefly
If there are no outside stimuli, the firefly flashes at a constant frequency w
✓ = !
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
• Suppose there’s a periodic stimulus (a flashlight), whose phase f satisfies
• If stimulus is ahead in the cycle, firefly speeds up. If stimulus is behind, firefly slows down
⇥ = ⌦
phase of the flashlight’s flashing rhythm
the instant when a flash is emitted by the stimulus
⇥(t)
⇥ = 0
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
✓ = ! +A sin(⇥� ✓)
sin(⇥� ✓) > 0 : if ⇥ ahead, then 0 < ⇥� ✓ < ⇡,
then firefly speeds up
sin(⇥� ✓) < 0 : if ⇥ behind, then ⇡ < ⇥� ✓ < 2⇡,
then firefly slows down
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
✓ = ! +A sin(⇥� ✓)
Let � = ⇥� ✓
=) � = ⇥� ✓
� = ⌦� ! �A sin�
✓ = !
⇥ = ⌦
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
✓ = ! +A sin(⇥� ✓)
Let � = ⇥� ✓
=) � = ⇥� ✓
� = ⌦� ! �A sin�
✓ = !
⇥ = ⌦
Change of variables(you’re not expected to know how to make
such variable changes)
redefine time to be At
define µ = ⌦�!A
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
✓ = ! +A sin(⇥� ✓)
Let � = ⇥� ✓
=) � = ⇥� ✓
� = ⌦� ! �A sin�
✓ = !
⇥ = ⌦
Change of variables(you’re not expected to know how to make
such variable changes) redefine time to be At
define µ = ⌦�!A
� = µ� sin�
Strogatz,NonlinearDynamicsandChaos(1994)
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µ =⌦� !
A
If µ is small, then frequencies are close to one another and we expect entrainment
If µ is large, we don’t expect entrainment
� = µ� sin�
Strogatz,NonlinearDynamicsandChaos(1994)
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Synchronization of Fireflies
µ =⌦� !
A
If µ is small, then frequencies are close to one another and we expect entrainment
If µ is large, we don’t expect entrainment
�
�
+
�
µ = 0
� = µ� sin�
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Synchronization of Fireflies
µ =⌦� !
A
If µ is small, then frequencies are close to one another and we expect entrainment
If µ is large, we don’t expect entrainment
�
�
+
�
� = µ� sin�
µ is small
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Synchronization of Fireflies
� = µ� sin� µ =⌦� !
A
If µ is small, then frequencies are close to one another and we expect entrainment
If µ is large, we don’t expect entrainment
�
�
+ +
µ is large
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Synchronization of Fireflies
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
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Synchronization of Fireflies
We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
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Synchronization of Fireflies
�
sin�
We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
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Synchronization of Fireflies
�
sin�
µ We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
![Page 26: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/26.jpg)
Synchronization of Fireflies
�
sin�
µ+
�+
� > 0 when µ > sin�
� < 0 when µ < sin�
We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
![Page 27: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/27.jpg)
Synchronization of Fireflies
�
sin�
+
�
µ�
� > 0 when µ > sin�
� < 0 when µ < sin�
We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
![Page 28: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/28.jpg)
Synchronization of Fireflies
�
sin�
�
µ
�
� > 0 when µ > sin�
� < 0 when µ < sin�
We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
![Page 29: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/29.jpg)
Synchronization of Fireflies
�
sin�µ
� > 0 when µ > sin�
� < 0 when µ < sin�
+ + We plot sinf and µseparately and find their intersections, which will be the fixed points
Fixed points when
� = µ� sin�
� = 0
µ� sin� = 0
µ = sin�
![Page 30: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/30.jpg)
Two-dimensional Systems
Example: spring-mass system
Transform into a system of coupled equation
Fixed points:
x+ !
2x = 0
(x = y
y = �!
2x
x = y = 0 ! x = y = 0
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Two-dimensional Systems
Example: spring-mass system
Transform into a system of coupled equation
Fixed points:
x+ !
2x = 0
(x = y
y = �!
2x
x = y = 0 ! x = y = 0
vector field: (x, y) = (y,�!
2x)
take specific points:
y = 0 ! (0,�!
2x)
x = 0 ! (y, 0)
x
y
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Love affairs
Romeo is in love with Juliet, but Juliet is a fickle lover: the more Romeo loves her, the more she wants to run away and hide. But when Romeo gets discouraged and backs off, Juliet begins to find him strangely attractive! Romeo on the other hand tends to echo her: he warms up when she loves him, and grows cold when she hates him.
Let R(t) be Romeo’s love/hate for Julietand J(t) be Juliet’s love/hate for Romeo
Strogatz,NonlinearDynamicsandChaos(1994)
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Love affairs(R = aJ
J = �bRa, b > 0
fixed points: R = J = 0 ! R = J = 0
vector field: (R, J = 0) = (aJ,�bR)
Strogatz,NonlinearDynamicsandChaos(1994)
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Love Affairs: general case(R = aR+ bJ
J = cR+ dJ
a > 0, b > 0
a < 0, b > 0
Romeo gets excited by Juliet’s love for him, and is further spurred by his own affection
Romeo gets excited by Juliet’s love for him, but he is turned off by his own affection
eager
cautious
Strogatz,NonlinearDynamicsandChaos(1994)
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Love affairs: Cautious Lovers
(R = aR+ bJ
J = bR+ aJ
a < 0, b > 0
measure ofcautiousness
measure ofresponsiveness
fixed points: R = J = 0 ! R = J = 0
Strogatz,NonlinearDynamicsandChaos(1994)
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Love affairs: Cautious Lovers
(R = aR+ bJ
J = bR+ aJ
a < 0, b > 0
measure ofcautiousness
measure ofresponsiveness
a = �2, b = 1
R
J
fixed points: R = J = 0 ! R = J = 0
Strogatz,NonlinearDynamicsandChaos(1994)
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Love affairs: Cautious Lovers
(R = aR+ bJ
J = bR+ aJ
a < 0, b > 0
measure ofcautiousness
measure ofresponsiveness
fixed points: R = J = 0 ! R = J = 0
a = �1, b = 2
R
J
Strogatz,NonlinearDynamicsandChaos(1994)
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Do opposites attract?(R = aR+ bJ
J = �bR� aJ
fixed points:
R = J = 0 ! R = J = 0
Strogatz,NonlinearDynamicsandChaos(1994)
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Do opposites attract?(R = aR+ bJ
J = �bR� aJ
fixed points:
a = 2, b = 1
R
JR = J = 0 ! R = J = 0
Strogatz,NonlinearDynamicsandChaos(1994)
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Do opposites attract?(R = aR+ bJ
J = �bR� aJ
fixed points:
a = 1, b = 2
R
JR = J = 0 ! R = J = 0
Strogatz,NonlinearDynamicsandChaos(1994)
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Do opposites attract?(R = aR+ bJ
J = �bR� aJ
fixed points:
R = J = 0 ! R = J = 0
a = 2, b = �1
R
J
Strogatz,NonlinearDynamicsandChaos(1994)
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Competition: Rabbits versus Sheep
Suppose both species are competing for the same food resources (say grass), and that the amount is limited. Ignore predators, seasonal changes, and other sources of food.
1. Each species would grow to its carrying capacity in the absence of the other species. So we can use the logistic model for each, and maybe assign rabbits a higher growth rate?
2. When rabbits and sheep encounter each other, trouble starts. Conflict occurs at a rate proportional to the size of each population. We assume conflicts reduce the growth rate for each species, with rabbits taking the bigger hit.
Strogatz,NonlinearDynamicsandChaos(1994)
![Page 46: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/46.jpg)
Rabbits versus Sheep
Recall the logistic equation:
Let R(t) be the population of rabbitsand S(t) be the population of sheep
N = rN
✓1� N
K
◆
(R = R(3�R� 2S)
S = S(2� S �R)
Strogatz,NonlinearDynamicsandChaos(1994)
![Page 47: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/47.jpg)
Rabbits versus Sheep
Recall the logistic equation:
Let R(t) be the population of rabbitsand S(t) be the population of sheep
N = rN
✓1� N
K
◆
(R = R(3�R� 2S)
S = S(2� S �R)
fixed points: R = S = 0
R(3�R� 2S) = 0
S(2� S �R) = 0
four fixed points:
(0, 0), (0, 2), (3, 0), (1, 1)Strogatz,NonlinearDynamicsandChaos(1994)
![Page 48: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/48.jpg)
Rabbits versus Sheep
Recall the logistic equation:
Let R(t) be the population of rabbitsand S(t) be the population of sheep
N = rN
✓1� N
K
◆
(R = R(3�R� 2S)
S = S(2� S �R)
four fixed points:
(0, 0), (0, 2), (3, 0), (1, 1)
R
S
Strogatz,NonlinearDynamicsandChaos(1994)
![Page 49: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/49.jpg)
Rabbits versus Sheep
(R = R(3�R� S)
S = S(2� S �R)
(R = R(3� 2R� S)
S = S(2� S �R)
(R = R(3� 2R� 2S)
S = S(2� S �R)
Strogatz,NonlinearDynamicsandChaos(1994)
![Page 50: Dynamical Systems: Lecture 2 - University of Oxfordpeople.maths.ox.ac.uk/.../2/dynamical_systems... · Dynamical Systems: Lecture 2. Recall: the logistic equation N˙ = rN](https://reader034.vdocuments.us/reader034/viewer/2022050520/5fa3de0c805aa44887616f10/html5/thumbnails/50.jpg)
Rabbits versus Sheep
(R = R(3�R� S)
S = S(2� S �R)
(R = R(3� 2R� S)
S = S(2� S �R)
(R = R(3� 2R� 2S)
S = S(2� S �R)
R
S
R
S
R
S
Strogatz,NonlinearDynamicsandChaos(1994)