how mathematics help us understand the world around uswtang/files/basistalk.pdfhow mathematics help...
TRANSCRIPT
How mathematics help us understand the
world around us
Wenbo Tang
School of Mathematical & Statistical Sciences,Arizona State University
2009-10 speaker series, Basis Scottsdale High School, Sept. 29th, 2009
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 1 / 14
The World As a Dynamical System
The world is in constant motion
Want to know the solution of a given variable at a given time
Solution often complex, yet dynamics may be governed by simple rules
E.g. Newton’s second law of motion
F = ma = md2x
dt2
Tricky part — F always non-constant!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 2 / 14
The World As a Dynamical System
The world is in constant motion
Want to know the solution of a given variable at a given time
Solution often complex, yet dynamics may be governed by simple rules
E.g. Newton’s second law of motion
F = ma = md2x
dt2
Tricky part — F always non-constant!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 2 / 14
The World As a Dynamical System
The world is in constant motion
Want to know the solution of a given variable at a given time
Solution often complex, yet dynamics may be governed by simple rules
E.g. Newton’s second law of motion
F = ma = md2x
dt2
Tricky part — F always non-constant!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 2 / 14
The World As a Dynamical System
The world is in constant motion
Want to know the solution of a given variable at a given time
Solution often complex, yet dynamics may be governed by simple rules
E.g. Newton’s second law of motion
F = ma = md2x
dt2
Tricky part — F always non-constant!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 2 / 14
The World As a Dynamical System
The world is in constant motion
Want to know the solution of a given variable at a given time
Solution often complex, yet dynamics may be governed by simple rules
E.g. Newton’s second law of motion
F = ma = md2x
dt2
Tricky part — F always non-constant!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 2 / 14
Mathematical Preparations Towards Dynamics
Motions change continuously (Calculus)
For physical problems, we need 4 dimensions (Multivariable Calculus)
Most important information is not on the axis (Linear Algebra)
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
Complex numbers denoting oscillation (Complex Variables)
d2x
dt2+ ω2x = 0; x = A cos(ωt) + B sin(ωt) = A1e
iωt + B1e−iωt
Ordinary and partial differential equations
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 3 / 14
Mathematical Preparations Towards Dynamics
Motions change continuously (Calculus)
For physical problems, we need 4 dimensions (Multivariable Calculus)
Most important information is not on the axis (Linear Algebra)
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
Complex numbers denoting oscillation (Complex Variables)
d2x
dt2+ ω2x = 0; x = A cos(ωt) + B sin(ωt) = A1e
iωt + B1e−iωt
Ordinary and partial differential equations
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 3 / 14
Mathematical Preparations Towards Dynamics
Motions change continuously (Calculus)
For physical problems, we need 4 dimensions (Multivariable Calculus)
Most important information is not on the axis (Linear Algebra)
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
Complex numbers denoting oscillation (Complex Variables)
d2x
dt2+ ω2x = 0; x = A cos(ωt) + B sin(ωt) = A1e
iωt + B1e−iωt
Ordinary and partial differential equations
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 3 / 14
Mathematical Preparations Towards Dynamics
Motions change continuously (Calculus)
For physical problems, we need 4 dimensions (Multivariable Calculus)
Most important information is not on the axis (Linear Algebra)
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
Complex numbers denoting oscillation (Complex Variables)
d2x
dt2+ ω2x = 0; x = A cos(ωt) + B sin(ωt) = A1e
iωt + B1e−iωt
Ordinary and partial differential equations
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 3 / 14
Mathematical Preparations Towards Dynamics
Motions change continuously (Calculus)
For physical problems, we need 4 dimensions (Multivariable Calculus)
Most important information is not on the axis (Linear Algebra)
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
1 0.5 0 0.5 11
0.5
0
0.5
1
x
y
Complex numbers denoting oscillation (Complex Variables)
d2x
dt2+ ω2x = 0; x = A cos(ωt) + B sin(ωt) = A1e
iωt + B1e−iωt
Ordinary and partial differential equations
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 3 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rules
I Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motion
I Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
How Is Mathematics Involved
Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)
These laws are translated into mathematical language
Differential equations are constructed to model processes
Equations usually appear in the form
dq
dt= F − D
Differential equations are solved using analytical or numericalprocedures
Simplifications make problems easier to solve
Solutions are interpreted in physical language
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 4 / 14
Example I: Modeling Population DynamicsThe logistic model has two important ingredients:
dN
dt= RN(1− N
K) N
NdNdt
t
I Exponential growth when population is smallI Growth rate decreases with size
R,K determined from experiments/observationsThis is overly simple, but captures the essential dynamicsDecrease can be in many ways, dynamical behaviors are similar
NN
G.Rate G.Rate
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 5 / 14
Example I: Modeling Population DynamicsThe logistic model has two important ingredients:
dN
dt= RN(1− N
K) N
NdNdt
t
I Exponential growth when population is smallI Growth rate decreases with size
R,K determined from experiments/observations
This is overly simple, but captures the essential dynamicsDecrease can be in many ways, dynamical behaviors are similar
NN
G.Rate G.Rate
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 5 / 14
Example I: Modeling Population DynamicsThe logistic model has two important ingredients:
dN
dt= RN(1− N
K) N
NdNdt
t
I Exponential growth when population is smallI Growth rate decreases with size
R,K determined from experiments/observationsThis is overly simple, but captures the essential dynamics
Decrease can be in many ways, dynamical behaviors are similar
NN
G.Rate G.Rate
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 5 / 14
Example I: Modeling Population DynamicsThe logistic model has two important ingredients:
dN
dt= RN(1− N
K) N
NdNdt
t
I Exponential growth when population is smallI Growth rate decreases with size
R,K determined from experiments/observationsThis is overly simple, but captures the essential dynamicsDecrease can be in many ways, dynamical behaviors are similar
NN
G.Rate G.Rate
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 5 / 14
Let’s Add Some Predation
In a modified model we considerloss of biomass by predation
dN
dt= RN(1−N
K)− BN2
A2 + N2
The functional response modelscertain predation behavior
This model supports two stablestates in some parameter range
Also for some parameters a largepopulation is the onlyequilibrium
The states are essentiallydetermined by the intersectionof two functions
Predation function and flow
N
N
P dNdt
Bifurcation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N
Fn
BNA2+N 2
R(1 ! NK )
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 6 / 14
Let’s Add Some Predation
In a modified model we considerloss of biomass by predation
dN
dt= RN(1−N
K)− BN2
A2 + N2
The functional response modelscertain predation behavior
This model supports two stablestates in some parameter range
Also for some parameters a largepopulation is the onlyequilibrium
The states are essentiallydetermined by the intersectionof two functions
Predation function and flow
N
N
P dNdt
Bifurcation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N
Fn
BNA2+N 2
R(1 ! NK )
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 6 / 14
Let’s Add Some Predation
In a modified model we considerloss of biomass by predation
dN
dt= RN(1−N
K)− BN2
A2 + N2
The functional response modelscertain predation behavior
This model supports two stablestates in some parameter range
Also for some parameters a largepopulation is the onlyequilibrium
The states are essentiallydetermined by the intersectionof two functions
Predation function and flow
N
N
P dNdt
Bifurcation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N
Fn
BNA2+N 2
R(1 ! NK )
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 6 / 14
Let’s Add Some Predation
In a modified model we considerloss of biomass by predation
dN
dt= RN(1−N
K)− BN2
A2 + N2
The functional response modelscertain predation behavior
This model supports two stablestates in some parameter range
Also for some parameters a largepopulation is the onlyequilibrium
The states are essentiallydetermined by the intersectionof two functions
Predation function and flow
N
N
P dNdt
Bifurcation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N
Fn
BNA2+N 2
R(1 ! NK )
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 6 / 14
Let’s Add Some Predation
In a modified model we considerloss of biomass by predation
dN
dt= RN(1−N
K)− BN2
A2 + N2
The functional response modelscertain predation behavior
This model supports two stablestates in some parameter range
Also for some parameters a largepopulation is the onlyequilibrium
The states are essentiallydetermined by the intersectionof two functions
Predation function and flow
N
N
P dNdt
Bifurcation
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
N
Fn
BNA2+N 2
R(1 ! NK )
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 6 / 14
Example II: A Mathematically Twisted Love Story
True dynamics of love is of course COMPLEX
But, let’s approach love mathematically with a weird example
Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.
Mathematical model:
dR
dt= aJ,
dJ
dt= −bR, a, b > 0
Differentiate left eq. w.r.t. t and use the right eq. we get
d2R
dt2+ abR = 0
The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 7 / 14
Example II: A Mathematically Twisted Love Story
True dynamics of love is of course COMPLEX
But, let’s approach love mathematically with a weird example
Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.
Mathematical model:
dR
dt= aJ,
dJ
dt= −bR, a, b > 0
Differentiate left eq. w.r.t. t and use the right eq. we get
d2R
dt2+ abR = 0
The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 7 / 14
Example II: A Mathematically Twisted Love Story
True dynamics of love is of course COMPLEX
But, let’s approach love mathematically with a weird example
Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.
Mathematical model:
dR
dt= aJ,
dJ
dt= −bR, a, b > 0
Differentiate left eq. w.r.t. t and use the right eq. we get
d2R
dt2+ abR = 0
The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 7 / 14
Example II: A Mathematically Twisted Love Story
True dynamics of love is of course COMPLEX
But, let’s approach love mathematically with a weird example
Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.
Mathematical model:
dR
dt= aJ,
dJ
dt= −bR, a, b > 0
Differentiate left eq. w.r.t. t and use the right eq. we get
d2R
dt2+ abR = 0
The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 7 / 14
Example II: A Mathematically Twisted Love Story
True dynamics of love is of course COMPLEX
But, let’s approach love mathematically with a weird example
Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.
Mathematical model:
dR
dt= aJ,
dJ
dt= −bR, a, b > 0
Differentiate left eq. w.r.t. t and use the right eq. we get
d2R
dt2+ abR = 0
The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 7 / 14
Example II: A Mathematically Twisted Love Story
True dynamics of love is of course COMPLEX
But, let’s approach love mathematically with a weird example
Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.
Mathematical model:
dR
dt= aJ,
dJ
dt= −bR, a, b > 0
Differentiate left eq. w.r.t. t and use the right eq. we get
d2R
dt2+ abR = 0
The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 7 / 14
Example III: Motion in the environment
Internal wave motions are important in several ways
1015 1016 1017 1018 1019 1020 10214000
3500
3000
2500
2000
1500
1000
500
0
z
Observations tell us that seawater or air density vary with height
We idealize a seawater drop as ping-pong ball, and neglect damping
The motion is governed by the buoyancy force, and for simplicity, weassume linear density stratification
ρ(z) = ρ0 − kz , k > 0
minus since seawater denser at depth, ρ0 is a reference density,chosen to be the density of the ping-pong ball
z = 0 at equilibrium density
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 8 / 14
Example III: Motion in the environment
Internal wave motions are important in several ways
1015 1016 1017 1018 1019 1020 10214000
3500
3000
2500
2000
1500
1000
500
0
z
Observations tell us that seawater or air density vary with height
We idealize a seawater drop as ping-pong ball, and neglect damping
The motion is governed by the buoyancy force, and for simplicity, weassume linear density stratification
ρ(z) = ρ0 − kz , k > 0
minus since seawater denser at depth, ρ0 is a reference density,chosen to be the density of the ping-pong ball
z = 0 at equilibrium density
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 8 / 14
Example III: Motion in the environment
Internal wave motions are important in several ways
1015 1016 1017 1018 1019 1020 10214000
3500
3000
2500
2000
1500
1000
500
0
z
Observations tell us that seawater or air density vary with height
We idealize a seawater drop as ping-pong ball, and neglect damping
The motion is governed by the buoyancy force, and for simplicity, weassume linear density stratification
ρ(z) = ρ0 − kz , k > 0
minus since seawater denser at depth, ρ0 is a reference density,chosen to be the density of the ping-pong ball
z = 0 at equilibrium density
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 8 / 14
Example III: Motion in the environment
Internal wave motions are important in several ways
1015 1016 1017 1018 1019 1020 10214000
3500
3000
2500
2000
1500
1000
500
0
z
Observations tell us that seawater or air density vary with height
We idealize a seawater drop as ping-pong ball, and neglect damping
The motion is governed by the buoyancy force, and for simplicity, weassume linear density stratification
ρ(z) = ρ0 − kz , k > 0
minus since seawater denser at depth, ρ0 is a reference density,chosen to be the density of the ping-pong ball
z = 0 at equilibrium density
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 8 / 14
Example III: Motion in the environment
Internal wave motions are important in several ways
1015 1016 1017 1018 1019 1020 10214000
3500
3000
2500
2000
1500
1000
500
0
z
Observations tell us that seawater or air density vary with height
We idealize a seawater drop as ping-pong ball, and neglect damping
The motion is governed by the buoyancy force, and for simplicity, weassume linear density stratification
ρ(z) = ρ0 − kz , k > 0
minus since seawater denser at depth, ρ0 is a reference density,chosen to be the density of the ping-pong ball
z = 0 at equilibrium density
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 8 / 14
Constructing Differential Equation
Motion is induced by buoyancy force
FB = ma = ρ0vd2z
dt2= [ρ(z)− ρ0]gv
→ d2z
dt2+
gk
ρ0z = 0
z
It’s an oscillation with frequency√
gkρ0
A lot more to learn before we can discuss internal gravity waves
Important to remember, we have idealized the problem a lot!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 9 / 14
Constructing Differential Equation
Motion is induced by buoyancy force
FB = ma = ρ0vd2z
dt2= [ρ(z)− ρ0]gv
→ d2z
dt2+
gk
ρ0z = 0
z
It’s an oscillation with frequency√
gkρ0
A lot more to learn before we can discuss internal gravity waves
Important to remember, we have idealized the problem a lot!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 9 / 14
Constructing Differential Equation
Motion is induced by buoyancy force
FB = ma = ρ0vd2z
dt2= [ρ(z)− ρ0]gv
→ d2z
dt2+
gk
ρ0z = 0
z
It’s an oscillation with frequency√
gkρ0
A lot more to learn before we can discuss internal gravity waves
Important to remember, we have idealized the problem a lot!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 9 / 14
Constructing Differential Equation
Motion is induced by buoyancy force
FB = ma = ρ0vd2z
dt2= [ρ(z)− ρ0]gv
→ d2z
dt2+
gk
ρ0z = 0
z
It’s an oscillation with frequency√
gkρ0
A lot more to learn before we can discuss internal gravity waves
Important to remember, we have idealized the problem a lot!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 9 / 14
Example IV: Synchronization
Synchronization in fireflies, clapping hands
We model cyclic behavior through angular speed, for Tom and TJ
dΘ
dt= Ω + K1 sin(θ −Θ)
dθ
dt= ω + K2 sin(Θ− θ)
To begin, we subtract the first equation from the second by writingφ = Θ− θ:
dφ
dt= Ω− ω − (K1 + K2) sinφ
Looking for dφ/dt = 0, we find that if |(Ω− ω)/(K1 + K2)| ≤ 1
φ? = arcsin(Ω− ω
K1 + K2)
Two equilibrium points, one stable, corresponding to phase lock
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 10 / 14
Example IV: Synchronization
Synchronization in fireflies, clapping hands
We model cyclic behavior through angular speed, for Tom and TJ
dΘ
dt= Ω + K1 sin(θ −Θ)
dθ
dt= ω + K2 sin(Θ− θ)
To begin, we subtract the first equation from the second by writingφ = Θ− θ:
dφ
dt= Ω− ω − (K1 + K2) sinφ
Looking for dφ/dt = 0, we find that if |(Ω− ω)/(K1 + K2)| ≤ 1
φ? = arcsin(Ω− ω
K1 + K2)
Two equilibrium points, one stable, corresponding to phase lock
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 10 / 14
Example IV: Synchronization
Synchronization in fireflies, clapping hands
We model cyclic behavior through angular speed, for Tom and TJ
dΘ
dt= Ω + K1 sin(θ −Θ)
dθ
dt= ω + K2 sin(Θ− θ)
To begin, we subtract the first equation from the second by writingφ = Θ− θ:
dφ
dt= Ω− ω − (K1 + K2) sinφ
Looking for dφ/dt = 0, we find that if |(Ω− ω)/(K1 + K2)| ≤ 1
φ? = arcsin(Ω− ω
K1 + K2)
Two equilibrium points, one stable, corresponding to phase lock
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 10 / 14
Example IV: Synchronization
Synchronization in fireflies, clapping hands
We model cyclic behavior through angular speed, for Tom and TJ
dΘ
dt= Ω + K1 sin(θ −Θ)
dθ
dt= ω + K2 sin(Θ− θ)
To begin, we subtract the first equation from the second by writingφ = Θ− θ:
dφ
dt= Ω− ω − (K1 + K2) sinφ
Looking for dφ/dt = 0, we find that if |(Ω− ω)/(K1 + K2)| ≤ 1
φ? = arcsin(Ω− ω
K1 + K2)
Two equilibrium points, one stable, corresponding to phase lock
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 10 / 14
Example IV: Synchronization
Synchronization in fireflies, clapping hands
We model cyclic behavior through angular speed, for Tom and TJ
dΘ
dt= Ω + K1 sin(θ −Θ)
dθ
dt= ω + K2 sin(Θ− θ)
To begin, we subtract the first equation from the second by writingφ = Θ− θ:
dφ
dt= Ω− ω − (K1 + K2) sinφ
Looking for dφ/dt = 0, we find that if |(Ω− ω)/(K1 + K2)| ≤ 1
φ? = arcsin(Ω− ω
K1 + K2)
Two equilibrium points, one stable, corresponding to phase lock
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 10 / 14
Reality Check
‘Realistic’ models are built on these simple models
Although independence is important, collaboration is crucial
Problems are interesting, walking through takes patience(constructing models, coding and debugging)
Important to keep track of things, as your memory power worksagainst you when you get older
You need to stay at the forefront, but being the lead expert on yourown field feels great!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 11 / 14
Reality Check
‘Realistic’ models are built on these simple models
Although independence is important, collaboration is crucial
Problems are interesting, walking through takes patience(constructing models, coding and debugging)
Important to keep track of things, as your memory power worksagainst you when you get older
You need to stay at the forefront, but being the lead expert on yourown field feels great!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 11 / 14
Reality Check
‘Realistic’ models are built on these simple models
Although independence is important, collaboration is crucial
Problems are interesting, walking through takes patience(constructing models, coding and debugging)
Important to keep track of things, as your memory power worksagainst you when you get older
You need to stay at the forefront, but being the lead expert on yourown field feels great!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 11 / 14
Reality Check
‘Realistic’ models are built on these simple models
Although independence is important, collaboration is crucial
Problems are interesting, walking through takes patience(constructing models, coding and debugging)
Important to keep track of things, as your memory power worksagainst you when you get older
You need to stay at the forefront, but being the lead expert on yourown field feels great!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 11 / 14
Reality Check
‘Realistic’ models are built on these simple models
Although independence is important, collaboration is crucial
Problems are interesting, walking through takes patience(constructing models, coding and debugging)
Important to keep track of things, as your memory power worksagainst you when you get older
You need to stay at the forefront, but being the lead expert on yourown field feels great!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 11 / 14
Roles And Challenges
Identify new dynamical processes and translate into mathematicallanguage
For complex problems, deal with one aspect, study certain range ofbehaviors
Many fronts in the application of mathematicse.g. Math Awareness Month themes: Mathematics and theEnvironment, Mathematics and Manufacturing, Mathematics andMedicine, Mathematics and Decision Making, Mathematics and theOcean, Mathematics and the Genome, Mathematics and Climate,Mathematics and Sports, ...
Holistic approach to modeling the environment, involvingatmosphere-ocean dynamics, natural resources, urban developmentand decision making, etc.
Uncertainties arise in nonlinear dynamical systems for the environment
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 12 / 14
Roles And Challenges
Identify new dynamical processes and translate into mathematicallanguage
For complex problems, deal with one aspect, study certain range ofbehaviors
Many fronts in the application of mathematicse.g. Math Awareness Month themes: Mathematics and theEnvironment, Mathematics and Manufacturing, Mathematics andMedicine, Mathematics and Decision Making, Mathematics and theOcean, Mathematics and the Genome, Mathematics and Climate,Mathematics and Sports, ...
Holistic approach to modeling the environment, involvingatmosphere-ocean dynamics, natural resources, urban developmentand decision making, etc.
Uncertainties arise in nonlinear dynamical systems for the environment
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 12 / 14
Roles And Challenges
Identify new dynamical processes and translate into mathematicallanguage
For complex problems, deal with one aspect, study certain range ofbehaviors
Many fronts in the application of mathematicse.g. Math Awareness Month themes: Mathematics and theEnvironment, Mathematics and Manufacturing, Mathematics andMedicine, Mathematics and Decision Making, Mathematics and theOcean, Mathematics and the Genome, Mathematics and Climate,Mathematics and Sports, ...
Holistic approach to modeling the environment, involvingatmosphere-ocean dynamics, natural resources, urban developmentand decision making, etc.
Uncertainties arise in nonlinear dynamical systems for the environment
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 12 / 14
Roles And Challenges
Identify new dynamical processes and translate into mathematicallanguage
For complex problems, deal with one aspect, study certain range ofbehaviors
Many fronts in the application of mathematicse.g. Math Awareness Month themes: Mathematics and theEnvironment, Mathematics and Manufacturing, Mathematics andMedicine, Mathematics and Decision Making, Mathematics and theOcean, Mathematics and the Genome, Mathematics and Climate,Mathematics and Sports, ...
Holistic approach to modeling the environment, involvingatmosphere-ocean dynamics, natural resources, urban developmentand decision making, etc.
Uncertainties arise in nonlinear dynamical systems for the environment
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 12 / 14
Roles And Challenges
Identify new dynamical processes and translate into mathematicallanguage
For complex problems, deal with one aspect, study certain range ofbehaviors
Many fronts in the application of mathematicse.g. Math Awareness Month themes: Mathematics and theEnvironment, Mathematics and Manufacturing, Mathematics andMedicine, Mathematics and Decision Making, Mathematics and theOcean, Mathematics and the Genome, Mathematics and Climate,Mathematics and Sports, ...
Holistic approach to modeling the environment, involvingatmosphere-ocean dynamics, natural resources, urban developmentand decision making, etc.
Uncertainties arise in nonlinear dynamical systems for the environment
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 12 / 14
Learn Good Math and Solve Important Problems!
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 13 / 14
Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 14 / 14