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Dr. Dylan McNamarapeople.uncw.edu/
mcnamarad
Fundamentals of Dynamical Systems / Discrete-Time Models
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Dynamical systems theory
• Considers how systems autonomouslychange along time– Ranges from Newtonian mechanics tomodern nonlinear dynamics theories
– Probes underlying dynamical mechanisms,not just static properties of observations
– Provides a suite of tools useful forstudying complex systems
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What is a dynamical system?
• A system whose state is uniquelyspecified by a finite set of variablesand whose behavior is uniquelydetermined by predetermined "rules"
– Simple population growth– Simple pendulum swinging– Motion of celestial bodies– Behavior of two “rational” agents in anegotiation game
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Mathematical formulations of dynamical systems
• Discrete-time model:xt = F(xt-1, t)
• Continuous-time model: (differential equations) dx/dt = F(x, t) xt: State variable(s) of the system at time t F: Some function that determines the rule
that the system’s behavior will obey
(difference/recurrence equations; iterative maps)
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Discrete-Time Models
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Discrete-time model
• Easy to understand, develop andsimulate– Doesn’t require an expression for the rateof change (derivative)
– Can model abrupt changes and/or chaoticdynamics using fewer variables
– Directly translatable to simulation in acomputer
– Experimentally, we often have samples ofsystem states at specific points of time
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Difference equation and time series
• Difference equationxt = F(xt-1, t)
produces series of values of variable x starting with initial condition x0:
{ x0, x1, x2, x3, … } “time series”
– A prediction made by the above model(to be compared to experimental data)
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Linear vs. nonlinear
• Linear:– Right hand side is just a first-orderpolynomial of variablesxt = a xt-1 + b xt-2 + c xt-3 …
• Nonlinear:– Anything else
xt = a xt-1 + b xt-22 + c xt-1 xt-3 …
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1st-order vs. higher-order
• 1st-order:– Right hand side refers only to theimmediate pastxt = a xt-1 ( 1 – xt-1 )
• Higher-order:– Anything else
xt = a xt-1 + b xt-2 + c xt-3 …
(Note: this is different from the order of terms in polynomials)
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Autonomous vs. non-autonomous
• Autonomous:– Right hand side includes only statevariables (x) and not t itselfxt = a xt-1 xt-2 + b xt-3
2
• Non-autonomous:– Right hand side includes terms thatexplicitly depend on the value of txt = a xt-1 xt-2 + b xt-3
2 + sin(t)
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Things that you should know (1)
• Non-autonomous, higher-orderequations can always be convertedinto autonomous, 1st-order equations– xt-2 → yt-1, yt = xt-1
– t → yt, yt = yt-1 + 1, y0 = 0
• Autonomous 1st-order equations cancover dynamics of any non-autonomoushigher-order equations too!
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Things that you should know (2)
• Linear equations– are analytically solvable– show either equilibrium, exponentialgrowth/decay, periodic oscillation (with>1 variables), or their combination
• Nonlinear equations– may show more complex behaviors– do not have analytical solutions in general
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Simulating Discrete-Time Models
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Simulating discrete-time models
• Simulation of a discrete-time modelcan be implemented by iteratingupdating of the system’s states
– Every iteration represents one discretetime step
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Exercise
• Implement simulators of the followingmodels and produce time series for t= 1~10
xt = 2 xt-1 + 1, x0 = 1
xt = xt-12 + 1, x0 = 1
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Exercise
• Simulate the following set ofequations and see what happens if thecoefficients are varied
xt = 0.5 xt-1 + 1 yt-1
yt = -0.5 xt-1 + 1 yt-1
x0 = 1, y0 = 1
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Building Your Own Model Equation
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Mathematical modeling tips
• Grab an existing model and tweak it
• Implement each assumption one by one
• Find where to change, replace it by afunction, and design the function
• Adopt the simplest form
• Check the model with extreme values
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Example: Saturation of growth
• Simple exponential growth model:xt = a xt-1
• Problem: How can one implement thesaturation of growth in this model?
• Think about a new nonlinear model:
xt = f(xt-1) xt-1
– Coefficient replaced by a function of x
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Modeling saturation of growth
xt = f(xt-1) xt-1
• f(x) should approach 1 (no netgrowth) when x goes to a carrying capacity of the environment, say K
• f(x) should approach the originalgrowth rate a when x is very small (i.e., with no saturation effect)
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What should f(x) be?
f(x)
x K
1
a
0
f(x) = – x + aa – 1 K
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A new model of growth
xt = f(xt-1) xt-1
= ( – (a – 1) xt-1 / K + a ) xt-1
• Using r = a – 1:
xt = ( – r xt-1 / K + r + 1 ) xt-1
= xt-1 + r xt-1 ( 1 – xt-1 / K ) Net growth
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Example: Logistic growth model
• N: Population• r: Population growth rate• K: Carrying capacity
• Discrete-time version:Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )
• Continuous-time version:dN/dt = r N ( 1 – N/K )
Nonlinear terms
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• Problem: Develop a nonlinear model ofa simple ecosystem made of predatorand prey populations
Modeling with multiple variables
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Think about how variables behave in isolation
Naturally grows to carrying capacity
if isolated
+ Naturally decays
if isolated
-
Rabbit Population : x
Fox Population : y
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Initial assumptions
• Rabbits will grow based on the logisticgrowth model, with carrying capacity= 1 for simplicity
• Foxes will decay exponentially
Rabbit: xt = xt-1 + a xt-1 (1 – xt-1) Fox: yt = b yt-1
(0<a, 0<b<1)
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Naturally grows to carrying capacity
if isolated
Naturally decays if isolated
Think about how variables interact with each other
+ -
Positive influence Foxes’ growth rate
increases with increasing rabbits +
Negative influence Rabbits’ survival rate
decreases with increasing foxes
- Rabbit Population : x
Fox Population : y
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Revised model • Introduced coefficient (1 – c yt-1) (0<c) to
the first term of x – Negative influence of foxes on rabbits’ survival
rate• Replaced b with (b + d xt-1) (0<d)
– Positive influence of rabbits on foxes’ growthrate
Rabbit: xt = (1 – c yt-1) xt-1+ a xt-1 (1 - xt-1)
Fox: yt = (b + d xt-1) yt-1 (0<a, 0<b<1, 0<c, 0<d)
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FYI: Lotka-Volterra model
• This model can be rewritten as:xt – xt-1 = α xt-1 (1-xt-1) – β xt-1 yt-1 yt – yt-1 = - γ yt-1 + δ xt-1 yt-1
– Known as the “Lotka-Volterra” equations (ofdiscrete-time version with carrying capacity)
– Models predator-prey dynamics in a generalform
– One of the most famous nonlinear systems withmultiple variables
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Analysis of Discrete-Time Models
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Equilibrium point
• A state of the system at which statewill not change over time– A.k.a. fixed point, steady state
• Can be calculated by solving
xt = xt-1
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Exercise
• Calculate equilibrium points in thefollowing models
Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )
xt = 2xt-1 – xt-12
xt = xt-1 – xt-22 + 1
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Phase Space Visualization
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Geometrical approach
• Developed in the late 19Cby J. Henri Poincare
• Visualizes the behavior ofdynamical systems astrajectories in a phase space
• Produces a lot of intuitive insights ongeometrical structure of dynamicsthat would be hard to infer usingpurely algebraic methods
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Phase space (state space)
• A theoretical space in which everystate of a dynamical system ismapped to a spatial location
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Phase space (state space)
• Created by “orthogonalizing” statevariables of the system
• Its dimensionality equals # ofvariables needed to specify thesystem state (a.k.a. degrees offreedom)
• Temporal change of the system statescan be drawn in it as a trajectory
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Attractor and basin of attraction
• Attractor:A state (or a set of states) fromwhich no outgoing edges or flowsrunning in phase space– Static attractors (equilibrium points)– Dynamic attractors (e.g. limit cycles)
• Basin of attraction:A set of states which will eventuallyend up in a given attractor
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• E.g. a simple vertical spring oscillator• State can be specified by two realvariables (location x, velocity v)
v
x Trajectory
(orbit)
Dynamics of continuous models can be depicted as “flow” in a continuous phase space
Phase space of continuous-state models
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Cobweb plot
• A visual tool to study the behavior of1-D iterative maps
• Take xt-1 and xt for two axes• Draw the map of interest (xt=F(xt-1))and the “xt=xt-1” reference line– They will intersect at “equilibrium points”
• Trace how time series develop froman initial value by jumping betweenthese two curves
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Cobweb Plot
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Cobweb Plot
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Rescaling Variables
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Rescaling variables
• Dynamics of a system won’t changequalitatively by linear rescaling ofvariables (e.g., x → α x’)
• You can set arbitrary rescalingfactors for variables to simplify themodel equations
• If you have k variables, you mayeliminate k parameters
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Exercise
• Simplify the logistic growth model byrescaling x → α x’
xt = xt-1 + r xt-1 (1 - xt-1/K)
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Linear Systems
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Dynamics of linear systems
• Some systems can be modeled aslinear systems– Their dynamics is described by a productof matrix and state vector
– Either in continuous or discrete time
• Dynamics of such linear systems canbe studied analytically
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Linear systems
• Linear systems are the simplest caseswhere states of nodes are continuous-valued and their dynamics aredescribed by a time-invariant matrix
• Discrete-time: xt = A xt-1
– A is called a “coefficient” matrix
– We don’t consider constants (as they canbe easily converted to the above forms)
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Asymptotic Behavior of Linear Systems
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xt = A xt-1
This equation gives the following exact solution:
xt = At x0
Where will the system go eventually?
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xt = A xt-1
• What happens if the system startsfrom non-equilibrium initial states andgoes on for a long period of time?
• Let’s think about their asymptoticbehavior lim t->∞ xt
Where will the system go eventually?
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Considering asymptotic behavior (1)
• Let { vi } be n linearly independenteigenvectors of the coefficient matrix (They might be fewer than n, but here we ignore such cases for simplicity)
• Write the initial condition usingeigenvectors, i.e.x0 = b1v1 + b2v2 + ... + bnvn
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Considering asymptotic behavior (2)
• Then:
xt = At x0 = λ1
t b1v1 + λ2t b2v2 + … + λn
t bnvn
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Dominant eigenvector • If |λ1| > |λ2|, |λ3|, …,
xt = λ1t { b1v1 + (λ2/λ1)t b2v2 + ... + (λn/λ1)t bnvn }
lim t->∞ xt ~ λ1t b1v1
If the system has just one such dominant eigenvector v1, its state will be eventually along that vector regardless of where it starts
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What eigenvalues and eigenvectors can tell us
• An eigenvalue tells whether aparticular “state” of the system(specified by its correspondingeigenvectors) grows or shrinks byinteractions between parts
– | λ | > 1 -> growing– | λ | < 1 -> shrinking
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Example • Phase space of a two-variable linear difference
equation with (a, b, c, d) = (1, 0.1, 0.1, 0.9)
x
y
Along these lines (called
invariant lines), the dynamics
can be understood as
simple exponential
growth/decay
|λ1|>1
|λ2|<1 v1
v2
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Example
x
y
This could be regarded as a very simple form of self-organization (though completely predictable); Order spontaneously emerges in
the system as time goes on
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Linear Stability Analysis of Nonlinear Systems
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Stability of equilibrium points
• If a system at its equilibrium point isslightly perturbed, what happens?
• The equilibrium point is called:– Stable (or asymptotically stable) ifthe system eventually falls back to the equilibrium point
– Lyapunov stable if the system doesn’tgo far away from the equilibrium point
– Unstable otherwise
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Question
• What is the stability of each of thefollowing equilibrium points?
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Linear stability analysis
• Studies whether a nonlinear system isstable or not at its equilibrium pointby locally linearizing its dynamicsaround that point
Linearization
Linearization
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Local linearization (1)
• Let ∆x be a small difference betweenthe system’s current state x and itsequilibrium point xe, i.e. x = xe +∆x
• Plug x = xe + ∆x into differentialequations and ignore quadratic or higher-order terms of ∆x (hence the name “linearization”)
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Local linearization (2)
• This operation does the trick toconvert the dynamics of ∆x into aproduct of a matrix and ∆x
• By analyzing eigenvalues of the matrix,one can predict whether xe is stableor not – I.e. whether a small perturbation (∆x)grows or shrinks over time
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Mathematically speaking…
• This operation is similar to “linearapproximation” in calculus
Taylor series expansion: F(x) = Σn=0~∞ F(n)(a)/n! (x-a)n
Let x → xe+∆x and a→xe , then F(xe+∆x) = F(xe) + F’(xe) ∆x
+ O(∆x2)Ignore
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Linearizing discrete-time models
• For discrete-time models:xt = F(xt-1)Left = xe + ∆xt
Right = F(xe + ∆xt-1) ~ F(xe) + F’(xe) ∆xt-1
= xe + F’(xe) ∆xt-1
Therefore, ∆xt = F’(xe) ∆xt-1
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First-order derivative of vector functions
• Discrete-time: ∆xt = F’(xe) ∆xt-1
This can hold even if x is a vector What corresponds to the first-order derivative in such a case:
F’(xe) = dF/dx(x=xe) =
∂ F1 ∂ F1 ∂ F1 ∂ x1 ∂ x2 ∂ xn
∂ F2 ∂ F2 ∂ F2 ∂ x1 ∂ x2 ∂ xn
∂ Fn ∂ Fn ∂ Fn ∂ x1 ∂ x2 ∂ xn
…
…
…
… … …
Jacobian matrix at x=xe
(x=xe)
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Eigenvalues of Jacobian matrix
• A Jacobian matrix is a linearapproximation around the equilibriumpoint, telling you the local dynamics:“how a small perturbation will grow,shrink or rotate around that point”– The equilibrium point serves as a localorigin
– The ∆x serves as a local coordinate– Eigenvalue analysis applies
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With real eigenvalues
• If all the eigenvalues indicatethat ∆x will shrink over time-> stable point
• If all the eigenvalues indicatethat ∆x will grow over time-> unstable point
• If some eigenvalues indicateshrink and others indicate growof ∆x over time-> saddle point (this is also unstable)
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With two complex conjugate eigenvalues (for 2-D systems) • If both eigenvalues indicate that
∆x will shrink over time-> stable spiral focus
• If both eigenvalues indicate that∆x will grow over time-> unstable spiral focus
• If both eigenvalues indicateneither shrink nor growth of ∆x-> neutral center (but this may or maynot be true for nonlinear models; furtheranalysis is needed to check if nearbytrajectories are truly cycles or not)
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• Each eigenvalue (or a pair of complexconjugate eigenvalues) tell you distinctdynamics simultaneously seen at theequilibrium point:
With real and complex eigenvalues mixed (for higher-dimm. systems)
All real eigenvalues (1 indicates growth; other 2 indicates shrink)
1 real eigenvalue indicates growth; other 2 indicates rotation (complex conjugates with no growth or shrink)
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Exercise
• Find all equilibrium points of thefollowing model, and study theirstability
xt = xt-1 yt-1
yt = yt-1 (xt-1 – 1)
87