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David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Dynamical Systems● A dynamical system is a system that evolves
in time according to a well-defined, unchanging rule.
● The study of dynamical systems is concerned with general properties of dynamical systems.
● We seek to classify and characterize the types of behavior seen in dynamical systems.
● We looked at two types of dynamical systems: iterated functions and differential equations.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Iterated Functions
●
● Example: Logistic Equation
● Given an initial condition, or seed, one repeatedly applies the function.
● The resulting sequence of numbers is the orbit, or itinerary.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Differential Equations
●
● Example: Newton's Law of Cooling:
● This is a rule for how the Temperature depends
on time. ● The rule is “indirect” since it involves the rate of
change of T and not T itself.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Solving Differential Equations
1. Analytic. Using calculus tricks to figure out a formula for x(t).
2. Qualitative. Draw graph of f(x) and use this to find fixed points and long-term behavior of solutions.
3. Numeric. Euler's method. dx/dt is changing all the time, but pretend it is constant for small time intervals ∆t.
We focused on Qualitative and Numeric solutions.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Uniqueness and Existence
● Given an initial condition, we can “obey the rule” and solve the iterated function or differential equation.
● Such a solution exists (provided that the right-hand side of the differential equation is well behaved.)
● Such a solution is unique. The initial condition and the rule determine the future behavior.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Chaos!A system is chaotic if:
1. The dynamical system is deterministic.
2. The orbits are bounded.
3. The orbits are aperiodic.
4. The orbits have sensitive dependence on initial conditions.
● The logistic equation, f(x) = rx(1-x) is chaotic for r=4.0.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
The Butterfly Effect
● For any initial condition there is another initial condition very near to it that eventually ends up far away.
● To predict the behavior of a system with SDIC requires knowing the initial condition with impossible accuracy.
● Systems with SDIC are deterministic yet unpredictable in the long run.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Randomness?● Algorithmic randomness: a random sequence
is one that is incompressible.● For the logistic equation with r=4.0, almost any
initial condition will yield a sequence that is random in the sense of incompressible.
● Thus the logistic equation is a deterministic dynamical system that produces randomness.
(This is a subtle and somewhat involved argument. I've omitted lots of details in this summary.)
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
1D Differential Equations vs. Iterated Functions
● Time is continuous● P is continuous● Cycles and chaos are
not possible● This is due to
determinism: for a given P the population can have only on dP/dt
● Time moves in jumps● x moves in jumps● Cycles and chaos
are possible
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Bifurcation Diagrams● A way to see how the behavior of a dynamical
system changes as a parameter is changed.● For each parameter value, make a phase line or
a final-state diagram.● “Glue” these together to make a bifurcation
diagram.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Bifurcation Diagrams: Logistic Equation with Harvest
● As the harvest rate is increased, the stable fixed point suddenly disappears.
● A continuous dynamical system has a discontinuous transition.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Bifurcation Diagrams: Logistic Equation
● There is a complicated but structured set of behaviors for the logistic equation.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Universality in Period Doubling● ● tells us how many times larger branch n is
than branch n+1
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Is Universal
●
● is the value of for large n.● delta is universal: it has the same value for all
functions f(x) that map an interval to itself and have a single quadratic maximum.
●
● This value is often known as Feigenbaum's constant.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Universality in Physical Systems
● The period doubling route to chaos is observed in physical systems
● delta can be measured for these systems.● The results are consistent with the universal
value 4.669... ● Somehow these simple one-dimensional
equations capture a feature of complicated physical systems
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Two-Dimensional Differential Equations
●
● Main example: Lotka-Volterra equations● Basic model of predator-prey interaction
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
The Phase Plane
● Plot R and F against each other● Similar to a phase line for 1D equations● Shows how R and F are related
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
No Chaos in 2D Differential Equations
● The fact that curves cannot cross limits the possible long-term behaviors of two-dimensional differential equations.
● There can be stable and unstable fixed points, orbits can tend toward infinity, and there can be limit cycles, attracting cyclic behavior.
● Poincaré–Bendixson theorem: bounded, aperioidc orbits are not possible for two-dimensional differential equations.
● Thus, 2D differential equations can not be chaotic.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Three-Dimensional Differential Equations
● Solutions are x(t), y(t), and z(t).
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Phase Space
● Instead of a phase plane, we have (3d) phase space.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Phase Space
● Curves in phase space cannot intersect.● But because the space is three-dimensional,
curves can go over or under each other.● This means that 3D differential equations are
capable of more complicated behaviors than 2D differential equations.
● 3D differential equations can be chaotic.● Chaotic trajectories in phase space often get
pulled to strange attractors.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Strange Attractors
● It is an attractor: nearby orbits get pulled into it. It is stable.
● Motion on the attractor is chaotic: orbits are aperiodic and have sensitive dependence on initial conditions.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Stretching and Folding
● The key geometric ingredients of chaos● Stretching pulls nearby orbits apart, leading to
sensitive dependence on initial conditions● Folding takes far apart orbits and moves them
closer together, keeping orbits bounded. ● Stretching and folding occurs in 1D maps as well
as higher-dimensional phase space.● This explains how 1D maps can capture some
features of higher-dimensional systems.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Strange Attractors
● Complex structures arising from simple dynamical systems.
● Three examples: Hénon, Rössler, Lorenz● The motion on the attractor is chaotic.● But all orbits get pulled to the attractor.● Combine elements of order and disorder.● Motion is locally unstable, globally stable.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Pattern Formation● We have seen that dynamical systems are
capable of chaos: unpredictable, aperiodic behavior.
● But dynamical systems can do much more than chaos.
● They can produce patterns, structure, organization...
● We looked at one example of a pattern-forming dynamical system, reaction-diffusion systems.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Reaction-Diffusion Systems
● Two chemicals that react and diffuse.● Chemical concentrations: u(x,y) and v(x,y).● The interactions are specified by f(u,v) and g(u,v).
● A deterministic, spatially-extended dynamical system.● The rule is local. The “next” value of u and v at a point
depends only on the present values of u and v and their derivatives at that point.
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Reaction Diffusion Results
● See program at the Experimentarium Digitale sitehttp://experiences.math.cnrs.fr/Structures-de-Turing.html
David P. Feldman Introduction to Dynamical Systemsand Chaos
http://www.complexityexplorer.org
Reaction Diffusion Results
● Belousov Zhabotinsky experiment● http://www.youtube.com/watch?v=3JAqrRnKFHo● Video by Stephen Morris, U Toronto.