Summarys3.amazonaws.com/.../DynamicsAndChaos/Unit+10/10Summary.pdfUnit Ten Summary Introduction to Dynamical Systems and Chaos

Unit Ten

Summary

Introduction to Dynamical Systems and Chaos

http://www.complexityexplorer.org

David P. Feldman Introduction to Dynamical Systemsand Chaos


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.



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.



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.



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.



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.



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.



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.



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.)



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



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.



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.



Bifurcation Diagrams: Logistic Equation

● There is a complicated but structured set of behaviors for the logistic equation.



Universality in Period Doubling● ● tells us how many times larger branch n is

than branch n+1



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.



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



Two-Dimensional Differential Equations

●

● Main example: Lotka-Volterra equations● Basic model of predator-prey interaction



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



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.



Three-Dimensional Differential Equations

● Solutions are x(t), y(t), and z(t).



Phase Space

● Instead of a phase plane, we have (3d) phase space.



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.



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.



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.



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.



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.



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.



Reaction Diffusion Results

● See program at the Experimentarium Digitale sitehttp://experiences.math.cnrs.fr/Structures-de-Turing.html



Reaction Diffusion Results

● Belousov Zhabotinsky experiment● http://www.youtube.com/watch?v=3JAqrRnKFHo● Video by Stephen Morris, U Toronto.

http://www.youtube.com/watch?v=3JAqrRnKFHo



Pattern Formation

● There is more to dynamical systems than chaos

● Simple, spatially-extended dynamical systems with local rules are capable of producing stable, global patterns and structures.

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Summarys3.amazonaws.com/.../DynamicsAndChaos/Unit+10/10Summary.pdfUnit Ten Summary Introduction to Dynamical Systems and Chaos