Unit Ten Summary Introduction to Dynamical Systems and Chaos

Transcription

1 Unit Ten Summary Introduction to Dynamical Systems

2 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.

3 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.

4 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.

5 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.

6 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 righthand side of the differential equation is well behaved.) Such a solution is unique. The initial condition and the rule determine the future behavior.

7 A system is chaotic if: Chaos! 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.

8 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.

9 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.)

10 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

11 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.

12 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.

13 Bifurcation Diagrams: Logistic Equation There is a complicated but structured set of behaviors for the logistic equation.

14 Universality in Period Doubling tells us how many times larger branch n is than branch n+1

15 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.

16 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 Somehow these simple one-dimensional equations capture a feature of complicated physical systems

17 Two-Dimensional Differential Equations Main example: Lotka-Volterra equations Basic model of predator-prey interaction

18 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

19 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.

20 Three-Dimensional Differential Equations Solutions are x(t), y(t), and z(t).

21 Phase Space Instead of a phase plane, we have (3d) phase space.

22 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.

23 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.

24 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.

25 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.

26 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 patternforming dynamical system, reaction-diffusion systems.

27 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.

28 Reaction Diffusion Results See program at the Experimentarium Digitale site

29 Reaction Diffusion Results Belousov Zhabotinsky experiment Video by Stephen Morris, U Toronto.

30 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.