1. Dynamical Systems and Chaos CAS Spring 2008

2. Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: –Has a notion of state, which contains all the information upon which the dynamical system acts. –A simple set of deterministic rules for moving between states. (Minor exception: stochastic dynamical systems). –Iterators. State diagrams that plot x t vs. x t+1 characterize a dynamical system. Example asymptotic behaviors –Fixed point –Limit cycles and quasi-periodicity –Chaotic Limit sets: The set of points in the asymptotic limit. Goal: Make quantitative or qualitative predictions of the asymptotic behavior of a system.

3. Introduction Two main types of dynamical systems: –Differential equations –Iterated maps (difference equations) Primarily concerned with how systems change over time, so focus on ordinary differential equations (one independent variable). Framework for ODE: Phase space is the space with coordinates We call this a n-dimensional system or an n-th order system.

4. Linear vs. Nonlinear A system is said to be linear if all x i on the right-hand side appear to the first power only. Typical nonlinear terms are products, powers, and functions of x i, e.g., –x 1 x 2 –(x 1 ) 3 –cos x 2 Why are nonlinear systems difficult to solve? –Linear systems can be broken into parts and nonlinear systems cannot. In many cases, we can use geometric reasoning to draw trajectories through phase space without actually solving the system.

5. Chaos Chaotic dynamical systems –Have complicated, often apparently random behavior –Are deterministic –Are predictable in the short term –Are not predictable in the long term –Are everywhere Turbulence Planetary orbits Weather ?disease dynamics, stock markets and other CAS?

6. Example Chaotic Dynamical System The Logistic Map Consider the following iterative equation: We are interested in the following questions: –What are the possible asymptotic trajectories given different x 0 for fixed r? Fixed points Limit cycles Chaos –How do these trajectories change with small perturbations? Stable Unstable –What happens as we vary r?

7. The Logistic Map cont. The logistic map: What is the behavior of this equation for different values of r and x 0 ? –For x t ==> 0 (stable fixed point) –For x t ==> stable fixed point attractor (prev. slide) note: x t = 0 is a second fixed point (unstable) –For x t ==> periodic with unstable points If r < 1/4 then x t+1 < x t However, consider what happens as r increases, between 1/4 and 3/4: –For an given r, system settles into a limit cycle (period) –Successive period doublings (called bifurcations) as r increases

8. Logistic Map State Diagram

10. Figure 10.2 goes here.

11. Transition to Chaos

12. Characteristics of Chaos Deterministic. Unpredictable: –Behavior of a trajectory is unpredictable in long run. –Sensitive dependence on initial conditions. Mixing : The points of an arbitrary small interval eventually become spread over the whole unit interval. –Ergodic (every state space trajectory will return to the local region of a previous point in the trajectory, for an arbitrarily small local region). –Chaotic orbits densely cover the unit interval. Embedded (infinite number of unstable periodic orbits within a chaotic attractor). –In a system with sensitivity there is no possibility of detecting a periodic orbit by running the time series on a computer (limited precision, round-off error). Bifurcations.

13. Predicting chaos The cascade of bifurcations can be predicted from the Feigenbaum constant The value of r at which logistic map bifurcates into period 2 n limit cycle is an d k = (a k - a k -1 )/(a k+1 -a k ) d approaches 4.669…so that the rate of time between bifurcations approaches a constant.

14. Information Loss Chaos as Mixing and Folding Information loss as loss of correlation from initial conditions

15. Shadowing Lemma to the rescue

16. Attractors Initially, a trajectory through a dynamical system may be erratic. This is known as the initial transient, or start-up transient. The asymptotic behavior of the system is known as equilibrium, steady state, or dynamic equilibrium. –This does not necessarily imply a static equilibrium or static state. –The only equilibrium states which can be observed experimentally are those modeled by limit sets which receive most of the trajectories. These are called attractors. Flake, Chapter 11 –Henon Map –Lorenz Equations The Lorenz Attractor Ruele and Takens (1971)

17. Attractor Basins (from Abraham and Shaw, 1984) Basin of attraction: The points of all trajectories that converge to a given attractor. –In a typical phase portrait, there will be more than one attractor. The dividing boundaries (or regions) between different attractor regions (basins) are called separatrices. –Any point not in a basin of attraction belongs to a separatrix.

18. Example Trajectories Linear Vector Fields Wikipedia, 2007

19. Henon Map x t+1 = a - x t 2 +by t y t+1 = x t x 0, y 0 between -1 and 1 Time series: x t vs t ?

20. ytyt xtxt State Space for the Henon Map Henon Map x t+1 = a - x t 2 +by t y t+1 = x t

21. Lorenz Equations

22. Chaos and Scientific Modeling Boiling water Predicting weather Predicting climate change How can we make the notion of chaos precise? How can we be sure that what looks like chaos is really chaotic and not just very complicated but perfectly predictable? –E.g., When we see a seemingly chaotic time series, how can we be sure that it is not periodic, with an extremely long period? How can chaos be measured? What is the value of numerical calculations in the presence of chaos?

23. Reading & References Reading For Monday, Flake Chapter 12 References Chaos and Fractals by by H. Peitgen, H. Jurgens, and D. Saupe. Springer-Verlag (1992). Nonlinear Dynamics and Chaos by S. H. Strogatz. Westview (1994). J. Gleick Chaos. Viking (1987). Robert L. Devaney An Introduction to Chaotic Dynamical Systems. Addison-Wesley (1989). Ralph Abraham and Christopher D. Shaw Dynamics-The Geometry of Behavior Vol. 1-3 (1984).

25. Review: Chaos and Strange Attractors Bifurcations leading to chaos: –In the 1 D logistic map, the amount by which r must be increased to get new period doublings gets smaller and smaller for each new bifurcation. –This continues until the critical point is reached (transition to chaos). Why is chaos important? –Seemingly random behavior may have a simple, deterministic explanation. –Contrast with world view based on probability distributions. A formal definition of chaos: –Chaos is defined by the presence of positive Lyapunov exponents. Working definition (Strogatz, 1994) –“Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.” Strange Attractors: chaotic systems with an asymptotic dynamic equilibrium. The system comes close to previous states, but never repeats them.

26. State Spaces: A Geometric Approach (Abraham and Shaw, 1984) An system of interest is observed in different states. These observed states are the target of modeling activity. State space: a geometric model of the set of all modeled states. Trajectory: A curve in the state space, connecting subsequent observations. Time series: A graph of the trajectory. The history of a real system is represented graphically. Example: Lotka-Volterra equations: population growth of 2 linked populations dF/dt = F(a-bS) dS/dt = S(cF-d)

27. Lotka Volterra 2 spp Lotka Voltera dF/dt = F(a-bS) dS/dt = S(cF-d) a is reproduction rate of Fish b is # of Fish a Shark can eat c is the energy of a Fish (fraction of a new shark) d is death rate of a shark Compare to single population logistic map Where is the equilibrium? Carrying capacity (density dependence, negative feedbacks) set by the environment vs another population Discrete vs continuous equations stochastic extinction with discrete equations dx i /dt = x i  A j ( 1-x j ) continuous chaos requires 3 dimensions (3 populations) A = A 11 A 12 A 13 matrix of coefficients that spp j has on spp I A 21 A 22 A 23 A 31 A 32 A 33

28. Lotka Volterra Time Series

29. Individual Models Implementing chaos as a quasi CA –Each individual is represented explicitly –Compare the sizes of the state spaces 3 floating point numbers vs 2 bits per individual x the number of individuals –What can we hope to predict in such a complicated system? –How can we hope to find ecosystem stability –Relate to Wolfram’s CA classes

30. Just one more little complication We’ve gone from simple 1 species population model To a model where multiple populations interact To a model where each individual is represented What if there are differences between individuals? Natural selection –Geometric increases in population sizes –Carrying capacity (density dependence) that limits growth –Heritable variation in individuals that results in differential survival –Populations become dynamical complex adaptive systems

31. Reading for MONDAY March 10 1. Flake, chapter 20 -20.02 (p 339-343) 2. Darwin, The Origin of Species, Introduction chapter http://www.literature.org/authors/darwin-charles/the-origin-of- species-6th-edition/introduction.html 3. The Central Dogma of Biology for Computer Scientists http://cnx.org/content/m11415/latest/