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+1 vs. x t 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.
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?
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: x 1 = f 1 (x 1, x 2,...,x n ) x n = f n (x 1, x 2,..., x n ) x i dx i /dt Phase space is the space with coordinates <x 1, x n > We call this a n-dimensional system or an n-th order system.
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.
Example Chaotic Dynamical System The Logistic Map Consider the following iterative equation: x t +1 = 4rx t (1 x t ) x t,r [0,1] 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?
The Logistic Map cont. The logistic map: x t +1 = 4rx t (1 x t ) What is the behavior of this equation for different values of r and x 0? For r 1 x t ==> 0 (stable fixed point) 4 For 1 x t ==> stable fixed point attractor (next slide) 4 < r < 3 4 note: x t = 0 is a second fixed point (unstable) For x t ==> periodic with unstable points and chaos r > 3 4 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
Logistic Map State Diagram x t +1 = 4rx t (1 x t )
Figure 10.2 goes here.
Transition to Chaos
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. Fractal regions in the bifurcation diagram
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.
Information Loss Chaos as Mixing and Folding Information loss as loss of correlation from initial conditions
Reading: Chapter 11, 12 for Monday Complexity in Climate Change Models for Wednesday
Transition to chaos in the Logistic Map http://en.wikipedia.org/wiki/file:logisticcobwebchaos.gif The Lorenz Equations http://cs.unm.edu/~bakera/lorenz.html
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. 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. The equilibrium states which can be observed experimentally are those modeled by limit sets which receive most of the trajectories. These are called attractors.
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.
Example Trajectories Linear Vector Fields Wikipedia, 2007
Shadowing Lemma
Chapter 12: Producer Consumer Dynamics 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. Example: Lotka-Volterra equations: population growth of 2 linked populations df/dt = F(a-bS) ds/dt = S(cF-d)
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 x t +1 = 4rx t (1 x t ) Where is the equilibrium?
Tuning parameters to find chaotic regimes Discrete vs continuous equations continuous chaos requires 3 dimensions (3 populations) A is a matrix of coefficients that spp j has on spp i A = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 A = 0.5 0.5 0.1-0.5-0.1 0.1 α 0.1 0.1
Lotka Volterra Time Series
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
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
Reading & 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).