Dynamical Systems and ChaosCAS Spring 2008

Introduction to Dynamical SystemsBasic 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 xt vs. xt+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.

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 <x1, … xn>

• We call this a n-dimensional system or an n-th order system.

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˙ x 1 = f1(x1, x2,..., xn )

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˙ x n = fn (x1,x2,..., xn )€

˙ x i ≡ dx i /dt

Linear vs. Nonlinear

• A system is said to be linear if all xi on the right-hand side appear to the first power only.

• Typical nonlinear terms are products, powers, and functions of x i, e.g.,

– x1x2

– (x1)3

– cos x2

• 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.

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?

Example Chaotic Dynamical SystemThe Logistic Map

• Consider the following iterative equation:

• We are interested in the following questions:– What are the possible asymptotic trajectories given different x0 for fixed r?

• Fixed points

• Limit cycles

• Chaos

– How do these trajectories change with small perturbations?• Stable

• Unstable

– What happens as we vary r?

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x t +1 = 4rx t (1− x t )

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x t ,r ∈ [0,1]

The Logistic Map cont.

• The logistic map:

• What is the behavior of this equation for different values of r and x0?

– For x t ==> 0 (stable fixed point)

– For x t ==> stable fixed point attractor (prev. slide)

note: xt = 0 is a second fixed point (unstable)

– For x t ==> periodic with unstable points

• If r < 1/4 then xt+1 < xt

• 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

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x t +1 = 4rx t (1− x t )

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r ≤1

4

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1

4< r <

3

4

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r >3

4

Logistic MapState Diagram

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.

Predicting chaos

• The cascade of bifurcations can be predicted from the Feigenbaum constant

• The value of r at which logistic map bifurcates into period 2n limit cycle is an

• dk = (ak - ak -1)/(ak+1 -ak)

• 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

Shadowing Lemma to the rescue

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)

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 TrajectoriesLinear Vector Fields

Wikipedia, 2007

Henon Mapxt+1= a - xt

2 +byt

yt+1 = xt

x0, y0 between -1 and 1

Time series: xt vs t ?

yt

xt

State Space for the Henon Map

Henon Mapxt+1= a - xt

2 +byt

yt+1 = xt

Lorenz Equations

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?

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).

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.

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 populationsdF/dt = F(a-bS)dS/dt = S(cF-d)

Lotka Volterra

2 spp Lotka VolteradF/dt = F(a-bS)dS/dt = S(cF-d)a is reproduction rate of Fishb is # of Fish a Shark can eatc 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 equationsstochastic extinction with discrete equations dxi/dt = xiAj

(1-xj)

continuous chaos requires 3 dimensions (3 populations) A = A11 A12 A13

matrix of coefficients that spp j has on spp I A21 A22 A23 A31 A32

A33

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x t +1 = 4rx t (1− x t )

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 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.html3. The Central Dogma of Biology for Computer Scientists

http://cnx.org/content/m11415/latest/