introduction to dynamical systems basic concepts of...
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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+1 vs. xt 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:
• 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 )
˙ x n = fn (x1,x2,...,xn )€
˙ x i ≡ dxi /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 xi, 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.
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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xt+1 = 4rxt (1− xt )
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xt ,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 xt ==> 0 (stable fixed point)
– For xt ==> stable fixed point attractor (next slide) note: xt = 0 is a second fixed point (unstable) – For xt ==> periodic with unstable points and chaos
• 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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xt+1 = 4rxt (1− xt )
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r ≤ 14
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14
< r <34
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r >34
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 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
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.
Chapter 12: Producer Consumer DynamicsState 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
Where is the equilibrium?
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xt+1 = 4rxt (1− xt )
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 = A11 A12 A13 A21 A22 A23
A31 A32 A33
A = 0.5 0.5 0.1 -0.5 -0.1 0.1
α 0.1 0.1
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).