Nonlinear Dynamics Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/
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Introduction: Dynamics of Simple Maps 3
Dynamical systems A dynamical system can be informally defined as any system where some fixed rule describes the time dependence of the position and velocity of a point in geometric space Examples Planet orbiting around a star Oscillating pendulum Chemical reaction Cellular automata (more about these later) 4
Fixed points There are many different types of motion For example, a moving object may reach a fixed point Fixed point (e.g., a pendulum coming to a complete stop due to friction). Limit cycle (the system state eventually repeats itself; e.g., planet orbiting around a star) Quasiperiodi orbit (the system is periodic, but its state does not precisely repeat; e.g., multiple planets orbiting a star with non-resonant orbits) 5
Chaos For a long time it was believed that every dynamical system had either a fixed point, a periodic orbit or a quasiperiodic orbit Now, we understand that there are plenty of examples of systems that do not fall in any of the above classes Turbulence in water or air Wobble of planets following complicate orbits Weather pattern Electric activity of the brain Double rod pendulum 6
Example 7
Logistic map x n+1=r x n (1 x n ), r [0, 4], x n [0,1] Simple model of population growth when the population size is small, the population will increase at a rate proportional to the current population. the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population Note: the book uses the slightly different formulation x n+1=4 r x n (1 x n ) we will not use this: we adopt the standard formulation at the top of this slide, as commonly used 8
Logistic map The equation is fully deterministic: apparently, nothing surprising can happen there x n+1= f ( x n )=r x n (1 x n ) y r/4 0 1/2 1 x 9
Fixed points We want to study the steady-state dynamics of the logistic map, for every value of r We start from a given x0 and iterate the recurrence xn+1 = f(xn) = rxn( 1 xn ) What are the fixed points of f? Those values x for which f(x) = x 10
Logistic map There are two fixed points: x = 0 and x = (r - 1) / r The second one is valid only if r 1 x n+1= f ( x n )=r x n (1 x n ) y y=x r/4 0 1/2 (r-1) / r 1 x 11
Iterates of the logistic map (r = 2.8) 12
Iterates of the logistic map (r = 3.2) 13
Iterates of the logistic map (r = 3.52) 14
Iterates of the logistic map (r = 4) 15
Classification of fixed points Let xf be a fixed point for function f If f '(xf) = 0 super-stable If f '(xf) < 1 attracting and stable If f '(xf) = 1 neutral If f '(xf) > 1 repelling and unstable 16
The logistic map r 1 1<r 3 Iterations eventually converge to the fixed point 0 (stable) Unstable fixed point 0 Stable fixed point (1 - r) / r r>3 Chaotic behavior, period-doubling bifurcations 17
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For those mathematically inclined Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Westview Press, 2003, ISBN 978-0813340852 19
Higher Dimensions 20
Higher dimensions Chaotic behavior can be observed by iterating some simple maps in higher dimensions Example: Arnold's cat map Γ :( x, y) ( (2x + y) mod 1,( x+ y) mod 1 ) 21
Arnold's cat map This (and similar) map is usually shown to illustrate the Poincaré recurrence theorem Certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state Iteration of the cat map eventually produces the initial image 22
Baker's map 23
Baker's map 24
Iterating the baker's map 25
Invariant image 26
Strange attractors Strange attractors are attractors with a fractal structure Let us consider the Hénon map (introduced by the French astronomer Michel Hénon) that maps two points (xt, yt) into a new pair of points (xt+1, yt+1), as follows: 2 t x t +1 =a x +b y t y t +1 =x t where a, b are two constants Again, this is a fully deterministic map 27
The Hénon map for a=1.29, b=0.3 (Only xt is shown, random initial values) xt t 28
The Hénon map for a=1.29, b=0.3 (Only xt is shown, random initial values) xt t 29
The Hénon map for a=1.29, b=0.3 (Only xt is shown, random initial values) xt x = 0.838486... is an unstable attractor t 30
The Hénon attractor viewed at different scales (plots of yt versus xt) 31
Hénon map attractor The Hénon map attractor is made of those points that map into the attractor In other words, the attractor is invariant in the Hénon map The Hénon map attractor can be computed by warping a square according to the Hénon map 32
Producer-Consumer Dynamics 33
Predator-Prey Model Volterra and Lotka df = F (a bs ) dt F = small fish population S = shark population a = reproduction rate of small fish b = number of small fish that a shark can eat c = amount of energy that a small fish supplies to a shark ds =S (cf d ) dt If c is large, csf will be large, meaning that the shark population increases d = death rate of sharks 34
Population Dynamics Fixed point at F = d/c, S = a/b 35
Generalization The Volterra-Lotka model can be extended to an arbitrary number n of species n dx i = x i Aij (1 x j ) dt j=1 where xi represents the i-th species and Aij represents the effect that species j has on species i 36
Example Sample Models Biology Wolf Sheep Predation 37