Nonlinear Dynamics Nonlinear Dynamics

Nonlinear Dynamics Nonlinear Dynamics

Moreno Marzolla

Dip. di Informatica—Scienza e Ingegneria (DISI) Università di Bologna

http://www.moreno.marzolla.name/

Complex Systems 2

Introduction: Dynamics of Simple Maps

Complex Systems 4

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)

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)

Complex Systems 6

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

Example

Complex Systems 8

Logistic map

● 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

we will not use this: we adopt the standard formulation at the top of this slide, as commonly used

x

= r x

( 1− x

) , r ∈[0, 4] , x

∈[ 0,1]

x

= 4 r x

( 1−x

)

Logistic map

● The equation is fully deterministic: apparently, nothing surprising can happen there

x

= f ( x

)= r x

( 1− x

)

0 1 / 2 1 x

y r / 4

Complex Systems 10

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 x_n+1 = f(x_n) = rx_n( 1 – x_n )

● What are the fixed points of f?

– Those values x for which f(x) = x

Logistic map

● There are two fixed points: x = 0 and x = (r - 1) / r

– The second one is valid only if r ≥ 1

x

= f ( x

)= r x

( 1− x

)

0 1 / 2 1 x

y r / 4

y = x

(r-1) / r

Complex Systems 12

Iterates of the logistic map (r = 2.8)

Iterates of the logistic map (r = 3.2)

Complex Systems 14

Iterates of the logistic map (r = 3.52)

Iterates of the logistic map (r = 4)

Complex Systems 16

Classification of fixed points

● Let x_F be a fixed point for function f

– If f '(x_F) = 0 super-stable

– If f '(x_F) < 1 attracting and stable

– If f '(x_F) = 1 neutral

– If f '(x_F) > 1 repelling and unstable

The logistic map

● r ≤ 1

– Iterations eventually converge to the fixed point 0 (stable)

● 1 < r ≤ 3

– Unstable fixed point 0

– Stable fixed point (1 - r) / r

● r > 3

– Chaotic behavior, period-doubling bifurcations

Complex Systems 18

For those mathematically inclined

Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Westview Press, 2003, ISBN 978-0813340852

Complex Systems 20

Higher Dimensions

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 )

Complex Systems 22

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

Baker's map

Complex Systems 24

Baker's map

Iterating the baker's map

Complex Systems 26

Invariant image

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 (x_t, y_t) into a new pair of points (x_t+1, y_t+1), as follows:

where a, b are two constants

● Again, this is a fully deterministic map

x

= a− x

+ b y

y

= x

Complex Systems 28

The Hénon map for a=1.29, b=0.3

t x_t

(Only x_t is shown, random initial values)

The Hénon map for a=1.29, b=0.3

t x_t

(Only x_t is shown, random initial values)

Complex Systems 30

The Hénon map for a=1.29, b=0.3

t x_t

(Only x_t is shown, random initial values)

x = 0.838486... is an unstable attractor

The Hénon attractor viewed at different scales

(plots of y

versus x

)

Complex Systems 32

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

Producer-Consumer Dynamics

Complex Systems 34

Predator-Prey Model

Volterra and Lotka

● 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

– If c is large, cSF will be large, meaning that the shark population increases

● d = death rate of sharks

dF

dt = F (a−bS ) dS

dt = S (cF −d )

Population Dynamics

Fixed point at F = d/c, S = a/b

Complex Systems 36

Generalization

● The Volterra-Lotka model can be extended to an arbitrary number n of species

where x_i represents the i-th species and A_ij represents the effect that species j has on species i

dx

dt = x

∑

j=1 n

A

( 1−x

)

Example

Sample Models → Biology → Wolf Sheep Predation