Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Logistic map: Period- 1 orbits
Map: x n +1 = rx n (1 − x n ) M ′ (x) = r(1 − 2x)
x = 0 : γ 1 = |M ′ (0)| = r x = 1 − 1/r : γ 2 =
|M ′ (1 − 1/r)| = |2 − r|
For 0 ≤ 0 ≤ 1: x = 0 is a stable orbit with the basin of attraction [0, 1]
For 1 ≤ 0 ≤ 3: x = 1 − 1/r is a
stable orbit with the basin of
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Dynamics
Dynamics → time evolution (of something)
Dynamical system → clear-cut mathematical prescription for evolving vector X forward in time.
There are two types of dynamical systems:
Differential equations: time is continuous, dX
dt = F (X )
Maps: time is discrete X n +1 = F (X n ).
Linear dynamical systems: F (X ) = AX + B
Nonlinear dynamical systems: otherwise
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Dynamics: geometric way of thinking
X ˙ =
dx 1 /dt = F 1 (x 1 , x 2 , x 3 ),
dx 2 /dt = F 2 (x 1 , x 2 , x 3 ),
dx 3 /dt = F 3 (x 1 , x 2 , x 3 ).
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Dynamical chaos
Poincare (1880)
“It so happens that small differences in the initial state of the
system can lead to very large differences in its final state. A
small error in the former could then produce an enormous one
in the latter. Prediction becomes impossible, and the system
appears to behave randomly”
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Dynamical chaos: the essence
a dynamical system entirely determined by its initial conditions (i. e., its evolution is deterministic ),
yet the state of the system at time t cannot be predicted.
Deterministic evolution + sensitivity to initial conditions
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Dynamical chaos: an example
Example: Moon & Holmes (1979)
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Example I
Is it chaotic or not?
dx /dt = sin(x)
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Example II
Lorenz system (1963)
X ˙ =
dx/dt = σ(y − x), dy/dt = x(ρ − z) − y, dx 3 /dt = xy − βz .
The parameters are: σ > 0 (Prandtl number), ρ > 0 (Rayleigh
number), β > 0 (just some tunable coefficient).
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Example II
Lorenz system (1963)
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Example II
Lorenz system: what about sensitivity to initial conditions?
The difference between two initial points is δx = 10 − 5
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Example II
Lorenz system: what about sensitivity to initial conditions?
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
Example II
Lorenz system as the model of weather dynamics
Difficulties in predicting the weather are not related to the
complexity of the Earths’ climate but to chaotic dynamics in
the climate equations!
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
One-dimensional maps
x n+1 = f (x n )
They seem to be easy but they are not!
Dynamics of nonlinear &
chaotic systems Lecture 4:
Nonlinear maps &
periodic orbits) S. Denisov
