Discrete Dynamic Systems

What is a Dynamical System?

Definition of Dynamical System

A dynamical system is characterized by a ‘rule’ (or ‘rules’) that determines how points in the state space of the system change over time

The particular form of a rule depends on the type of dynamical system being studied, and the type of dynamical system depends on the space–time structure of the system.

Two Geometrical Cases of Dynamical Systems

Continuous

t is independent variable

Does not always have analytical solutions

Maps R → R

Discrete

n is independent variable

Uses a numerical method

Discretizes the system

Creates sequences through iterations

Maps R → N

Sequences, Bifurcations, and Period Doubling

Sequences in the Logistic Family

Example of Period Doubling with Logistic Equation

First two samples act ‘normally’, while the third oscillates between two values

R n x R n x R n x0.4 1 0.6 0.6 1 0.6 0.8 1 0.6

2 0.384 2 0.576 2 0.768

3 0.37847 30.58613

8 30.57016

3

40.37636

9 40.58219

3 40.78424

7

50.37554

5 50.58378

6 50.54145

2

60.37521

7 60.58315

2 60.79450

2

70.37508

7 70.58340

6 7 0.52246

80.37503

5 80.58330

4 80.79838

6

90.37501

4 90.58334

5 90.51509

1

100.37500

6 100.58332

9 100.79927

1

110.37500

2 110.58333

5 110.51339

8

120.37500

1 120.58333

3 120.79942

6

13 0.375 130.58333

4 130.51310

2

14 0.375 140.58333

3 140.79945

1

15 0.375 150.58333

3 150.51305

4

16 0.375 160.58333

3 160.79945

5

17 0.375 170.58333

3 170.51304

6

18 0.375 180.58333

3 180.79945

5

19 0.375 190.58333

3 190.51304

5

20 0.375 200.58333

3 200.79945

5

Period Doubling in the Logistic Family

Plots from IterateMapApp.java

Red – R = 0.4

Green – R = 0.6

Blue – R = 0.8

Note how blue seems to oscillate between two equilibria

Figure 1: IterateMapApp.java

Bifurcations in the Logistic Family

Bifurcation Diagram of

Figure 2: BifurcateApp.java

Chaos

What is Chaos?

“Chaos: When the present determines the future, but the approximate present does not approximately determine the future,” Edward Lorentz

Chaos occurs when there is no repeating sequence in later time

How Do We Quantify Chaos?

Lyapunov Exponents

Figure 3: BifurcateAppDDP3.java

Weakly, Strongly, and Non Chaotic

The continuous and discrete equations describe the same system. However… 1) In continuous we only need to worry about the roughness in the inputs/ initial conditions. But in discrete we also have to worry about the roughness in our time step.