1. Discrete Dynamic Systems

2. What is a Dynamical System?

3. 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.

4. Two Geometrical Cases of Dynamical Systems

5. Sequences, Bifurcations, and Period Doubling

6. Sequences in the Logistic Family RnxRnxRnx 0.410.6 1 0.810.6 20.38420.57620.768 30.3784730.58613830.570163 40.37636940.58219340.784247 50.37554550.58378650.541452 60.37521760.58315260.794502 70.37508770.58340670.52246 80.37503580.58330480.798386 90.37501490.58334590.515091 100.375006100.583329100.799271 110.375002110.583335110.513398 120.375001120.583333120.799426 130.375130.583334130.513102 140.375140.583333140.799451 150.375150.583333150.513054 160.375160.583333160.799455 170.375170.583333170.513046 180.375180.583333180.799455 190.375190.583333190.513045 200.375200.583333200.799455

7. 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

8. Bifurcations in the Logistic Family Figure 2: BifurcateApp.java

9. Chaos

10. 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

11. How Do We Quantify Chaos?  Lyapunov Exponents Figure 3: BifurcateAppDDP3.java

12. 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.