An Orderly Approach to Chaos

1. An Orderly Approach to Chaos Presented by Trevor Nash and Peter Renn

2. Outline: • Chaos theory is a tremendously interesting but often misunderstood phenomena which manifests itself in myriad real world systems. • What are the requirements of chaos? • Examples illustrating each requirement • Logistic map • Example of fully chaotic system of ODEs: • Driven Damped Pendulum • Bifurcations of Chaos • Conclusion

3. What is Chaos? Qualitative Description • “Chaos: When the present determines the future, but the approximate present does not approximately determine the future” • Systems that are completely deterministic but which exhibit behavior which appears unpredictable • A dynamical system is chaotic if its long term behavior cannot be predicted given initial conditions with some degree degree of uncertainty • At some point your model will breakdown

4. What is Chaos? Mathematical Description • Most non-technical accounts of chaos focus solely on sensitivity to initial conditions • The true definition is much more nuanced and interesting • Chaotic systems generally all satisfy these three conditions: • Sensitivity to initial conditions • Topological mixing • Dense periodic orbits

5. Sensitivity to initial conditions • Small changes in initial conditions lead to large changes in future values • Given a point in phase space and its associated trajectory there exists another point arbitrarily close whose associated trajectory is significantly different • Characterised quantitatively by the Lyapunov Exponent:

6. Topological mixing • A mapping is considered to be topologically mixing if for every pair of non-empty open sets there exists an integer n such that: • What does this mean? • Given an initial condition with associated trajectory and an arbitrary point there exists a point in time at which the trajectory is arbitrarily close to the arbitrary point • In other words, if you give me a solution curve and a random point in phase space, I can give you a time value at which the solution curve is as close as you want to the random point

7. Dense periodic orbits • A subset A of a topological space X is said to be dense in X if every point in X either belongs to A or is a limit point of A • Basically, if every point in X is either in A or arbitrarily close to a point in A then A is dense in X • In the context of a chaotic dynamical system, X is the phase space and A is the set of points which lie on closed or periodic orbits • Thus, given a point in the phase space of a chaotic system there exists a periodic orbit which is arbitrarily close • The points of a chaotic system are dense around periodic orbits

8. Why the non-technical description is wrong • Sensitivity to IC’s alone does not lead to chaos • Consider the recurrence relation: • This system is sensitive to IC’s because slightly different starting values quickly diverge • Yet this system is clearly by no means chaotic • All solutions tend to either negative or positive infinity

9. Graphs trajectories with IC’s of 1 and 1.1 Various trajectories

10. Chaotic discrete dynamical systems • Differential equations are not the only systems which can exhibit chaotic behavior • Discrete dynamical systems or difference equations can be chaotic as well • Discrete analogs to differential equations which express the value of a variable after one time step in terms of its current value • Useful for modeling certain real world phenomena where time is not thought of as continuous • Ex. population dynamics where most births or deaths happen at one point in time

11. Logistic Map • Consider the difference equation: • This is a discrete analog of the logistic population model • takes on values between 0 and 1 representing the ratio between the existing population and the largest possible population • Looking at the time series plot this equation appears to be chaotic • Does it meet the three conditions for chaotic behavior?

12. Sensitivity to initial conditions • Examining a time series plot with trajectories for two slightly different seeds shows that the trajectories quickly diverge • Thus, the system is sensitive to initial conditions

13. Topological mixing • We must show that a given trajectory becomes arbitrarily close to any point in the phase space • Essentially, the solution must visit every region of space • This can be assessed by using a histogram • The phase space is partitioned into subintervals • The height of each bar reflects how many times the trajectory enters that sub interval • Looking at the histogram for the logistic map it is clear that every sub interval is visited many times • This suggests that it is topologically mixing

14. Dense periodic orbits • This system has an orbit of period 2: • By Sharkovski’s theorem it must have periodic points of all other periods as well • Thus, any point in the phase space will be arbitrarily close to a periodic point • This suggests that the system has dense periodic orbits

15. Pendulum Pandemonium: • Chaotic system relating to ODEs - Driven Damped Pendulum • What is special about a driven damped pendulum? • Several Forces • Driven • F(t) = F0cos(ωt) • Damped • Air Resistance - bv • Gravity • Gravitational Force -mg • One of the simplest physical systems where chaos occurs Φ F(t) bv mg

16. Equation of Motion • ⍵0 is the natural frequency: • β is the damping constant: • γ is the drive strength:

17. Solutions to the DDP • Traditionally, an attempt to linearize the system would be made • In doing so, the chaotic nature is lost • Chaos was “hidden” by linearization • Solving a nonlinear ordinary differential equation is not easy • Impossible to solve analytically • Numerical solutions can be found with Mathematica • Not chaotic for all parameter values • Controlling parameter • Bifurcation points

18. Bifurcations of Chaos • Remember ɣis the drive strength • Ratio of the driving force to the force of gravity • Why is this important? • Drive strength parameter affects chaotic nature of solutions: Graphs for γ = 0.2,Δɸ(0) = 1 y(t) y(t) t t Parameters: ω = 2π; ω0 = 1.5ω = 3π; β = 0.25ω0 = 0.75π. Note: The period here is equal to one.

19. Larger Drive Strength • Increasing the drive strength, and decreasing the difference in initial conditions y(t) y(t) t t Graphs for γ= 1, Δɸ(0) = 0.0001

20. Larger Drive Strength • Not chaotic, yet y(t) y(t) t t Graphs for γ = 1, Δɸ(0) = 0.0001

21. Critical Drive Strength • Increasing the drive strength further, the critical value is reached • The system is now chaotic y(t) y(t) t t Graphs for γ= 1.084, Δɸ(0) = 0.0001

22. Critical Drive Strength • As time goes on, the difference between the solutions grows y(t) y(t) t t Graphs for γ = 1.084, Δɸ(0) = 0.0001

23. Increasing the Drive Strength Further • At an even larger drive strength, it appears as though the solutions are still diverging y(t) y(t) t t Graphs for γ = 1.13, Δɸ(0) = 0.0001

24. Increasing the Drive Strength Further • As time increases, similar shapes appear y(t) y(t) t t Graphs for γ = 1.13, Δɸ(0) = 0.0001

25. Increasing the Drive Strength Further • These solutions are not chaotic, but rather periodic. It may take time for transient behavior to die out y(t) y(t) t t Graphs for γ = 1.13, Δɸ(0) = 0.0001

26. Increasing the Drive Strength Further • These solutions are not chaotic, but rather periodic y(t) y(t) t t Graphs for γ = 1.13, Δɸ(0) = 0.0001

27. What is happening? • The system has bifurcation points where the system changes its behavior • In this case, the parameter causing this change is the drive strength • This means that the system is not always chaotic above the critical drive strength • There are spans of parameter values which are mostly chaotic, and spans which are not • Need to find a way to illustrate how varying a parameter affects the behavior of a function

28. Bifurcation Diagrams for Chaotic Systems • Period Doubling Cascade • As the drive strength increases, every other maximum value decreases • Maximum value reached half as frequently, effectively doubling the period • This period doubling continues as drive strength increases • Successive bifurcation points can be found using the Feigenbaum number, δ: Where, δ = 4.6692016

29. Bifurcation Diagrams for Chaotic Systems • Period Doubling Cascade • As the drive strength increases, every other maximum value decreases • Maximum value reached half as frequently, effectively doubling the period • This period doubling continues as drive strength increases • Successive bifurcation points can be found using the Feigenbaum number, δ: Where, δ = 4.6692016

30. Critical Value • The critical value is where the system becomes chaotic • Can be ‘calculated’ using the limit of the Feigenbaum number equation: Where the critical value is found as the limit: • Adjacent bifurcation points are times closer than the previous set of bifurcation points. This means the distance goes to zero in the limit as n goes to infinity

31. Critical Value • This method makes sense, since each additional n value doubles the period • As n goes to infinity, so does the period • The limit suggests that the function becomes chaotic when the period goes to infinity • The period-doubling cascade is not present in every chaotic system, however it is a common “route to chaos”

32. Conclusion • While chaos is often characterised as sensitivity to IC’s its true nature is much more nuanced and interesting • Even deterministic systems are guaranteed to become unpredictable • Chaos can manifest itself into seemingly simple physical systems • The parameters of a system can dictate whether or not it’s chaotic

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