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 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 mg bv
F(t)
b is the drag coefficient and v is the velocity

16. Equation of Motion ⍵0 is the natural frequency:
β is the damping constant:
γ is the drive strength:
b is the drag coefficient. 2nd Order nonlinear ODE. Nonlinear because of sine phi

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
fix

18. Bifurcations of Chaos Remember ɣ is 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
Graphs for change in initial position equal to one line up as time goes on.
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
You can increase the driving strength and still not reach a chaotic system
Graphs for γ = 1, Δɸ(0) =

20. Larger Drive Strength Not chaotic, yet
y(t)
y(t) t t
Larger time frame that becomes clear
Graphs for γ = 1, Δɸ(0) =

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
At this drive strength, Within a 10 second interval, the two solutions already appear to diverge
Graphs for γ = 1.084, Δɸ(0) =

22. Critical Drive Strength
As time goes on, the difference between the solutions grows
y(t)
y(t) t t
As time goes on more it diverges even more, showing the effect of the liapanov exponent. Note: the liaponov exponent is an approximate relation and allows for the difference in solutions to oscillate
Graphs for γ = 1.084, Δɸ(0) =

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
It appears initially that this is just going to be another chaotic set of solutions. Seems to already diverge within 10 periods
Graphs for γ = 1.13, Δɸ(0) =

24. Increasing the Drive Strength Further
As time increases, similar shapes appear
y(t)
y(t) t t
But as time increases a pattern emerges
Graphs for γ = 1.13, Δɸ(0) =

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
This pattern continues. Transient Behaviors may stick around
Graphs for γ = 1.13, Δɸ(0) =

26. Increasing the Drive Strength Further
These solutions are not chaotic, but rather periodic
y(t)
y(t) t t
Slightly different attractors accounts for the height difference. Important to note that this shows how the critical drive strength is not the minimum drive strength for chaotic behavior in the sense that not all drive strengths above the minimum display chaotic behavior
Graphs for γ = 1.13, Δɸ(0) =

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
Now I’m going to talk a little bit about what is happening here. Not all parameter values create systems meeting the requirements Trevor discussed.

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:
Not your typical bifurcation diagram. The points don’t represent sources/sinks like the bifurcation diagrams we are familiar with. Rather, each point is representative of a fixed height. Single point represents pre-bifurcation point 1
Where, δ =

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:
Period doubling cascade is occurring by changing the number of unique maxima and minima required for each period to be acheived. Happens naturally. The points it occurs are bifurcation points
Where, δ =

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
By now you might be asking what do these bifurcation values have to do with chaos? Well the critical value is just a bifurcation value when n is at 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”
The graph on the left is first period. The graph on the

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

33. References
Fordyce, Rachel Frost. "Chaotic Waterwheel." Thesis. The Division of Mathematics and Natural Sciences Reed College, Print.
Lipa, Chris. "Determinism and Chaos." Introduction to Chaos. Cornell University, n.d. Web. 13 Nov
Taylor, John R. Classical Mechanics. Sausalito, CA: U Science, Print.
Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, Print.
Carroll, Mark C. "More about Dense Periodic Orbits." Good Math Bad Math. N.p., 26 Jan Web. 13 Nov
Carroll, Mark C. "The End of Defining Chaos: Mixing It All Together." Good Math Bad Math. N.p., 7 Feb Web. 13 Nov
Gleick, James. Chaos: Making a New Science. New York, NY: Penguin, Print.
Abbasi, Nassar M. “Chaotic Motion of a Damped Driven Pendulum: Bifurcation, Poincare Map, Power Spectrum, and Phase Portrait”, Wolfram,
n.d. Web. 12 Oct. 2011
Luque, Bartolo. "Analytical Properties of Horizontal Visibility Graphs in the Feigenbaum Scenario - INSPIRE-HEP." Analytical Properties of
Horizontal Visibility Graphs in the Feigenbaum Scenario - INSPIRE-HEP. Inspire, Web. 30 Nov