An Orderly Approach to Chaos Presented by Trevor

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

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 2

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 of uncertainty At some point your model will breakdown 3

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 4

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: 5

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 6

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 7

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 8

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

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 10

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? 11

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 12

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 13

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 14

Pendulum Pandemonium: ● Chaotic system relating to ODEs - Driven Damped Pendulum ● What is special about a driven damped pendulum? ○ Several Forces ■ Driven ● F(t) = F 0 cos(ωt) ■ Damped ● Air Resistance - bv ■ Gravity ● Gravitational Force - mg ○ One of the simplest physical systems where chaos occurs Φ bv F(t) mg 15

Equation of Motion ● � 0 is the natural frequency: ● β is the damping constant: ● γ is the drive strength: 16

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 17

Bifurcations of Chaos ● Remember ɣ is the drive strength ○ Ratio of the driving force to the force of gravity y(t) ● Why is this important? y(t) Graphs for γ = 0. 2, Δɸ(0) = 1 ○ Drive strength parameter affects chaotic nature of solutions: t t Parameters: ω = 2π; ω0 = 1. 5ω = 3π; β = 0. 25ω0 = 0. 75π. Note: The period here is equal to one. 18

Larger Drive Strength ● Increasing the drive strength, and decreasing the difference in initial conditions y(t) t Graphs for γ = 1, Δɸ(0) = 0. 0001 t 19

Larger Drive Strength ● Not chaotic, yet y(t) t Graphs for γ = 1, Δɸ(0) = 0. 0001 t 20

Critical Drive Strength y(t) ● Increasing the drive strength further, the critical value is reached ● The system is now chaotic y(t) t t Graphs for γ = 1. 084, Δɸ(0) = 0. 0001 21

Critical Drive Strength ● As time goes on, the difference between the solutions grows y(t) t t Graphs for γ = 1. 084, Δɸ(0) = 0. 0001 22

Increasing the Drive Strength Further y(t) ● At an even larger drive strength, it appears as though the solutions are still diverging y(t) t Graphs for γ = 1. 13, t Δɸ(0) = 0. 0001 23

Increasing the Drive Strength Further ● As time increases, similar shapes appear y(t) t Graphs for γ = 1. 13, t Δɸ(0) = 0. 0001 24

Increasing the Drive Strength Further y(t) ● These solutions are not chaotic, but rather periodic. It may take time for transient behavior to die out y(t) t Graphs for γ = 1. 13, t Δɸ(0) = 0. 0001 25

Increasing the Drive Strength Further ● These solutions are not chaotic, but rather periodic y(t) t Graphs for γ = 1. 13, t Δɸ(0) = 0. 0001 26

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 27

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 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

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 30

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” 31

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 32

References Fordyce, Rachel Frost. "Chaotic Waterwheel. " Thesis. The Division of Mathematics and Natural Sciences Reed College, 2009. Print. Lipa, Chris. "Determinism and Chaos. " Introduction to Chaos. Cornell University, n. d. Web. 13 Nov. 2015. Taylor, John R. Classical Mechanics. Sausalito, CA: U Science, 2005. Print. Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, 2012. Print. Carroll, Mark C. "More about Dense Periodic Orbits. " Good Math Bad Math. N. p. , 26 Jan. 2010. Web. 13 Nov. 2015. Carroll, Mark C. "The End of Defining Chaos: Mixing It All Together. " Good Math Bad Math. N. p. , 7 Feb. 2010. Web. 13 Nov. 2015. Gleick, James. Chaos: Making a New Science. New York, NY: Penguin, 2008. 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, 2012. Web. 30 Nov. 2015. 33