1. Introduction to chaotic dynamics
PHYS Dynamics, Chaos, and Special Relativity
Guido Pupillo
uibk.ac.at
UQ, Brisbane, 17/09/2008

2. Simple pendulum
Oscillate
Rotate

3. Simple pendulum
Oscillate
Rotate
Double pendulum

4. See: Wheatland at Univ. of Sidney

5. Sensitivity to initial conditions
“Characteristic time”
Sensitivity to initial conditions

6. Sensitivity to initial conditions
“Characteristic time”
Sensitivity to initial conditions
Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions

7. Introduction to chaotic dynamics
Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions
The simplest example: One-dimensional non-linear maps
- the logistic map
Phenomenology:
- Initial conditions, fixed points and linear stability
- Bifurcation analysis, period doubling
- Bifurcation diagrams
- Chaos
Conclusions
Analysis:
Lyapunov exponents
Stretching and folding
Add something else in the “Analysis” section?

8. Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions
Strogatz, Steven H. Nonlinear Dynamics and Chaos. Cambridge MA: Perseus, 1994.
Let V be a set. The mapping f: V → V is said to be chaotic on V if:     1. f has sensitive dependence on initial conditions,     2. f is topologically transitive     3. periodic points are dense in V.
MAYBE SUBSTITUTE THIS WITH AN EXAMPLE!!! DOUBLE PENDULUM TO SHOW THE SENSITIVITY?

9. One-dimensional maps One-dimensional maps, definition:
- a set V (e.g. real numbers between 0 and 1)
- a map of the kind f:VV
Linear maps:
- a and b are constants
- linear maps are invertible with no ambiguity  no chaos
1) Why linear maps are not good for chaos? Check EXACTLY what goes wrong
2) Add a formal definition of a map: given a set V={x_n}, a map is f:VV
Non-linear maps: The logistic map

10. One-dimensional maps Non-linear maps: The logistic map with
Discretization of the logistic equation for the dynamics
of a biological population x
Motivation:
b: birth rate (assumed constant)
cx: death rate depends on population (competition for food, …)
How do we explore the logistic map?
1) STRTCHING AND FOLDING??!!!!

11. Geometric representation1
Evolution of a map:
1) Choose initial conditions
2) Proceed vertically until you hit f(x)
3) Proceed horizontally until you hit y=x
4) Repeat 2)
5) Repeat 3) . : x
f(x)
0.5 1
Evolution of the logistic map
fixed point ?
1) In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation… WHY??!!!

12. Phenomenology of the logistic mapb)
y=x
f(x) 1
0.5
y=x
f(x) 1
0.5
fixed point
fixed point c) 1
0.5 1
0.5 d)
chaos?
2-cycle?
In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation… WHY??!!!
Recall the DEFINITION OF A FIXED POINT, from Pag.7 of the notes (“special point where the velocity field is zero”. The system is then in mechanical equilibrium)
What’s going on? Analyze first a)  b)
b)  c) , …

13. Geometrical representationx
f(x) 1
0.5 x
f(x) 1
0.5
Evolution of the logistic map
fixed point
In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation… WHY??!!!
Recall the DEFINITION OF A FIXED POINT, from Pag.7 of the notes (“special point where the velocity field is zero”. The system is then in mechanical equilibrium)
How do we analyze the existence/stability of a fixed point?

14. Fixed points Stability Logistic map? - Condition for existence:
- Notice: since
the second fixed point exists only for
Stability
- Define the distance of from the fixed point
- Consider a neighborhood of
- The requirement implies
Logistic map?
Taylor expansion
1) In the motivation (Pag.40 of notes) it is written that if species have a definite reproductive season, then one has a discrete map, and not a differential equation… WHY??!!!

15. Stability and the Logistic Map
- Stability condition:
- First fixed point: stable (attractor) for
- Second fixed point: stable (attractor) for
No coexistence of 2 stable fixed points for these parameters
(transcritical biforcation) x
f(x) 1
0.5 x
f(x) 1
0.5
What about ?

16. Evolution of the logistic map
Period doubling 1
Observations: x
1) The map oscillates
between two values of x
f(x)
0.5 1
0.5 1
Evolution of the logistic map
Check WHAT IS THE EXACT DEFINITION OF PITCHFORK BIFURCATION??!!! FROM egwald-online. MAKE A LIST OF POSSIBLE BIFURCATIONS?!
Add that since x_{n+2}=x_n it is natural to regards these fixed points as stable fixed points (attractors) for f(f(x)).
2) Period doubling:
What is it happening?

17. Why do these points appear?
Period doubling
- At the fixed point becomes unstable,
since
Observation: an attracting 2-cycle starts 
(flip)-bifurcation
The points are found solving the equations 1
0.5
>
and thus:
Why do these points appear?
These points form a 2-cycle for
However, the relation suggests
they are fixed points for the iterated map
Stability analysis for :
and thus:
For , loss of stability and bifurcation to a 4-cycle
Now, graphically..
Rifare the stability analysis as in pag.22 of Rasband
EXPLAIN SOMEWHERE HOW TO GET THE FEILGELBAUM NUMBER, \mu_{\infty}, AND SHOW THE TABLE WITH THE BIFURCATION NUMBERS AS IN Rasband PAG.23

18. Bifurcation diagram Plot of fixed points vs
Check WHAT IS THE EXACT DEFINITION OF PITCHFORK BIFURCATION??!!! FROM egwald-online. MAKE A LIST OF POSSIBLE BIFURCATIONS?!
_DEFINE_ PITCHFORK BIFURCATIONS!! AS IN PAG.31 OF Rasband 3) SHOW THAT THE LYAPUNOV EXPONENT SHOWS CHAOS..

19. Bifurcation diagram Plot of fixed points vs What is general?
Observations:
Infinite series of period doublings
at pitchfork-like (flip) bifurcations
After a point
chaos seems to appear
3) Regions where stable periodic
cycles exist occur for
What is general?
Check WHAT IS THE EXACT DEFINITION OF PITCHFORK BIFURCATION??!!! FROM egwald-online. MAKE A LIST OF POSSIBLE BIFURCATIONS?!
_DEFINE_ PITCHFORK BIFURCATIONS!! AS IN PAG.31 OF Rasband 3) SHOW THAT THE LYAPUNOV EXPONENT SHOWS CHAOS..

20. Bifurcation diagram How do we characterize/quantify chaos?
General points:
Period doubling is a quite general route to chaos (other possibilities, e.g. intermittency)
Period doublings exhibit universal properties, e.g. they are characterized by certain numbers that do not depend on the nature of the map. For example, the ratio of the spacings between consecutive values of at the bifurcation points approaches the universal “Feigenbaum” constant. The latter occurs for all maps that have a quadratic maximum
Thus, we can predict where the cascade of period doublings ends, and something else starts
The something else looks chaotic, however, can we quantify how chaotic really is?
How do we characterize/quantify chaos?
Chaos: rapid divergence of nearby points in phase space
Measure of divergence: Lyapunov exponent

21. Lyapunov exponent One dimensional systems
One-dimensional system with initial conditions
and
with
After n iterations, their divergency is approximately
- If there is convergence  no chaos
- If there is divergence  chaos
One dimensional systems
After n steps
Logistic map
Thus:
Prove the “CHAIN RULE”
(chain rule)

22. Stretching and folding
Beginning of the lecture: “Chaos: is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions ”
However, in general it is necessary to have a mechanism to keep chaotic trajectories within a finite volume of phase-space, despite the expoential divergence of neighboring states
1/2 1
“stretching” (divergence) for (0,1/2)
“folding” (confinement) for (0,1/2)
- “stretching+folding” is responsible for loss of information on initial conditions as the iteration number (time) increases
- for 1D maps, non-linearity makes “time”-inversion ambiguous  loss of information
Prove the “CHAIN RULE”
1/2 1

23. Conclusions Chaos - the logistic map Phenomenology:
- Initial conditions, fixed points and linear stability
- Bifurcation analysis, period doubling
- Bifurcation diagrams
- Chaos
Prove the “CHAIN RULE”
Analysis:
Lyapunov exponents
Stretching and folding
Conclusions