Introduction to chaotic dynamics PHYS2100 - Dynamics, Chaos, and Special Relativity Guido Pupillo guido.pupillo @ uibk.ac.at UQ, Brisbane, 17/09/2008

Simple pendulum Oscillate Rotate

Simple pendulum Oscillate Rotate Double pendulum

See: Wheatland at Univ. of Sidney

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

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

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?

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

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??!!!!

Geometric representation 1 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??!!!

Phenomenology of the logistic map b) 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) , …

Geometrical representation x 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?

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??!!!

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 ?

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?

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

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..

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..

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

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)

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

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