Dynamical Chaos Igor Volobouev Computational Physics (PHYS 4301/5322) Texas Tech University

A Long-Held Assumption Left alone, the motion of a conventional classical (i.e., deterministic) bounded dynamic system will converge to either Steady state (a point in the phase space) Periodic motion (a closed loop in the phase space called a limit cycle) Quasi-periodic state (an n-dimensional torus in the phase space)

Is this statement in agreement with your intuition?

Poincaré’s Homoclinic Tangle In late 1880s, Henri Poincaré studied the dynamical stability of the solar system He simplified the problem down to a three body system in which the mass of the third body (e.g., the Earth) was much smaller than the mass of the other two (e.g., the Sun and the Jupiter), treating the effect of the third body presence as a perturbation What he found was that the motion of the third body itself was unstable – arbitrarily small changes in its position and velocity could lead to arbitrarily large changes in the motion Poincaré understood that he discovered a fundamentally new type of dynamical motion. However, the numerical tools of the time did not allow him to explore this discovery in detail.

The Butterfly Effect Another system with unpredictable properties was discovered by MIT meteorologist Edward Lorenz in early 1960s. He performed simplified atmospheric model calculations on a computer (LGP-30) with whopping 16 KB of memory running at 120 kHz clock rate. Lorenz’s paper “Deterministic nonperiodic flow”, published in 1963 in the Journal of Atmospheric Sciences, started the modern field of chaos research. In 1972, Lorenz gave a talk at the 139th meeting of the AAAS entitled “Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?” The connection with Poincaré’s findings did not become apparent until later

The Lorenz Model A highly simplified model of convection (a.k.a. Rayleigh-Bénard problem) x represents the intensity of the convective motion y is proportional to the temperature difference between the ascending and descending convective currents z describes the deviation of the vertical temperature profile from linearity Parameter r is the Rayleigh number (roughly, a measure of the temperature difference between the top and the bottom of the fluid) Parameters σ and b are related to fluid properties (such as viscosity)

dx = σ ( y − x) dt dy = −x z + r x − y dt dz = x y −b z dt

The Lorenz Attractor Attractor is a set of points in phase space to which the solution of an equation evolves long after transients have died out

From the May 2013 article by A.E. Motter and D.K. Campbell in Physics Today

Lorenz Model Simulation Go to the course web page: http://highenergy.phys.ttu.edu/~igv/Computational Physics/Lectures/Lecture5Examples/

Chaotic Attractors Phase space trajectories in chaotic attractors (a.k.a. strange attractors) never retrace themselves (otherwise the motion would be periodic) The sets of points occupied by orbits in chaotic attractors form fractals To possess a chaotic attractor, a dynamical system must be nonlinear  Chaotic solutions often occur in systems with dissipation and influx of energy from outside. These conditions, however, are not strictly necessary.

Fractal Dimension There is no unique way to define the dimensionality of a fractal Box-counting dimension: divide the attractor into spheres/boxes of radius ε and ask how many boxes do we need to cover all the points in the data set. If we evaluate this number N(ε) as a function of ε as it becomes small, then we define the box-counting ln(𝑁𝑁(ε)) dimension by 𝑑𝑑 = lim ε→0 ln(1/ε)

Sierpinski triangle: d = ln(3)/ln(2) ≈ 1.585

The Koch Curve Fractal dimension is defined from Leff = NsLs ~ L1-d d = ln(4)/ln(3) ≈ 1.262 Self-similarity (exact or statistical) is typical in fractal structures

Lyapunov Exponent The difference between two dynamical trajectories with slightly different initial conditions often exhibits exponential dependence on time: ∆𝜃𝜃(𝑡𝑡) ≤ 𝑒𝑒 λ𝑡𝑡 ∆𝜃𝜃(0) (starting with infinitesimally small ∆𝜃𝜃(0) ). The parameter λ is called “Lyapunov exponent”. Classical chaotic systems have at least one positive Lyapunov exponent. In such cases the system evolution is deterministic but the state of the system at times 𝑡𝑡 ≫ 1/λ is unpredictable due to extreme sensitivity to initial conditions. Note the difference between chaotic and stochastic systems (e.g., a canonical ensemble in statistical mechanics). Apart from initial conditions, there is no inherent source of randomness in chaotic systems.

The Shadowing Lemma As small perturbations are exponentially amplified, do numerical simulations, with their unavoidable round-off errors, make sense? The answer depends on the purpose of the study.  Simulations of individual trajectories work only within a limited “prediction time” on the order of 1/λ (this is how Lorenz discovered that his simulations are chaotic).  Simulations of attractors works due to the “shadowing lemma”: exact mathematical solutions “shadow” noisy solutions (stay infinitely close to them) because exact orbits are “dense” on the attractor. Thus a noisy solution looks, for some length of time Δt, like a real solution started from a different point (and like yet another real solution for some other Δt, etc).

Identifying Chaotic Behavior The time-dependent attractor shape in the phase space could be complicated and not easy to visualize, especially for highdimensional dynamical systems. The analysis of the system motion can be aided by Poincaré sections Bifurcation diagrams Spectral diagrams

Poincaré Sections Invented by Henri Poincaré as a means of simplifying phase space diagrams of complicated systems Useful for dynamic systems for which one can identify a natural frequency (driven pendulum, planet orbiting around a star, etc) Poincaré section is simply a collection of attractor slices at times that differ by the period corresponding to the natural frequency (a.k.a. stroboscopic phase space diagram). These slices are visualized using 2-d scatter plots. One can explore all possible pairs of dynamic variables.

Bifurcation Diagrams Bifurcation is a sudden change of system properties in response to a small change in the system parameter  A simple example: the number of real solutions of a quadratic equation as a function of the constant term

Bifurcation diagrams relevant to the studies of chaotic systems are displays of possible values of one of the dynamic variables in a Poincaré section as a function of some system parameter

Spectral Diagrams The Fourier power spectra of chaotic systems exhibit characteristic complex structure both above and below the natural frequency Extracting information about attractors from chaotic time series in not trivial. Google “chaotic time series analysis” for references.

The Logistic Map Perhaps, the simplest system that exhibits rich chaotic behavior xn +1 = µ xn (1 − xn ) One iteration of the map corresponds to one “natural” period in a dynamic system Can be interpreted as a simplified model of population growth. The natural period is the time between consecutive generations. Verhulst continuous model:

𝑑𝑑𝑁𝑁 𝑑𝑑𝑡𝑡

= 𝑟𝑟𝑟𝑟 1 −

𝑁𝑁 𝐾𝐾

Logistic Map Evolution

From the October 2013 American Journal of Physics article by J. Groff • A: Steady-state fixed-point solution • B: Period-4 oscillations • C: Chaos

Bifurcation Diagram for the Logistic Map

Source: Wikipedia

μ

Why Period Doubles? At a steady-state point, we must have • 𝑥𝑥 = 𝑓𝑓 𝑥𝑥 = µ𝑥𝑥(1 − 𝑥𝑥) (fixed point) •

𝑑𝑑 𝑓𝑓 𝑑𝑑𝑑𝑑

𝑥𝑥