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Chapter 1 A global overview of nonlinear dynamical systems and chaos

I.1 Definitions Dynamical system –A system of one or more variables which evolve in time according to a given rule –Two types of dynamical systems: Differential equations: time is continuous Difference equations (iterated maps): time is discrete

I.1 Definitions Linear vs. nonlinear –A linear dynamical system is one in which the rule governing the time-evolution of the system involves a linear combination of all the variables. EXAMPLE: –A nonlinear dynamical system is simply… …not linear

I.1 Definitions Chaos: Poincaré: (1880) “ It so happens that small differences in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.”

I.1 Definitions Chaos: Poincaré: (1880) “ It so happens that small differences in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.”

I.1 Definitions THE ESSENCE OF CHAOS! a dynamical system entirely determined by its initial conditions (deterministic) but which evolution cannot be predicted in the long-term

I.2 Examples of chaotic systems the solar system (Poincare) the weather (Lorenz) turbulence in fluids (Libchaber) solar activity (Parker) population growth (May) lots and lots of other systems…

I.2 Examples of chaotic systems the solar system (Poincare)

I.2 Examples of chaotic systems the solar system (Poincare)

I.2 Examples of chaotic systems the weather (Lorenz) Difficulties in predicting the weather are not related to the complexity of the Earths’ climate but to CHAOS in the climate equations!

I.2 Examples of chaotic systems the weather (Lorenz)

I.2 Examples of chaotic systems turbulence in fluids

I.2 Examples of chaotic systems turbulence in fluids (Libchaber) convection in liquid helium.

I.2 Examples of chaotic systems turbulence in fluids (similar experiment)

I.2 Examples of chaotic systems turbulence in fluids (similar experiment)

I.2 Examples of chaotic systems turbulence in fluids - period doubling to chaos

I.2 Examples of chaotic systems solar activity

I.2 Examples of chaotic systems solar activity

I.2 Examples of chaotic systems solar activity

I.2 Examples of chaotic systems solar activity

I.2 Examples of chaotic systems solar activity

I.2 Examples of chaotic systems solar activity

I.2 Examples of chaotic systems population growth (May) –discrete, apparently simple dynamical systems can exhibit a rich array of behaviour –examples in nature are ubiquitous (population dynamics, epidemic propagation, …) –discretization corresponds to breeding cycle, seasonal recurrence, … –EXAMPLE: the logistic equation

Mathematical detour: cobweb diagrams for 1D maps

I.2 Examples of chaotic systems Logistic equation r = 2 r = 3.3

I.2 Examples of chaotic systems Logistic equation r = 3.5r = 3.9

I.2 Examples of chaotic systems Logistic equation

I.3 Fractals Fractals appear everywhere in the study of dynamical systems Characteristics: –a fractal is a geometric object which can be divided into parts, each of which is similar to the original object. –they possess infinite detail, and are generally self- similar (independent of scale) –they can often be constructed by repeating a simple process (a map, for instance) ad infinitum. –they have a non-integer dimension!

I.3 Fractals A section through the Lorenz dynamical system yields a fractal somewhat like the Cantor set an infinity of points, yet none are connected!

I.3 Fractals Bifurcation diagrams are also fractal

I.3 Fractals Bifurcation diagrams are also fractal

I.3 Fractals Also appear in nature.. What is the length of a coastline? (Mandelbrot)

I.3 Fractals Fractals naturally appear in iterated maps. –EXAMPLE: the Koch curve

I.3 Fractals Appear to be the fate of most 2D iterated maps –Example: The Mandelbrot set is built from the quadratic map where z n and c are complex numbers.

I.3 Fractals