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Louis Maddox edited this page Sep 10, 2016 · 8 revisions

Nonlinear Dynamics: Mathematical and Computational Approaches

1 - Introduction

  • a determ. sys. is nonrand, cause/effect linked and current state determines future
  • dynamic(al) evolves with time
  • nonlinear sys is one where variables that matter are not linear e.g. gas needle in car
  • nonlinear dyn and chaos not rare
    • of all systems in universe vast majority nonlinear
    • traditional training assumes linearity
  • eqns describing chaotic systems need comp. solution
  • distinguishes this course from others [incl. Strogatz's] - applied

Dynamical systems are about change with time: derivatives

  • classical mechanics [masses, springs, pulleys, springs]
  • Cassini video: flew by Saturn's moon Hyperion
    • Hyperion is a strange shape, so tumbles chaotically
    • chaos in how planets move thru space
  • solutions can only be conic sections: ellipsis, parabola, hyperbola

Conics (circles, ellipses, parabolas, and hyperbolas) involves a set of curves that are formed by intersecting a plane and a double right cone (probably too much information!). But in case you are interested, there are four curves that can be formed, and all are used in applications of math and science:

  • Systems with 3 or more bodies can be chaotic

    • solar system is chaotic, but stable in a sense

  • field dates back to Henri Poincaré in late 19C, but really got going in 1960s with Ed Lorentz's paper, Deterministic Nonperiodic Flow

    • first to notice patterns of chaos, and sensitivity of evolution of those patterns

  • in 70s, paper by Li and York was first to use word 'chaos' in conjunction with this behaviour (Period Three Implies Chaos)

  • in 70s and 80s the 'chaos cabal' at U Cal and Santa Cruz got very interested in NLD, approached with 'beating roulette' (modelling ball on roulette wheel), after which really took off

  • imagine dropping wood chip on eddy in creek

  • patches trace out swirl of water in eddy

  • weather is NL and chaotic, small changes ⇒ large effect (sensitive dependence on initial effects)

  • marine invert's use chaotic mixing, used in car fuel injection design etc

  • driven nonlinear oscillators show it e.g. pendulum, heart (normally periodic, ventricular fibrillation is chaotic state), fireflies

  • lots in classical mechanics, from 3 body problem to how black holes move around each other

Do not email course staff - many thousands of students !

2 - Maps and difference equations

  • Maps are systems that operate in discrete time
    • time proceeds in clicks
    • modelled with difference equation
  • Flows are continuous time systems
  • This distinction may be new to you: flow is e.g. pendulum, dynamic that operates continuously in time and space
    • imagine shining strobe light at pendulum
    • strobe is a map: time only exists in dynamical sys at discrete intervals
    • doesn't make sense to ask what state of sys is in between samples
    • monthly interval statistics e.g., or low fps video
  • study maps first: their dynamics are representative: a good example of what can happen in NLD systems, but the math is a lot easier
  • most NLD courses take this path for that reason
    • circle back around ideas with flows
  • the map is a mathematical operator that take current state and tells what next is
  • f is the map in x_n+1 = f(x_n)
  • iterates of map converge to a fixed point of the map in a cosine map
  • the logistic map: x_n+1 = Rx_n(1-x_n)
  • x = state, n = time, r = parameter
    • very simple population model (x is like 'the ratio of foxes to rabbits in back yard)
  • note passage thru 'transient' state to fixed point [convergence]
  • changing r value (e.g. hunger of foxes, birthrate of rabbits) understandably changes the end result (stabilisation of fixed point)

3 - Transients and attractors

  • recall: progression of iterates converging to asymptote

  • for different values of x_0 and parameter r

  • the notion of a progression of iterates, x_0, x_1 and so on is called an orbit or a trajectory of the dynamical system

  • an orbit or a trajectory is a sequence of values of the state variables of the system

    • a logisttic map has 1 state variable, x, other systems may have >1 state variable e.g. pendulum (position and velocity of both bobs of the pendulum in order to say what state it's in
    • the starting value of state variable is called initial condition
    • fixed point = asymptote

  • technically fixed point is a state of the system that doesn't move under the influence of the dynamics

    • i.e. the fixed point to which the logistic map orbit converges is an "attracting" fixed point

  • "states that don't move under influence of dynamics" as definition of fixed points captures both stable (attracting) and unstable (repelling) attractors

  • dynamical systems have several different kinds of asymptotic behaviours

  • subsets to which things converge as time goes to infinity

  • attracting fixed points are 1 kind of attractor, there are 3 other kinds

  • basin of attraction: metaphor for water flow over terrain

    • consider then meaning of: flood, water table
    • "what do you think will happen to a rain drop that falls exactly on that basin boundary?"

  • "overshoot" which grows as the value of r grows [while x_0 initial condition fixed]

    • fixed point moves, transient lengths differ (takes longer with greater r, i.e. convergence takes longer)
    • 'overshoot' gets more pronounced: orbit is still converging to a fixed point, but in oscillatory fashion cf. 1-sided
      • like if you push down on hood of car and it bounces before settling

4: Parameters and bifurcations

  • exploring what happens to attractors in dynamics of logistic map as parameter r changes

  • if you continue to raise r, in one of the cases which showed oscillatory behaviour in the transient section up to the convergence at a fixed point, you observe the oscillation's 2 sides appearing very tightly bunched, though close enough that they appear to be converging on a fixed point still

  • raising r yet further, there is a clear splitting of these two lines, to what appear to be separate and almost parallel lines: convergence seems to have disappeared

  • there are several types of attractors, and this is another one

  • a periodic orbit, or a limit cycle (synonyms)

  • the period is 2, an attractive periodic orbit

  • note that a map is a discrete time system, so lines between the points are not appropriate

    • time makes no sense between iterates. can't think about what it might mean to be x of 0.5: it does not exist

  • raising r further: 2-cycle no longer exists, converged to a 4-cycle

    • also a limit cycle/periodic orbit (period 4)

  • changes, from fixed point to 2-cycle and 2-cycle to 4-cycle, with changes to r parameter are called bifurcations

  • english definition suggests "forking in 2" but mathematics use is more general: means that there's been a change in the topology of the attractor

  • a bifurcation is a qualitative change in the attractor

    • not just that the fixed point moves, but that it vanishes, or that the period of a periodic orbit changes
    • r is a logistic parameter in the bifurcation map: affects dynamics in a fundamental way, causing those qualitative changes

  • continuing to increase parameter, there's no seeming pattern in the values: no convergence, not to a fixed point not to the periodic orbit.

    • longer plot: no seeming orbit
    • the patterns starting to see are a chaotic attractor (a.k.a. strange attractors)

  • varying the initial conditions, the trajectory always converged to the same attractor: it does that here [with a chaotic attractor] too, but it can be hard to see in a time-domain plot like this.

  • the points are tracing out the same subset of states but in a different order

(!!!)

  • ...
    • it's kind of like dropping a wood chip into an eddy. if you drop a wood chip in different parts the wood chip will trace out the same structure but in a different order
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