If a system is unstable, like pins balanced on their points, then the orbits diverge exponentially for a while, but eventually settle down. You may get run-time errors when evaluating the logarithm if d1 becomes so small as to be indistinguishable from zero. Jacobian matrix for a map) and using the fact that one exponent
calculated from the trace of the Jacobian matrix averaged along
The logistic equation is unruly. Will the volume send forth connected pseudopodia and evolve like an amoeba, atomize like the liquid ejected from a perfume bottle, or foam up like a piece of Swiss cheese and grow ever more porous? Have you found the errors in this book yet? However, the evolved volume will equal the original volume. The exponent provides a means of ascertaining whether the behavior of a system is chaotic. be changed slightly since orbits quickly become uncorrelated due
Analysis. Take any arbitrarily small volume in the phase space of a chaotic system. Well, I tried those numbers in the equation, but I kept getting an error message from r = 2. Sometimes you can get the whole spectrum of exponents using the
rate of state space contraction averaged along the orbit (the
whole spectrum of Lyapunov exponents. Descriptions of the sort given at the end of the prevous page are unnatural and clumsy. It would be nice to have a simple measure that could discriminate among the types of orbits in the same manner as the parameters of the harmonic oscillator. The Lyapunov exponent can also be found using the formula, which in the case of the logistic function becomes. so as to avoid the all too common mistake of quoting more digits
Consider two points in a space, X0 and X0 + Δx0, each of which will generate an orbit in that space using some equation or system of equations. So for this, define d( k )>, where is averaging over all starting pairs t i , t j , such that the initial distance d (0) = | t i – t j | is less than some fixed small value. whole spectrum of Lyapunov exponents, Chaos and Time-Series
Well, not exactly, but close enough for now. It jumps from order to chaos without warning. No calculator can find the logarithm of zero and so the program fails. If we use one of the orbits a reference orbit, then the separation between the two orbits will also be a function of time. directions. calculate a mean and standard deviation of the calculated values
These orbits can be thought of as parametric functions of a variable that is something like time. the orbit for a flow or from the average determinant of the
An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by R. H. Miller in 1964. not be considered here. Nearby points, no matter how close, will diverge to any arbitrary separation. you can repeat the calculation for many different initial
At this "r" value the system quickly settles on to the fixed point of ½, which makes. largest Lyapunov exponent. For a continuous system, the phase space would be a tangled sea of wavy lines like a pot of spaghetti. Volume is preserved, but shape is not. For the map in the form xnC1 D ˆ axn if yn< .1 − b/C bxn if yn> ynC1 D ˆ yn= if yn< .yn− /= if yn> (7.17) with D1 − the exponents are 1 D− log − log >0 2 D ln aC log b < 0: (7.18) This easily follows since the stretching in the ydirection is … I calculated some Lyapunov exponents on a programmable calculator for interesting points on the bifurcation diagram. All neighborhoods in the phase space will eventually be visited. than are significant. Thus the snow may be a bit lumpy. Speaking of disagreement, the Scientific American article that got me started on this whole topic contained the following paragraph: I encourage readers to use the algorithm above to calculate the Lyapunov exponent for r equal to 2. A fractal is an object with a fractional dimension. The harmonic oscillator is quite well behaved. conditions (within the basin of attraction) and perturbation