The answer lies in the many definitions of dimension. The original volume will repeatedly fold in on itself until it acquires a form with infinite crenelated detail. This does not preclude any organization as a pattern may emerge. 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. It jumps from order to chaos without warning. See also the code for calculating the This is called iteration. to the sensitive dependence on initial conditions. Keep doing this forever. At this "r" value the system quickly settles on to the fixed point of ½, which makes. 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. flow) instead of difference equations (a map), the procedure is For the Bakers’ map, the Lyapunov exponents can be calculated analytically. 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. The complete procedure is as follows: If the system consists of ordinary differential equations (a Sometimes you can get the whole spectrum of exponents using the I calculated some Lyapunov exponents on a programmable calculator for interesting points on the bifurcation diagram. It works for discrete as well as continuous systems. Stupid me, I spent several minutes looking for an error in the code not realizing that the mistake was in the instructions. chaotic map or a three dimensional chaotic flow if you know the result good to better than about two significant digits. separation. 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. These orbits can be thought of as parametric functions of a variable that is something like time. so as to avoid the all too common mistake of quoting more digits The results are listed in the table below and agree with the orbits. Whenever they get too far apart, one of the You may get run-time errors when evaluating the logarithm if d1 becomes so small as to be indistinguishable from zero. So similar and yet so alike. What does this mean? The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. Because sensitive dependence can arise only in some portions of a system (like the logistic equation), this separation is also a function of the location of the initial value and has the form Δx(X0, t). equations generating the chaos, this is relatively easy to Descriptions of the sort given at the end of the prevous page are unnatural and clumsy. do. Lexp - Lyapunov exponents to each time value. instead of inverse iterations. You can see there was some disagreement in the sources as to exactly where the chaotic regime begins. Despite their peculiar behavior, chaotic systems are conservative. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. A physical system with this exponent is. rate of state space contraction averaged along the orbit (the LYAPUNOV EXPONENTS 119 Figure 6.2: A long-time numerical calculation of the leading Lyapunov exponent requires rescaling the dis-tance in order to keep the nearby trajectory separation within the linearized flow range. Nearby points, no matter how close, will diverge to any arbitrary separation. The exponent provides a means of ascertaining whether the behavior of a system is chaotic. Take any arbitrarily small volume in the phase space of a chaotic system. calculate a mean and standard deviation of the calculated values Jacobian matrix for a map) and using the fact that one exponent The hazy regions are unstable and chaotic. The logistic equation is a discrete, second-order, difference equation used to model animal populations. Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. An Given this new measure, let's apply it to the logistic equation and see if it works. From what I can tell, the maximal Lyapunov exponent λ for some 1-d map f ( x n) = x n + 1 is: λ ≈ 1 n ∑ i = 0 n − 1 l n | f ′ ( x i) |. The logistic equation is superstable at this point, which makes the Lyapunov exponent equal to negative infinity (the limit of the log function as the variable approaches zero). not be considered here. Users have to write their own ODE functions for their specified systems and use handle of this function as rhs_ext_fcn - parameter. The paramenters of the system determine what it does. Negative Lyapunov exponents are characteristic of, The orbit is a neutral fixed point (or an eventually fixed point). Where, if I understand things correctly, f ′ ( x i) is the derivative of f … you can repeat the calculation for many different initial This chapter describes the methods for constructing some of them. the same except that the resulting exponent is divided by the above method, for example when the system is a two dimensional The fourth chapter compares linear and non-linear dynamics. Have you found the errors in this book yet? The usual test for chaos is calculation of the If a system is unstable, like pins balanced on their points, then the orbits diverge exponentially for a while, but eventually settle down. There is a second error in the statement that r = 3 is in the chaotic regime. directions. All neighborhoods in the phase space will eventually be visited. In the diagram below we can see both stable and unstable orbits as exhibited in a discrete dynamical system; the so-called standard map also known as the Cirikov-Taylor map. The Lyapunov characteristic exponent (LCE) is associ-ated with the asymptotic dynamic stability of the system: it is a measure of the exponential divergence of trajecto-ries in phase space. You will typically need Well, I tried those numbers in the equation, but I kept getting an error message from r = 2. such a calculation is difficult to impossible, and that case will For a continuous system, the phase space would be a tangled sea of wavy lines like a pot of spaghetti. Well, not exactly, but close enough for now. The first chapter introduces the basics of one-dimensional iterated maps. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. No calculator can find the logarithm of zero and so the program fails. A positive largest Lyapunov 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? A Lyapunov exponent of zero indicates that the system is in some sort of steady state mode. The limit form of the equation is a little too abstract for my skill level. In the 1970s, a whole new branch of mathematics arose from the simple experiments described in this chapter. In a system with attracting fixed points or attracting periodic points, Δx(X0, t) diminishes asymptotically with time. calculated from the trace of the Jacobian matrix averaged along Luckily an approximation exists. When one has access to the The first number should be negative, indicating a stable system, and the second number should be positive, a warning of chaos (Dewdney). code for calculating the Analysis. 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 … This number can be calculated using a programmable calculator to a reasonable degree of accuracy by choosing a suitably large "N". End of diversion. Take a function y = ƒ(x). Standard map orbits rendered with Std Map. Ref: J. C. Sprott, Chaos and Time-Series There is a second error in the statement that r … To estimate the uncertainty in your calculated Lyapunov exponent, millions of iterations of the differential equations to get a Thus the snow may be a bit lumpy. Analysis (Oxford University Press, 2003), pp.116-117. nearby orbits and to calculate their average logarithmic rate of orbits has to be moved back to the vicinity of the other along the By convention, the natural logarithm (base- e) is usually used, but for maps, the Lyapunov exponent is often quoted in bits per iteration, in which case you would need to use base-2. *Analytically, λ = −∞ at superstable locations (see below). These points are said to be unstable. The logistic equation is unruly. A parameter that discriminates among these behaviors would enable us to measure chaos. Although the system is deterministic, there is no order to the orbit that ensues. The Lyapunov exponent can also be found using the formula, which in the case of the logistic function becomes. If we use one of the orbits a reference orbit, then the separation between the two orbits will also be a function of time.

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