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ISSN 1320-0682 | |||
| Volume 03 | April 1996 | |||
D. J. Jefferies and M. J. Crawford
http://www/ee/surrey.ac.uk/Personal/D.Jefferies/index.html
Simple traps in one-dimensional piecewise-linear maps are described, and their properties under iteration are illustrated. Experimental electronics are described which iterate these maps in real time. Trapping consists of a sudden change of behaviour when a chaotic transient ceases and regular behaviour ensues. Experimentally, system noise is always important and can eject the system from a trap. The statistics of time to trapping are presented, and traps in higher dimensional systems are described. In the light of such simple experiments on trapping systems, the irregular and sometimes suddenly catastrophic events which occur in complex nonlinear systems which may have high dimensionality become more understandable.
A piecewise-linear [1, 2, 3] one-dimensional map relates the output real variable y to the input real variable x in the ranges (0,1), (0,1) by connected straight line segments in a rectangular Cartesian graph of y against x. The example taken as the starting point for this investigation is shown in Figure 1.
Figure 1: A piecewise-linear trapped map.
In a region where the slope dy/dx of the straight line section has magnitude greater than unity (that is, more than +1 or less than -1) any error in x results in a greater error in y. A piecewise-linear map which has slope everywhere magnitude greater than unity, and is also multiple valued in the sense that there are regions where given a value of y, there corresponds more than one value of x, that gives rise to unending chaos when iterated. To iterate a map, one transfers the output to the input, calculates a new output, and so on. This is particularly easy to do electronically, and the piecewise-linear characteristic can be synthesised from a collection of linear operational amplifiers. A circuit diagram and a functional block diagram will be shown at the meeting. There are a number of subtle implementation details, including the avoidance of trapping regions at the turning points of the transfer characteristic, which will be communicated.
The multiple valued property is loosely referred to as folding, and the slope magnitude greater than unity is referred to as stretching. There is a set of measure zero of starting values of x which result in closed trajectories which are periodic. In such a system the starting value repeats exactly after a finite number of iterations only when the starting values lie within this very restricted set [4] Roughly speaking, the chance of entering a periodic trajectory is zero, or at least vanishingly small. Any added noise, however small, will destroy the periodicity. Such an iterating system we term a chaotic map into which we introduce traps.
Traps consist of regions of the piecewise linear characteristic of slope zero (parallel to the x axis) or nearly zero. The iteration line is the line of unit slope linking (x,y) = (0,0) to (1,1). If the chaotic map has a trap on the iteration line, then eventually the chaotic motion will cease suddenly (Figure 2) and the system will thereafter remain at the fixed point where the trap crosses the iteration line. If the trap does not intersect the iteration line (Figure 3), then it is likely that the trapped motion will be periodic (Figure 4).
Figure 2: A fixed point trapping.
Figure 3: A periodic trap.
Figure 4: A periodic trapping.
It is possible that the chaotic motion, usually termed a chaotic transient [5] , can last a very long time. For the map of Figure 1, simulations of 100,000 experiments running to trapping gave the statistics shown in Figure 5 .
Figure 5: Trapping time distribution.
Although we have not calculated the slope of the exponential fall-off, this in principle should be quite easy, being related to the Liapunov exponent [4] and the structure of the map with its trap.
In the case where the trap lies above or below the iteration line, a little thought shows that the periodicity (number of iterations per cycle) in the trap will vary discontinuously and in a fractal manner as the trap is raised or lowered. If the slope of the trap section is non-zero, then intermittencies occur; the trapped region becomes noisily periodic as in the classic intermittent behaviour described by Pomeau and Manneville [6].
When conducting real experiments with electronics, the ratio of random noise
to the notional (0,1) ranges of the iterating variables is of the order
to 1. For
small trapping regions, one observes the phenomenon of untrapping, where the
noise releases the system from the fixed point, and it then sets off on another
chaotic transient until the next trapping event. We shall play a tape of such a
system. One can add noise in the simulation by summing 12 random numbers in the
range (0,1) and subtracting 6. By the central limit theorem, the resulting
variable will be nearly Gaussianly distributed with unity standard deviation,
and it may be scaled to the trap size. Figure 6
shows the visual behaviour of such a simulated noisy system.
Figure 6: A trapping iterated system with added noise.
In the electronic version of the trapping system, two identical (as far as practically realisable) transfer function amplifiers were constructed. The trap size and piecewise linear breakpoints were controllable by 10-turn potentiometers with scales, 1% accuracy, and the settings were precise to 1 part in 1000 of full scale. Two sample-and-hold circuits, externally clocked, transfer the output values to the input on a clock edge. Thus, one starts from two uncoupled chaotic trap systems with variables called, say, u and v at the inputs and w and z at the respective outputs. A circuit simulates a rotation matrix which takes part of the output w, adds and subtracts it to part of z, and applies the resulting two outputs to the inputs u and v via the sample-and-hold circuits.
There are some details here concerning setting the ranges of the variables, and the additional contraction of the mapping produced by the rotation circuitry needs to be zero. There is no space to discuss these. Experimentally, this two-dimensional system takes (as might be expected) longer to trap for a trap having a given size within one of the transfer characteristic circuits. However, the added noise need only release the system from the trap in one branch or dimension. Thus, the probability of untrapping is very nearly the same as for a one-dimensional version. The tape played is taken from this two-dimensional trap. No extra noise has been added above the inherent circuit noise.
Although the complexity of the electronics precludes experiments in large numbers of dimensions, it becomes clear that if this system is generalised to increasing numbers of dimensions, the probablity of trapping will decrease rapidly with the dimension. Therefore, reasonably large traps (large per dimension) can be introduced, compared to the noise level, and one can in principle construct an electronic circuit which will have a very large mean time to trapping, and a negligibly small chance of being released from the trap by noise. Alternatively, in a noisy real world environment, one can envisage sizeable traps larger than the noise which nevertheless are entered with low probability.
Suppose one calls the trap size, with respect to the range of the variable,
. Typically,
in our simulations and experiments,
was of the
order of 0.001. In an N-dimensional system the probability of trapping,
per iteration, will be of the order
. Thus, in
our system of examples in Figures 2 and 5, where it is possible to have
iterations
to trapping, in a ten-dimensional system we would find experiments taking
iterations
to trapping. Thus, in the real world examples, considering the global behaviour
of high-dimensional systems which are intrinsically quite noisy, traps might be
seen only rarely. However, if they do exist, as we have seen they may be
entered suddenly and without warning, and therefore sudden naturally occurring
events (extinctions?) may be a consequence of the natural dynamics and have no
prime cause other than the nature of the dynamics.
Considering the much-studied [4] logistic map, which is not piecewise-linear but consists of an inverted parabola of height a, it is possible to embed traps in a region which is only visited during the initial transient. On the chaotic attractor, there is no chance of trapping. Whether or not such a system traps will then require the initial state to lie in a certain region, which may be fractal; then, if such a system is knocked off its final attracting chaotic solution by some external stimulus, the transient may be re-entered and the chance of trapping revived. Thus, in a real world example where attracting solutions are the exception rather than the norm, and where most of the chaotic behaviour consists of repeatedly excited transients, traps outside the chaotically attracting regions may be important.
It is also possible to envisage a regression of nested maps; a small logistic map may be constructed within the main logistic map with the new zero on the fixed point where the iteration line intersects the main map. One can then have trapping which is itself chaotic; and the added noise will perturb the motion from the smaller chaotic region to the larger. Thus, we can have fixed point traps, limit cycle traps and chaotic traps. These can lie within or outside a chaotic attractor.
The concept of trapping seems to have applications in informing thought about complex chaotic systems. Firstly, trapping happens suddenly with no warning. Secondly, in a high-dimensional system it can happen with small probability per iteration, but relatively ultimate certainty even though the system is noisy. Thirdly, it gives insight into bursty behaviour that is not necessarily linked to classical intermittencies. In Figure 7 we show these effects in the simulation of a double potential well [7, 8] system; the system consists of a resonant circuit or a linear spring-mass arrangement with two separated centres, and is closely related to impacting systems. Further interesting phenomena suggestive of chaotic trapping are to be found in the paper [9] on spatio-temporal chaos of Eilbeck and Scott.
Figure 7: Trapping in a double well oscillator.
It is possible to construct relatively simple electronic circuits to study these trapping effects empirically. The behaviour of such circuits is striking; the possible types of behaviour become much more vivid when observed directly than they are in thought experiments or simulations.
Of great importance to modern technology is the likely behaviour of large arrays of parallel computing processes. With a single computer, a crash is usually fatal until human intervention restores the operation. In a large network of computers, an individual crash may cause the network to respond by resetting the computer at that node. The question arises as to the reliability of such a system. It is possible that the study of trapping nonlinear systems may inform this kind of engineering.
In the study of the global behaviour of complex systems, it is sometimes difficult to trace back the local causes of catastrophic events. The authors hope that this study of low-dimensional traps will give food for thought in this endeavour.
Traps in Chaotic Systems
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