WCCM V Fifth World Congress on Computational Mechanics July 7–12, 2002, Vienna, Austria Eds.: H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner
Escape Rate of Transient Chaotic Phenomena in Digitally Controlled Systems G´abor Csern´ak, G´abor St´ep´an Department of Applied Mechanics Budapest University of Technology and Economics, H-1521 Budapest, Hungary e-mail:
[email protected]
Key words: nonlinear dynamics, transient chaos, escape rate, digital control Abstract We analyse the digitally controlled stationary motion of a 1 DOF mechanical system. The control we use is assumed to have two deficiencies: the computer samples the velocity at discrete time instances, and the velocity measurement has a finite resolution. Taking into consideration these effects, we arrive at a 1D map, which describes the dynamics of the system. This map is referred to as the micro-chaos map, since it has a chaotic attractor in some parameter domains. In the case of a modified map, transient chaotic behaviour can be found, too. Our goal is to find a method, which can be used to calculate the mean lifetime of such transients exactly in some system parameter domains. We developed a method for this purpose, which is quite different from the usual escape rate calculations [2, 3, 4], and applied it for the modified micro-chaos map.
G´abor Csern´ak, G´abor St´ep´an
1 Introduction Chaotic oscillations often disappear suddenly – this phenomenon is referred to as transient or metastable chaos. This transient behaviour cannot be characterized by conventional damping factors since the transient chaotic oscillation does not produce an exponential decay in amplitudes, it rather disappears unexpectedly. The duration of these oscillations varies stochastically, so the estimation of the life expectancy of the transient motion needs extensive simulation work and statistical analysis. There are several numerical methods, which were introduced to give lifetime estimations quickly [1, 2, 3, 4], these methods are based on the assumption, that the probability that a solution has not yet escaped from a given set – the so-called chaotic repeller – decays exponentially with time. The exponent is called uniformly distributed initial escape rate . A more technical definition of is the following: Consider points in a region containing the repeller. By iterating trajectories from the initial points, many will leave the region in a few steps. Let denote the number of trajectories staying inside after steps. One finds asymptotically
(1)
where is the escape rate. The reciprocal of the escape rate is usually considered to be the mean lifetime of the transient chaotic phenomenon. This approach is not exactly correct, for the following reasons:
In the case of continuous flows with exponential rate of escape the mean lifetime chaotic behaviour can be calculated as follows:
of the transient
!#" (2) This result corresponds to the usually used relation. However, in the case of maps, we obtain another expression for the mean number of iteration steps : $ %&% + ')')( ( * (3) "
#,
In the calculation above, we exploited the assumption that the probability of staying in a set decays exponentially. The set in question is the so-called chaotic repeller or chaotic saddle [3]. It can be considered as a dense but measure zero set of repelling points, which form a strange structure, a “maze” for the solution curves. If a solution arrives at this maze, a very long time might be needed to leave it. The exponential decay, mentioned above, can be valid in the small neighborhood the repeller only, where the behaviour is chaotic, indeed. Thus, the calculation of the escape rate does not provide a correct mean lifetime in problems where the repeller and the set of possible initial conditions do not coincide. The exponential rate of escape is asymptotically fulfilled, only. Although the convergence is usually quite fast, this systematic error cannot be always neglected.
Our goal is to find a method which does not show these deficiencies, and can be used to calculate the mean lifetime exactly, taking into account the non-chaotic part of the motion, too. In the case of some piecewise linear maps such calculations can be performed. We developed a method for this purpose, and applied it for a model of a digitally controlled system.
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WCCM V, July 7–12, 2002, Vienna, Austria
2 Mechanical Model Consider the motion of a one-degree-of-freedom mechanical system, which is subject to some negative velocity-dependent friction force. An example for such systems can be seen in Figure 1. This mechanical
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Figure 1: The mechanical model
model consists of a block sliding on a rough surface near a prescribed velocity, , under the action of an electric motor. As also can be seen in the figure, the friction force characteristic is locally decreasing at low speeds. Because of this negative dissipation, the stationary motion at the speed is unstable.
to act on the To stabilize the stationary motion, we apply a control force
denotes a differential gain, with which the strength of the control can be tuned. The system. Here, linearized equation of motion of the controlled system assumes the following form:
,
, , , , "
If we use an analogue control system, the appropriate choice of ,
,
,
(4)
results in the asymptotic stability of the stationary motion. However, we would like to use digital control, where one encounters two deficiencies, that are the quantization in time and in space:
at discrete time instances, !#"$% the computer samples the velocity difference %'&)( &+* -, , where ( stands for the sampling time,
,
" " "
,
! ,
"
and the velocity measurement has a finite resolution, . .
Taking into consideration the effect of these discretizations, we arrive at the following linearized, * nondimensional equation of motion, which is valid between two successive sampling instants, / &)( '&0 1( : :
2 3 2 .5416)798 (5)
,
, ,
. @ ! where refers to the negative dissipation, denotes the velocity difference at the &CBED , sampling instant and 416)7 F stands for the integer-part function. One finds that the velocity difference at '&G 1( can be expressed by 2'&)( : : ,
: IH ( :JK 3 LJIK .
(7)
This map can serve as a simple cartoon model for other digitally controlled systems, too.
3 Chaos and Transient Chaos 3.1
The modified micro-chaos map
It was shown in [5], that the solutions of system (7) may behave chaotically in finite parameter domains. The amplitude of this chaotic vibration is very small, it is in the order of the resolution of velocity measurement . , so this is what the name of the map refers to.
/
at the origin, which may capture In more realistic situations, there might exist an attracting domain @ the trajectory. This motivates us to introduce the modified micro-chaos map :
Here,
(
H
,
416)7
if if
> , and
> " @
is less than unity, assuring that the origin is an attracting point, indeed. The exact value of not important, since the trajectories are not followed inside the domain of attraction.
(8) is
1.5
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1
0.5
0 0
0.5
1
1.5
2
Figure 2: The modified micro-chaos map
,
"
, , , (thick In Figure 2, the graph of the map can be seen for the parameters , lines), with a quite long trajectory (thin lines), which eventually arrives at the domain of attraction of the . is a new parameter, it characterizes the size of the sub-interval, origin, whose size is 4
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WCCM V, July 7–12, 2002, Vienna, Austria
which is directly reachable from the right by a trajectory – see dashed lines in the figure. In what follows, will refer to this sub-interval itself, too.
/
. The left border is defined by The chaotic repeller of the map is confined in the interval the border of the domain of attraction, while the value for the right border comes from the following consideration: a solution, starting from a greater initial value, first tends to the left, and sooner or later it takes a value less than 1. It can be seen from definition (8), and also in Figure 2, that if a solution arrives below 1, later it cannot take a value greater than . It means, that the mixing of possible trajectories – a necessary condition for chaos – can only be realized below this value.
The usual numerical escape rate calculations can only be performed if the initial conditions are distributed within . However, this is not required for the application of our procedure, as introduced in the next section. 3.2
Life expectancy calculation
/
There are some domains of the interval of initial conditions " @ -, , namely the first pre-images of , from which the number of iterations needed to reach the interval is known. These intervals and numbers will be referred to as fundamental escape intervals and kickout numbers, respectively. In Figure 3, the fundamental escape intervals are shown, with numbers denoting the kickout numbers.
Figure 3: The structure of the initial conditions
The remaining intervals – referred to as fractal intervals –, over which the shaded coloumns are drawn, have a complex structure. These intervals are mapped onto the set denoted by in a few steps, thus their structure is the same. The exploration of this structure plays a central role in our method. In the following, we will perform calculations for a fractal interval of unit length. It is just a rescaled version of . As Figures 3 and 4 show, the set contains sub-intervals, so does . The lengths of the sub-intervals in will be denoted by greek letters. In the following, these letters will refer to the intervals, as well.
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G´abor Csern´ak, G´abor St´ep´an
Escape intervals and number of steps needed to escape
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6
β
5
δ
4
α
4
γ
3
ε
Number of steps needed to be mapped onto
Figure 4: Structure of the set
(i.e. onto itself)
As it can be seen in Fig. 4, the fractal interval also contains subsequent pre-images of . These intervals will be referred to as escape intervals. There are three escape sub-intervals in : 6, 5 and 4 steps are needed to escape from the domains , and , respectively. The other two sub-intervals and are mapped onto the whole set in 4 and 3 steps, respectively. It means, that these sub-intervals are , thus the “fractal intervals” have fractal structure, indeed. This self-similarity similar to the interval provides the base of a recursive calculation [6]. Consider the escape intervals, from which the trajectories escape via the sub-interval (see Figure 4). Besides the interval itself, such intervals can be found in the fractal intervals and , too, with lengths and , respectively, since some parts of and are mapped onto . Similarly, there are intervals, which are mapped onto at the second step and , intervals mapped onto at the BED step. The lengths of all the intervals of this kind can be represented in a tree-structure of the graph (see Figure 5).
α αδ
αβ αβ
2
αβδ
αδβ
αδ
Figure 5: The tree-structure
2
Let us denote the joint length of these intervals by , too. The sums of the intervals in the sub-trees and with roots and will be denoted by and , respectively. Since , , the quantities , , and can be determined easily.
By multiplying the lengths of the intervals by the kickout numbers of the intervals, we arrive at the refers to a sub-interval in weighted tree-structure , shown in Figure 6. Here, for example, G , which is mapped onto in 4 steps. Then 6 steps are needed for the solutions to escape from . If we substract the sextuple of the elements of the tree from the elements of the tree , the resulting structure is more treatable (see Figure 7). Let us denote the sum of the elements of the two sub-trees in the new tree by and , respectively, and let denote their union. It is easy to see, that the following equations are fulfilled:
,
,
6
"
(9)
WCCM V, July 7–12, 2002, Vienna, Austria
6α (6+4)αβ (6+4+4)αβ
2
(6+3)αδ
(6+4+3)αβδ
(6+3+3)αδ
(6+3+4)αδβ
2
Figure 6: Weighted tree
0
4αβ (4+4)αβ
2
3αδ
(4+3)αβδ
(3+4)αδβ
(3+3)αδ
Figure 7: Modified weighted tree
2
Solving these equations, we obtain the sum of the elements of the tree . The sum of the elements of . Performing a similar calculation for the other two escape intervals can be obtained as and , too, the sum of the resulting weighted sums gives the mean kickout number for the set , since
6C7 is zero. it is a Cantor-like set, and the measure of
/
Now, the kickout number of for initial conditions in " @ -, can be calculated easily: We sum the lengths of the fundamental escape intervals multiplied by the corresponding kickout numbers, then add to this sum the weighted sum of the lengths of the fractal intervals. The weight is the kickout number for the set ( with normalized length) plus the number of steps needed to reach the set from the given interval. After a division by the size of , we obtain the mean kickout number.
Unfortunately, the calculation above cannot be performed, if the intervals on the right and left side of are fractal intervals. However, we could find escape intervals there by detecting more pre-images of , but the calculation would be more complex.
In Figure 8, the mean kickout numbers are shown at different values of the parameter . Vertical dashed lines denote the parameter domains, where we were able to apply the above described method by taking into consideration the first few pre-images of only (see Figure 3). The solid and the dashed curves in Fig. 8 represent the results of numerical simulations and the above calculation, respectively. As it can be seen, the two sets of results almost coincide within the domains of applicability of this “first order” method. Here the curves of kickout numbers are linear in . The intervening intervals consist of several small domains, where this linear relation is fulfilled, too. By taking into consideration more pre-images of , this higher order fractal structure could be explored. Note, that at greater values of no fractal intervals – and consequently no chaotic behaviour – occur, thus the kickout number can be calculated more easily.
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G´abor Csern´ak, G´abor St´ep´an
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Simulations Parameter borders Analytic results
Mean kickout number
7
6
5
4
3
2
1 0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Figure 8: Results
Acknowledgements The publication of this paper was supported by the Rubik International Foundation (Magyar M´ern¨okakad´emia Alap´ıtv´any – Rubik Nemzetk¨ozi Alap´ıtv´any) and the Hungarian Scientific Research Foundation OTKA Grant No. T030762/99
References [1] H. Kantz, P. Grassberger, Repellers, Semi-attractors, and Long-lived Chaotic Transients, Physica 17D, (1985), pp. 75-86 [2] T. T´el, Escape Rate from Strange Sets as an Eigenvalue, Physical Review A, 36(3) (1987) [3] T. T´el, Transient Chaos, Published in: Directions in Chaos, 3, Experimental Study and Characterization of Chaos, edited by Hao Bai-lin, World Scientific Publishing Company, Singapore, (1990), pp. 149–211 [4] P. Cvitanovi´c, R. Artuso, R. Maineri, G. Tanner and G. Vattay, Classical and Quantum Chaos, www.nbi.dk/ChaosBook, Niels Bohr Institute, Copenhagen, (2001) [5] G. Haller G. St´ep´an, Micro-Chaos in Digital Control, J. Nonlinear Sci., 6, (1996) pp. 415-448 [6] G. Csern´ak, G. St´ep´an, Life Expectancy Calculations of Transient Chaotic Behaviour Using Approximate 1D Maps, Meccanica, 35(6), (2000)
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