=Paper=
{{Paper
|id=Vol-1987/paper66
|storemode=property
|title=Control of Chaos in Strongly Nonlinear Dynamic Systems
|pdfUrl=https://ceur-ws.org/Vol-1987/paper66.pdf
|volume=Vol-1987
|authors=Lev F. Petrov,Nikolay Pogodaev,Boris T. Polyak,Andrey A. Tremba,Yuri S. Popkov,Larisa Rybak,Elena Gaponenko,Viktoria Kuzmina,Olga N. Samsonyuk,Natalya Sedova,Vadim I. Shmyrev,Ruslan Yu. Simanchev,Inna V. Urazova,Alexander K. Skiba,Stepan P. Sorokin,Maxim V. Staritsyn,Alexander S. Strekalovsky,Anna Tatarczak,Nikolay P. Tikhomirov,Dmitry V. Aron,Alexey A. Tret'yakov,Sergey Trofimov,Aleksey Ivanov,Yury Fettser,Tatiana S. Zarodnyuk,Alexander Yu. Gornov,Anton S. Anikin,Evgeniya A. Finkelstein,Elena S. Zasukhina,Sergey V. Zasukhin,Vitaly Zhadan,Anna V. Zykina,Olga N. Kaneva
}}
==Control of Chaos in Strongly Nonlinear Dynamic Systems==
https://ceur-ws.org/Vol-1987/paper66.pdf
Control of Chaos in Strongly Nonlinear Dynamic
Systems
Lev F. Petrov
Plekhanov Russian University of Economics
Stremianniy per., 36,
115998, Moscow, Russia
lfp@mail.ru
Abstract
We consider the dynamic processes models based on strongly nonlin-
ear systems of ordinary differential equations and various stable and
unstable solutions of such systems. The behavior and evolution of so-
lutions are studied when the system parameters and external influences
change. As a tool for controlling the behavior of solutions is considered
a system parameters change. Particular attention is paid to the param-
eter of dissipation as a tool for controlling chaos. For simple systems, a
controlled transition from stable solutions to chaos and back is consid-
ered. The generalization of the solutions chaotic behavior for complex
systems and the criteria for optimal level of chaos in such systems are
discussed.
1 Introduction
For complex systems such as Economics, social relations, communication, chaos is a natural form of behavior.
For these systems mathematical models allow qualitatively, but not quantitatively evaluate the system behavior.
In real complex systems, both a regular ordered behavior and chaotic states of systems are observed.
Complete elimination of chaos in a complex system, as a rule, leads to system degradation. For the economy
this is stagnation, lack of competition, initiatives and new prospective. For social relations, this is the lack of
democracy. For information transfer systems, this is regulated and metered information and misinformation,
information vacuum, the lack of the possibility of obtaining information from alternative sources.
The lack of complex systems regulation leads to an excessive level of chaos. This usually has negative conse-
quences for the behavior of the real complex system. In the economy, this manifests in the form of chaos, mass
evasion from paying taxes, economic crises, excessive volatility of indicators. Excessive chaos in social relations
leads to anarchy, the emergence of local groups and leaders who do not submit to control from the center, conflicts
between individual groups and the center. The complete lack of regulation in information transmission systems
leads to the use of such systems for the exchange of undesirable information. You can observe numerous sad
examples of the implementation of such scenarios in real complex systems.
Therefore we can conclude that for complex systems there is an optimal level of chaos. In the economy, not
suppressed initiative, originate new business, but complied with the rules of economic behavior. In a society, a
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of
the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org
460
democratic regime is implemented without elements of anarchy and dictatorship. The information transmission
systems are used for free exchange with the blocking of unwanted information. The blocking can be realized
by censorship or ignoring of undesirable information by the majority of participants. Unfortunately now only
the first steps are being taken to develop mathematical models that allow us to quantify the level of chaos in
complex systems.
For simple nonlinear dynamical systems, the situation is radically opposite. For such systems, the absence of
the solution chaotic component does not lead to the system degradation, but is one of the main natural forms of
behavior. At the same time, for simple nonlinear dynamical systems, chaos is also a natural form of solutions.
For such systems, an interactive algorithm for the qualitative and quantitative study of regular and chaotic
solutions has been developed [Petrov, 2015] , which allows investigating both types of solutions and transitions
between them. For this, periodic solutions are calculated and their stability is analyzed. The periodicity of the
solution is determined by the discrepancy between the initial and final points of the trajectory on the solution
period. To construct periodic solutions, the concept of minimizing this discrepancy is used. This strategy of
finding periodic solutions for essentially nonlinear systems of ordinary differential equations can be interpreted as
the search for a global minimum of the distance between the beginning and end of the solution trajectory in one
period. Based on the numerical results of essentially nonlinear systems of ordinary differential equations study,
we will consider the tools for controlling chaos in simple nonlinear dynamical systems and discuss qualitative
analogies with complex nonlinear dynamical systems.
2 Statement of the Problem
As a model problem, consider the essentially nonlinear generalized Duffing equation
ẍ + (1 − P )ω02 x(t) + b1 (ẋ(t)) + γ1 (x(t)) = W cos(ωt), (1)
where b1 (ẋ(t)) is, in the general case, a nonlinear dissipative function, γ1 (x(t)) - function that determines the
nonlinear characteristics of the system, W cos(ωt) - a periodic external influence. It is only required of functions
b1 (ẋ(t)) and γ1 (x(t)) that the methods of the Cauchy problem numerical solution for equation (1) used in the
algorithm for global optimization of periodic discrepancy allow to construct a solution on one period with a
given accuracy. In the classic version of the problem statement b1 (ẋ(t)) = bẋ(t) and γ1 (x(t)) = γx3 (t). Without
limiting the generality, we will take the same form b1 (ẋ(t)) and γ1 (x(t)).
We note a fundamentally different character of the system for P > 1 and P < 1. For the equations of both
these classes, the regular solutions, chaotic behavior and strange attractor are known [Kovacic et al., 2011] ,
[Holmes, 1979] and etc.
For P > 1 the simplest prototype corresponding to this equation is the forced oscillations of the ball in
the profile with two symmetrical holes ( Figure 1 ). The same equation describes the oscillations of some
mechanical systems with a jump, in particular, forced transverse vibrations of the beam, which lost its static
stability [Holmes, 1979], [Petrov, 2013]. This is oscillator with negative linear stiffness. The phase space has
three singular points. The source of chaos is the presence two (or several in the general case) of the minimum
potential energy points and the possibility a solutions jumping to another potential well.
Figure 1: The simplest prototype object and phase trajectories of equation (1) solutions with P > 1
For P < 1 the classical version of the Duffing equation is realized. The amplitude-frequency characteristic has
three lines (Figure 2). The upper and lower lines correspond to stable solutions, the middle line – to unstable
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solutions. The source of chaos for such system is the possibility of solutions hopping from one line of the
amplitude-frequency characteristic to another.
Figure 2: The simplest prototype object, phase trajectories of solutions and amplitude-frequency characteristic
of equation (1) with P < 1
Based on physical considerations, we will apply the dissipative characteristics of the system, determined
by the dissipation coefficient b, the amplitude W and the frequency ω of the periodic external influence as
parameters for controlling chaos. The interactive computational numerical-analytical algorithm for ordinary
differential equations systems periodic solutions constructing and analyzing the stability [Petrov, 2015] was used
for research.
3 Results of Numerical Experiments
As the basic model, we adopted equation (1) with the following values of the parameters [Holmes, 1979] : P = 2,
ω02 = 10, gamma = 100, W -variable parameter (the base value W = 1.5 ), b -variable parameter (the base value
b = 1 ), ω -variable parameter (the base value ω = 3.76). At basic values of the parameters, the solution is
chaotic (Figure 3).
3.1 Chaos Control Via Amplitude of External Periodic Influence
Study the oscillations dependence on the external influences amplitude allow to detect the range of values at
which the solution has the character of a strange attractor [Holmes, 1979]. It is additionally established that
the transition from stable periodic solutions to chaos at the boundaries of the strange attractor zone is realized
according to the scenario of period doubling bifurcations [Feigenbaum, 1983] , [Petrov, 2010] (Figure 4).
Figure 3: Chaos in the simple system (1)with P = 2, ω02 = 10, ω = 3.76, b = 1, gamma = 100, W = 1.5 : Phase
portrait, the solution and three-dimensional representation
Outside the zone of the strange attractor, stable solutions correspond to external influences – for small W
oscillations around one of the two singular points (Figure 1, curves 1 and 2), for large values W oscillations of a
larger amplitude, spanning both singular points (curve 3).Chaos is realized when all of these solutions become
unstable ( Figure 3).
Thus, it can be concluded that the amplitude of external influence W is one of the tools for controlling the
system behavior in terms the chaos presence. For large amplitude values W , the external influence determines
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Figure 4: Period doubling bifurcation of the solutions of equation (1) with P = 2, ω02 = 10, ω = 3.76, b =
1,gamma = 100, W = 0.7; W = 0.886; W = 0.926; W = 0.936; W = 0.9379
the nature of the solution, and chaos does not appear. For small values W , it is not enough of external energy
to realize a chaotic state.
We note that within the zone of the strange attractor on the parameter W , anomalous existence of stable
periodic (6π/ω and 10π/ω ) solutions detected [Holmes, 1979] , [Petrov, 2010] ( Figure 5 ).
Figure 5: Phase trajectories of pairwise symmetric stable solutions of equation (1) with period 6π/ω at P = 2,
ω02 = 10, γ = 100, b = 1, ω = 3.76, W = 1.7
We note the symmetry in the pair stable solutions given in Figure 5.
3.2 Chaos Control Via Dissipation
The influence of dissipation on chaos appears in a different form. In the zone where there was originally the
strange attractor, as the dissipation coefficient b increases, stable solutions begin to be determined. At first they
have a rather complicated form ( Figure 6 ), but with a further increase in the parameter b the solutions become
simpler ( Figure 7 , curve 1). In this case, the simplest solutions already exist ( Figure 7 , curve 2), but they are
still unstable. With further growth of dissipation coefficient b , only the simplest harmonic solutions that are
similar to (Figure 7, curve 2) remain stable. Large dissipation leads to the elimination of deterministic chaos. A
decrease in the dissipation coefficient b leads to the appearance of stable solutions with a large amplitude (Figure
1 , curve 3 ). In this case, solutions with small amplitude (Figure 1, curve 1 and 2 ) are unstable. Thus, both
increase and decrease of dissipation is a means for suppressing chaos.
3.3 Chaos Control Via Frequency of External Periodic Influence
The dependence of the solutions form of equation (1) on the external periodic influence frequency ω has not so
obvious a physical meaning as on the amplitude W and dissipation b. At low frequencies of the external influence
( 0.3 < ω < 0.6 ), stable and unstable solutions have high-frequency components (Figure 8). At high frequencies
of external influence, other behavior of solutions is observed (Figure 9).
3.4 Chaos Control in Multidimensional Systems
In addition to the chaos control tools discussed above for dynamical systems with several degrees of freedom
the chaotic behavior of solutions may depend on the interaction of oscillations in different degrees of freedom
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Figure 6: Phase trajectory and one 8π/ω period of the equation (1) stable solution with at P = 2, ω02 = 10,
γ = 100, b = 1.81, ω = 3.76, W = 1.5
Figure 7: Phase trajectories and solutions of equation (1) with P = 2, ω02 = 10, γ = 100, b = 2, ω = 3.76,
W = 1.5 1 - stable 2 - unstable
[Petrov, 2013]. It is possible to suppress chaos in one of the oscillation forms due to the influence of regular
oscillations in another form. This is a generalization of the linear effect of oscillations dynamic damping on
nonlinear multidimensional oscillations with possible chaos.
4 Optimization of Simple and Complex Dynamic Systems in Terms of Chaos
To formulate the problem of optimizing a dynamic system by the chaos criterion, it is necessary to determine the
quantitative expression of the objective function. The measure of chaos, including for complex systems, is the
fractal dimension of the solution [Peters, 1996] . After determining the chaos control techniques can be formulate
the optimization problem of the chaos level with these control tools.
For simple systems the objective function can be formulated on the basis of technical requirements -a minimum
value of the solution fractal dimension ( when necessary the ordered behavior of the system), a maximum of this
dimension (when required the disordered chaotic behavior of the system).
For complex systems the chaos level must be within a certain range, corresponding to optimal or acceptable
conditions for the system functioning and development. Qualitatively, these constraints were discussed above.
For definiteness, the middle of the range of optimal (acceptable) values of this dimension can be chosen as the
value of the objective function (solution fractal dimension) for a complex dynamical system.
For the optimization problem of nonlinear dynamic system by the level of chaos we have formulated the
objective function and tools for chaos control. This approach allows us to use optimization methods to find the
values of the parameters of a nonlinear dynamic system that provide the target level of chaos in the system.
5 Qualitative Generalizations
The results of our numerical experiments on control the solutions behavior of essentially nonlinear dynamical
systems demonstrate the possibility of realizing the transition to chaos and back using the management tools
listed above. It is possible as predictable system response to control actions (e.g., bifurcation of period doubling),
and unpredictable behavior (for example, the existence of stable solutions in a small region in the area of the
strange attractor).
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Figure 8: Solutions of the equation(1) with P = 2, ω02 = 10, γ = 100, b = 1, W = 1.5 at low frequencies of
external influence 1 - ω = 0.3 (stable), 2 and 3 - ω = 0.5 (symmetric, unstable) , 4- ω = 0.6 (stable)
Figure 9: Solutions of the equation(1) with P = 2, ω02 = 10, γ = 100, b = 1, W = 1.5 at high frequencies of
external influence 1 - ω = 5.5; 5.0 (stable), 1 - ω = 4.5 (symmetric, unstable),2 - ω = 4.7 (stable) , 3 - ω = 4.6
(stable) ,4- ω = 4.5 (chaos)
For relatively simple essentially nonlinear dynamical systems, the intensity of external influence, dissipative
characteristics, other parameters of the system can serve as an instrument for controlling the chaotic behavior of
solutions. At the same time, the results of the investigation are of a quantitative nature with a known error in
the calculations and are confirmed by experiments. We note that chaos in simple nonlinear systems is realized
when there are several singular points in the phase space. From the point of view of complex systems it is a
choice between several approximately equivalent states, realizations, and possibilities.
To qualitatively evaluate the tools for controlling chaos in complex systems, let us assume the following
analogies:
Dissipation in the economy – transfer of an economic asset to another form – taxes, losses, suppression of
economic initiative, etc.
Dissipation in social relations – suppression of initiative, intimidation, over-regulation, self-restraint, etc. Note
that the standard of living can manifest itself as an analog of dissipation – at an extremely low standard of living,
the social system tends to go into chaos, with high living standards, chaos in society generally does not manifest
itself.
Dissipation in information – censorship and self-censorship, silencing, etc.
External influence in the economy – adding external economic asset.
External influence in social relations – economic influence, agitation, propaganda, fashion, etc.
External influence in information systems – massive information and disinformation with possible blocking of
alternative sources of information.
The above analogies are of a debatable nature, but there are already approaches and mathematical models
[Milovanov, 2001] and others, which in time will be able to confirm or disprove these assumptions.
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