A.Yu. Shvets, V.A. Sirenko

Transcription

1 ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ, СИСТЕМНИЙ АНАЛІЗ ТА КЕРУВАННЯ 45 UDC 59.67, 54.4 A.Yu. Shvets, V.A. Sirenko EW WAYS OF TRASITIO TO DETERMIISTIC CHAOS I OIDEAL OSCILLATIG SYSTEMS We considered nonideal dynamical system with a five-dimensional hase sace. Questions of occurrence of deterministic chaos in such systems are investigated. By using the reviously develoed by the authors the techniue of comuter simulation of deterministic chaos, we discovered and described a number of new scenarios of transition to deterministic chaos. In this research analyzed in detail the hase ortraits, signatures of sectrum of Lyaunov characteristic exonents and distribution of the invariant measure in various regular and chaotic attractors. In articular, there is found the transition to chaos by the scenario of generalized intermittency with two laminar hases. Succeed to identify the transition to chaos, which begins by the Feigenbaum scenario and ends through intermittency. The role of symmetry of attractors in such transitions is exlored. Identified the transitions to chaos, by scenario of the generalized intermittency, with two coarse-grained laminar hases. Also managed to find the scenario of generalized intermittency, in which taken lace the transition from hyer-attractor of one tye to another tye of hyer-attractor. Keywords: nonideal dynamical system, chaotic attractor, scenarios of transitions to chaos. Introduction Desite the infinite variety of the dynamic systems, it is ossible to divide all scenarios of transitions to deterministic chaos into three basic grous [, ]. To the first grou belongs Feigenbaum scenario of transition to chaos through an infinite cascade of eriod-doubling bifurcations of limit cycles. To the second grou of scenarios belong transitions to chaos through an intermittency by Pomeau Manneville of various tyes. At last, to the third grou of scenarios belong transitions to chaos through destruction of uasi-eriodic attractors (invariant toruses). However, the descrition of the scenarios of transition to chaos in dynamical systems is found in initial stages of the study. Many uestions about the relationshi between the different scenarios of transition to chaos remain unexlored []. Statement of the roblem All oscillatory dynamic systems consist of two main arts, a source of vibration excitation and vibrational loads. If the ower of the excitation source is comarable to the ower consumed by the vibrational load, then such a system is called non-ideal by Sommerfeld Kononenko []. If the ower of the excitation source considerably exceeds the the ower consumtion of the vibrational load, then such a system is called an ideal by Sommerfeld Kononenko. Objectives of the global energy saving, forced to the maximum to minimize the ower of various sources of vibration excitation. Therefore, the vast majority, the real modern oscillatory systems are nonideal. In mathematical modeling of nonideal systems is necessary to consider the influence of vibrational loads on the functioning of the source of vibration excitation. eglecting the influence of loads on excitation source may lead to serious errors in the descrition of the dynamic behavior of the system. In articularly, may be comletely lost the information about real existing deterministic vibrational regimes []. Consider the following nonlinear system of differential euations: d =α β+ ( ) + d τ B( ) ; d =α + β+ ( ) + dτ B( ) + ; dβ = + β μ ; () dτ d =α [ β+ ( )] dτ B( ) ; d d τ =α + [ β+ ( )] B( ), where,, β,, hase coordinates, τ time, AB,, α, μ,, some arameters. As was established in works [, 4, 5] the system of

2 46 Наукові вісті НТУУ "КПІ" 5 / euations () is used to describe the fluid oscillations in cylindrical tanks, for the study of endulum systems with a vibrating susension oint, for the simulation of vibrations of thin shells and a number other toical roblems of nonlinear dynamics. The arameters AB,, α, μ,, have one or another hysical or geometrical meaning deending on the considered alication task. Phase variables,,, are the generalized coordinates of the vibrational subsystem and hase variable β describes the oeration of the source of excitation of oscillations. System of euations () is a nonlinear deterministic dynamical system with a five-dimensional hase sace. In aers [, 6, 7] it is esteblished that there are several tyes of regular and chaotic attractors of this system and show that the transition to deterministic chaos can be carried out on all three grous of scenarios. The aim of this work is the identification and descrition of new scenarios of transition to deterministic chaos in dynamical systems of the form (). We shall investigate, as the scenarios of transition from regular attractors to chaotic attractors, as the scenarios of transitions between different tyes of chaotic attractors. Identification and descrition different scenarios of transition to chaos Since the system of euations () is a nonlinear system of ordinary differential euations of the fifth order, the construction of attractors of such a system is only ossible through the alication of different numerical methods. Large series comuter simulation of the system of euations () was carried out in the sace of the arameters ( α, AB,,,, μ ) of this system. The methodology of such a comuter simulation is described in detail in the works [, 6]. As a result of such a comuter simulation was able to identify and describe some of the new scenarios of transition to chaos. Let the arameters of the system () takes the value: A =, ; B =,5 ; μ = ; = ; α =,. As bifurcation arameter we choose,5,5,5,5 a b c,5,5,5,5 d Fig.. Projections of hase ortraits of limit cycles at =,645 (a, b), =,668 (c, d) and chaotic attractor at =,695 (e) e

3 ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ, СИСТЕМНИЙ АНАЛІЗ ТА КЕРУВАННЯ 47 the arameter. When changing in limits,656 < <,6 in system exists two single-stroke stable limit cycles. The rojections of the hase ortraits of such limit cycles, built at =,645 are shown in Fig., a, b. Their rojections are symmetrical relative to the axis of abscissa =. Signatures of the sectrum of the Lyaunov characteristic exonents (LCE) of such cycles are of the form,,,,, which is tyical for eriodic regimes []. At increasing the value of the arameter, there are simultaneous eriod doubling of the existing limit cycles. The rojections of hase ortraits arising -stroke cycles, built at =,668 are shown in Fig., c, d. Their rojections are also symmetrical relative to the axis of abscissa. With further increasing the value of the arameter going infinite cascade of erioddoubling of the existing limit cycles. This cascade ends with the emergence of a chaotic attractor (Fig., e). The rojection of this attractor consists of two symmetrical about a horizontal axis arts, a c Fig.. Projections of hase ortraits of limit cycles at =,8 (a, b), the rojection of the hase ortrait (c) and and the distribution of the invariant measure (d) of the chaotic attractor at =,8 b d and occuies the region of hase sace in which symmetrical limit cycles exist. The signature of the LCE sectrum of chaotic attractor has the form +,,,,. In this case, unlike the regular attractors, chaotic attractor has several imortant distinguishing features. Maximal Lyaunov exonent of the chaotic attractor is ositive, all the trajectories of the attractor unstable by Lyaunov and moment of time of the Poincarй returns of trajectories are unredictable. The motion of the reresentative oint of the trajectory on a chaotic attractor can be divided into two hases. In a first hase, a trajectory commits chaotic wandering along the to or bottom turns of the attractor. At unredictable times trajectory moves in the oosite symmetric art of the attractor, and there continue a chaotic wandering in this art of chaotic attractor. This rocess is reeated an infinite number of times. Thus, in this case, the transition to chaos united the eculiarities of Feigenbaum scenario (infinite cascade of eriod doubling bifurcations of limit cycles), and the eculiarities of Pomeau Manneville scenario (unredictable intermittency between the uer and lower arts of chaotic attractor). Consider the features of the transition to deterministic chaos at the exiting of the arameter out of the boundaries of the interval, 5 < <,89. At (,5;,89) attractors of the system () will be symmetrical, relative to the axis of abscissa, limit cycles. The rojections of the hase ortraits of a tyical air of limit cycles, constructed for =, 8, are shown in Fig., a, b. These cycles have more comlex structure than reviously considered in Fig., a, b. If the increasing (or decreasing) of arameter leads to exit the value of this arameter out of the boundaries of the interval, 5 < < <, 8 9 both limit cycles are shown in Fig., a, b are vanishing. Full-blown chaotic attractor arise in system. Tyical rojection of the hase ortrait of the chaotic attractor of this form is shown in Fig., c. In Fig., d is shown the distribution of the invariant measure by the hase ortrait of the chaotic attractor.

4 48 Наукові вісті НТУУ "КПІ" 5 / Heavily drawn area in Fig., d by the form reminiscent of the slice symmetric limit cycles. The distribution of the invariant measure by the hase ortrait of the chaotic attractor clarifies the mechanism of its occurrence. The emergence of chaos in this case has the characteristic features of intermittency. Tyical motion of trajectories by the attractor consists three hases, two laminar and turbulent. In each laminar hase the trajectory commits a uasieriodic motion in a small neighborhood of uer or lower of vanishing limit cycles. At unredictable moments of time a turbulent burst occurs and the trajectory goes to remote regions of hase sace. Turbulent hase of motion corresond aler areas of distribution of the invariant measure in Fig., d. Process of motion of trajectory by attractor consists following hases: one of the laminar hases turbulent hase one of the laminar hases. This rocess reeated endlessly. Moments of burst of the trajectory in turbulent hase and the switches between the two laminar hases are unredictable and reresent a chaotic seuence of moments of times. In this case, the transition to chaos reminiscent of the classic scenario intermittency of Pomeau Manneville. However, in this case, the transition to deterministic chaos consists not one, but two laminar hases. At comuter modeling and numerical analysis of the existing system of dynamic regimes, it was,96 < <,89 in sys- found that when tem exist symmetric stable limit cycles (Fig., a, b). At increasing of arameter, namely when =,89, the simultaneous eriod doubling bifurcation of existing cycles take lace (Fig., c, d). Further increase value of the arameter generates an infinite cascade of eriod doubling of symmetric cycles, which ends the aearance of symmetric chaotic attractors at (Fig., e, g). Arising of each of the symmetric chaotic attractors occurs by Feigenbaum scenario. It is worth noting that the obtained chaotic attractors (Fig., d, e) are searate and have different basins of attraction. In this case there is no henomenon of slice chaotic attractors, which was observed in the event of chaotic attractors in Fig., e and Fig., c. Let us analyze the scenario of transition to chaos at decreasing and exit it through the left border of interval,96 < <,87. As mentioned above, for values of the arameter,96 < <,89 in the system exist stable symmetric limit cycles (Fig., a, b). At =,96 the symmetric limit cycles disaear and in system arise two symmetric chaotic attractors (Fig., e, f ). Transition from regular regime to chaotic one for both symmetric chaotic attractors occur by classical scenario of intermittency by Pomeau Manneville. This fact clearly illustrated the distrubution of invariant measure by the hase ortrait of the chaotic attractors (Fig., g, h). Heavily drawn areas in these figures which are located in a neighbourhood of disaearing symmetrical limit cycles, corresond to laminar hase of an intermittency. In this hase a trajectories of chaotic attractors carry out uasi-eriodic motions in a small neighbourhood of disaearing limit cycles. More ale arts of distributions (Fig., g, h) corresond to a turbulent hase of intermittency. Arising symmetrical chaotic attractors exist in small interval of a changing of arameter. At further decrease of _ at =,9 7 in system occurs bifurcation chaos chaos. The symmetric chaotic attractors (Fig., e, f ) are disaeared after transiting of a oint of a bifurcation. In the system () arise the chaotic attractor of other tye with more comlicated structure that shown in Fig., k. The chaotic attractor of such tye reresents slice of disaearing symmetrical chaotic attractors (Fig., e, f ). Its structure is similar to the chaotic attractors considered above (Fig., e, Fig., c). Transition to chaos in this case haens absolutely under other scenario which considerably differs from the revious scenarios. The distribution of invariant measure for comlicated chaotic attractor is shown in Fig., l. The two more dark symmetrical areas of this distribution have the geometrical form similar to corresonding geometrical form symmetric chaotic attractors from Fig., e, f. Analysis the dynamics of rocess shown in Fig., l allows to state that transition to chaos here haens under the scenario of the generalised intermittency which is described for the first time in [, 6]. Moving of a tyical trajectory in attractor consists of three hases. Two of them are termed coarse-grained laminar [, 6, 8]. In these hases trajectory carry out chaotic motions in one of the dark areas of Fig., l, where were located the disaeared chaotic attractors of the revious tye. In the third hase turbulent, attractor trajectories leave these areas and move to remote areas of a hase sace. The more ale arts in Fig., l corresond to turbulent hase. After that the trajectories return to

5 ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ, СИСТЕМНИЙ АНАЛІЗ ТА КЕРУВАННЯ 49,5,5,5,5,5,5 a b с,5,5,5,5,5 d e f,5,5,5,5,5 g h k l Fig.. Projections of hase ortraits of limit cycles at =,9 (a, b), =,89 (c, d), and chaotic attractors at =,855 (e, f ), =,97 (k); distributions of invariant measures of chaotic attractors at =,96 (g, h), =,97 (l )

6 5 Наукові вісті НТУУ "КПІ" 5 / areas of coarse-grained laminar hases. Transition of a trajectory from one of coarse-grained laminar hases to turbulent hase is unredictable in regard to a time and reeats the infinite number of times. Also are ossible unredictable direct transitions from one of coarse-grained laminar hase to other coarse-grained laminar hase. It is necessary to notice that here, unlike described in [, 6, 8], generalised intermittency consist of two coarse-grained laminar hases. At last we will describe scenarios of transitions regular regime chaos and chaos other chaos at following values of arameters of system (): A =,; B =,5; = ; μ = 4,5; =. As bifurcation arameter we will choose arameter α. Comuter simulation of the system () revealed that at values,54 < α <, in system exists a single-stroke stable limit cycle. At α =,54 existing limit cycle loses stability,5,5,5 and the system, as a result of bifurcation eimark, uasi-eriodic attractor arises with the signature sectrum LCE,,,,. Presence in sectrum ЛХП of two zero exonents testifies about uasieriodicity of attractor []. In fig. 4, b is shown the rojection of a hase ortrait of such uaa b с,5,5,5 5 5,5,5 d e f р g h Fig. 4. Projections of hase ortraits of limit cycles at α =, 5 (a), α =,5 (c); a uasieriodic attractor α =, (b); a chaotic attractor at α =, 8 (d ); small α =, 88 (e) and large α =, 4 ( f, g, h) hyerchaotic attractors

7 ІНФОРМАЦІЙНІ ТЕХНОЛОГІЇ, СИСТЕМНИЙ АНАЛІЗ ТА КЕРУВАННЯ 5 sieriodic attractor built at α =,. Trajectories of uasieriodic attractor everywhere dense coat toroidal-shaed surface of attractor and are always returned in arbitrary small neighbourhood of attractor with some almost eriod. At the further decrease of value of arameter, namely at α =,5 there occurs destruction of torus. Trajectory of system carry out finite number of turnovers in torus and becomes closed, thus in system arise a resonance limit cycle in torus (Fig. 4, c) []. At α =, 8 5 a resonance limit cycle is disaear and in the system () arises a chaotic attractor. The rojection of the hase ortrait of the chaotic attractor of such tye built at α =, 8 is shown in Fig. 4, d. In this case the scenario of transition to chaos through destroy uasieriodic regimes [] is realised. At the value of arameter α =, 8 8 bifurcation chaos-hyerchaos takes lace in system (). As result of this bifurcation arise the hyer-chaotic attractor (Fig. 4, e) with the signature of sectrum LCE +, +,,,. Phase ortrait of an originating hyerchaotic attractor is similar to hase ortrait of adisaearing chaotic attractor. However hyer-chaotic attractor has two ositive Lyaunov's exonents, therefore close hase trajectories of a hyer-chaotic attractor diverge in two directions of a hase sace. While close hase trajectories of a chaotic attractor diverge only in one direction of a hase sace. Extremely interesting bifurcation occurs in the system at α =, 4. Existing at α <, 4 hyer-chaotic attractor disaears and in the system aears hyer-chaotic attractor of an entirely different tye. The different rojections of the hase ortrait of the hyer- chaotic attractor of this tye, built at α =, 4, are shown in Fig. 4, f, g. This attractor is localized in a much larger volume of hase sace than the attractor that shown in Fig. 4, e. In Fig. 4, g is shown an enlarged fragment of the central art of Fig. 4, f. Therefore, the attractor which shown in Fig. 4, e, is called called small, and in Fig. 4, f, g is called big. In [, 6, 8] has been described a scenario of generalized intermittency in transitions chaos chaos. A careful study of Fig. 4, f, g, h shows that in this case is realized scenario of generalized intermittency for transition hyer-chaos hyerchaos. Coarse-grained laminar hase of such intermittency are motion trajectories of big hyerchaotic attractor in the neighborhood of disaeared small hyer-chaotic attractor. Accordingly, the turbulent hase of generalized intermittency is the dearture of trajectories in remote areas of hase sace. It should be noted, that the realization of the scenario of generalized intermittency for transition hyer-chaos hyer-chaos for the system () is detected for the first time. Conclusions Thus in the work are identified and described the new scenarios of transition to deterministic chaos in some, imortant for alications, nonideal dynamical system. In further research is lanned detection of such new scenarios for other tyes of dynamical systems. Also great interest calls the roblem of finding the boundaries of the basins of attraction of considered regular and chaotic attractors. References. S.P. Kouznetsov, Dynamic chaos. Russia, Moscow: Physmatlit, 6, 56.. T.S. Krasnoolskaya and A.Yu. Shvets, Regular and chaotical dynamics of systems with limited excitation. Russia, Moscow: R&C Dynamics, 8, 8.. V.O. Kononenko, Vibrating System with a Limited Powersuly. Great Britain, London: Iliffe, 969, I.O. Lukovsky, Mathematical models of nonlinear dynamics solid bodies with a liuid. Ukraine, Kyiv: aukova Dumka,, R.A. Ibrahim, Liuid Sloshing Dynamics: Theory and Alications. Cambridge University Press, 5, 97. Рекомендована Радою фізико-математичного факультету НТУУ КПІ 6. T.S. Krasnoolskaya and A.Yu. Shvets, Dynamical chaos for a limited ower suly oscillations in cylindrical tanks, J. Sound Vibr., vol.,. 5 55, A.Yu. Shvets and V.O. Sirenko, Peculiarities of Transition to Chaos in onideal Hydrodynamics Systems, CMSIM, no.,.,. 8. A.Yu. Shvets, Generalized scenario of intermittency in the transition to deterministic chaos, Reorts of AS of Ukraine. Math, Science, Engineering Sciences, no. 5,. 5,. Надійшла до редакції 6 жовтня 4 року