[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA

SAIKI, Yoshitaka

YAMADA, Michio

Kyoto University Research Information Repository

Title<Poster Presentation 16>Relations between statistical valuesalong unstable periodic orbits in differential equation systemsAuthor(s)SAIKI, Yoshitaka; YAMADA, MichioCitationIUTAM Symposium on 50 Years of Chaos : Applied andTheoretical (2011): 148-149Issue Date2011-12URL http://hdl.handle.net/2433/163087RightType BookTextversionpublisherKyoto UniversityP16 IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical Kyoto, Japan, November 28- December 2, 2011 148 Relations between statistical values along unstable periodic orbits in differential equation systems Yoshitaka SAIKI1 , Michio YAMADA2 1 Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan, saiki@math.sci.hokudai.ac.jp 2 Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan, yamada@kurims.kyoto-u.ac.jp Introduction A chaotic dynamical system includes an infinite number of unstable periodic orbits in general. Therefore, in order to capture statistical values of chaos weighted average along a series of unstable periodic orbits (UPOs), which is sometimes called a cycle expansion, has been proposed and used to study low dimensional systems [1, 2, 3]. Recently, in some turbulence systems in fluid dynamics, it has been shown that even only a few UPOs with relatively low periods can capture mean properties of chaotic motions [4, 5]. For example, for the turbulent Couette flow of rather low Reynolds number of the full Navier-Stokes system, Kawahara and Kida [4] obtained a remarkable agreement of an averaged velocity profile along a single UPO with that along a chaotic orbit in phase space of turbulent Couette flows. In the above studies, it seems that only a few UPOs with relatively low periods are enough to capture some mean properties of a chaotic solution. However, on the other hand, it appears that an UPO with longer period gives a better approximation to the statistical properties of chaotic solutions. So we may have a question why in the above systems even a small number of UPOs with rather low periods can give a remarkably good approximation to the various chaotic mean values. Some studies have been concerned with this problem. For example, Saiki and Yamada [6] employed three chaotic systems described by low dimensional ODEs and investigated the difference between the average of a dynamical quantity along an UPO and that along a chaotic orbit, especially with an attention focused on the dependence of the variance of the averaged value on the period of the UPOs. It was found that for each chaotic system the distributions of a time average of a dynamical variable along UPOs with lower and higher periods are similar to each other and the variance of the distribution is small, in contrast with that along chaotic segments. In this paper, we employ two chaotic systems described by differential equation systems and investigate the relation between the averages of dynamical quantities along UPOs detected numerically. Relation between statistical values along UPOs Lorenz system UPOs in the Lorenz system (dxjdt = a(y - x), dyjdt = rx - y - xz, dzjdt = xy - bz) with the classical parameter values (a= 10, b = 8/3, r = 28) have been extensively studied. Although the Lorenz system is not uniformly hyperbolic, it is recently proved by the aid of numerical calculation with guaranteed accuracy that the Lorenz attractor is chaotic and includes an infinite number of UPOs densely. Here we focus our attention to the relations between time averaged values of dynamical variables along UPOs of the Lorenz system. We found more than 1000 UPOs of the periods from 1.558 through 16.445, corresponding respectively from 2 through 23 rotations around a wing of the Lorenz attractor. The number of rotations corresponds to the period (PERIOD N) of the Poincare map defined by the Poincare section z = r - 1, dz / dt > 0. Figure 1 shows the relation of time average of z ( (z)) and the variance of z (Var of z) along UPOs (N = 11), along segments of chaotic orbits (chaotic segments) (N = 1, · · ·, 11) and along a long chaotic orbit. It shows that whereas the chaotic segments give scattering points in ( (z), Var of z)-plane, the UPOs give an almost straight line. In addition, a chaotic orbit gives a point on the line. From Figure 2 we can see that also in (-\,(z) )-plane the UPOs give points on a straight line, where ,\ represents the Lyapunov exponent. Figures 1 and 2 show that if we choose a special UPO whose Lyapunov exponent approximates that of a long chaotic orbit, the UPO also gives various macroscopic statistical quantities of chaos, even if the period is not large enough. This result implies that there are UPOs which can capture various types of macroscopic statistical quantities of chaos. K uramoto-Sivashinsky system The Kuramoto-Sivashinsky (KS) system can be written as Ut = -Uxx- VUxxxx- (u2 )x, where the "viscosity" control parameter v is fixed as 0.02991. We assume periodic boundary conditions u(x, t) = u(x + 27r, t) and through a symmetry reduction u(x, t) = -(i/2) ~ak(t)eikx we obtain the equations in Fourier space [7]: -1 k-1 N ak = (k 2 - vk4 )ak + ~( 2: a_mak-m - 2: amak-m + 2: amam-k), k = 1, ... , 16. m=k-N m=l m=k+1 IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical Kyoto, Japan, November 28 - December 2, 2011 N '+-0 1.. ro > 100 80 60 40 20 o~~~--~~~--21 22 23 24 25 26 27 <z> A N v L~ I I Llo f i I I I LJ f I n.11.____.___._1 ___.__1 _.__I _.__I -'----JI 1~.1 ~.10 ~.~ ~.~0 ~.~ ~.~0 1 1.~0 A A ct) ro v 0.15 ,-----.,...--1-.,...--1-.,---1-0.125 0.1 0.075 0.05 '---___._I _ __.__-----'------' -0.075 -0.07 -0.065 -0.06 -0.055 Figure 1: ((z),Var of z) for each UPO Figure 2: (A, (z)) for each Figure 3: ((a2 ), (a3 )) for each UPO (N = (N = 1, · · ·, 11) (green) and each chaotic UPO (N = 1, · · ·, 11) (red) 1, · · ·, 12) (red) and each chaotic segment (N = segment (N = 11) (red) with chaos aver- with chaos averages (blue) 8) (green) with chaos averages (blue) for the KS ages (blue) for the Lorenz system. for the Lorenz system. system. For the case of the KS system we obtain a similar result on the relation between time averaged values along UPOs. We see in Figure 3 that ( (a2), (a3)) for each UPO with PERIOD 8 (Poincare map is defined by the section a1 = 0, dad dt > 0) is on a single almost straight line, whereas those along chaotic segments scatter in the plane. We should also remark that averages along a chaotic orbit are on the line. Concluding Remarks By employing two chaotic systems described by differential equation systems, we study relations between time averaged values of some variables along UPOs, and find that there is almost a one to one and linear corre-spondence. In addition, statistical values along a long chaotic orbit are on the same constraint as those along UPOs. This implies that UPO which approximates one of the statistical quantities along a long chaotic orbit approximates another statistical quantity, and suggests that longer UPOs of two systems can be approximable by some UPOs with low periods at least in the statistical sense. Moreover, we found in the Lorenz system that when the system possesses a tangency structure as a parameter changes, another class of UPOs appear outside the linear correspondence discussed above. References [1] T. Kai and K. Tomita, Statistical Mechanics of Deterministic Chaos, Prog. Theor. Phys. 64 (1980) 1532. [2] P. Cvitanovic, Invariant measurement of strange sets in terms of cycles, Phys. Rev. Lett. 61 (1988) 2729. [3] C. Grebogi, E. Ott and J. A. Yorke, Unstable periodic orbits and the dimensions of multifractal chaotic attractors, Phys. Rev. A 37 (1988) 1711. [4] G. Kawahara and S. Kida, Periodic motion embedded in plane Couette turbulence, J. Fluid Mech., 449 (2001) 291. [5] S. Kato and M. Yamada, Unstable periodic solutions embedded in a shell model turbulence, Phys. Rev. E 68 (2003) 025302(R). [6] Y. Saiki and M. Yamada, Time averaged properties of unstable periodic orbits and chaotic orbits in ordinary differential equation systems, Phys. Rev. E 38 (2009) 015201(R). [7] E. L. Rempel and A. C.-1. Chian, Intermittency induced by attractor-merging crisis in the Kuramoto-Shivashinsky equation, Phys. Rev. E 71 (2005) 016203. 149 

<Poster Presentation 16>Relations between statistical values along unstable periodic orbits in differential equation systems

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<Poster Presentation 16>Relations between statistical values along unstable periodic orbits in differential equation systems

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