Amplitude Equation of Higher-Dimensional Nikolaevskii Turbulence

Abstract

The scaling structure of higher-dimensional Nikolaevskii turbulence described bẏ φ(r, t) = −∇ 2 { − (∇ 2 + k 2 0 ) 2 }φ + (∇φ) 2 is investigated for small (> 0). This is done by extending the Matthews-Cox scaling in a one-dimenional system to higher dimensions. By deriving the amplitude equation for small in higher dimensions and carrying out the numerical integration of their amplitude equations for a two-dimensional system, it is shown that the same scaling structure found in the one-dimensional case exists also in two dimensions. * )

Non-linear Evolution of P-waves in Viscous–Elastic Granular Saturated Media

“…Tribelsky) The theory (Tribelsky and Tsuboi 1996;Tribelsky and Velarde 1996;Xi et al 2000) resulted in invariance of asymptotic oscillations to initial conditions but the instability was found that corresponded to special spatial-temporary chaos. This chaos (without critical Reynolds number) was observed in electro-hydrodynamic convection in nematic liquid crystals, phase transitions and in a three-variable reaction diffusion system, see references in Tribelsky (1997), Tribel'skii (1997, Fujisaka (2003), Tanaka and Kuramoto (2003), Tanaka (2003), that are linked with the combination of the instability interval and neutral stability of waves of zero wave number that is with the distinction feature of Fig. 6.…”

Instability of spatially periodic patterns due to a zero mode in the phase-field crystal equation

“…This is because the longrange repulsive interaction of monomer concentration deviation is not screened [21], hence the zero mode is always absolutely stable in this system. The systems that should be excluded include also the phenomena described by the Nikolaevskii equation [7,22]. The symmetries of that equation are quite different from those of the PFC equation, and its behavior is correspondingly quite different.…”

High-Dimensional Chaotic and Attractor Systems

“…Over the past decade or so, synchronization in chaotic oscillators [FY83,PC90] has received much attention because of its fundamental importance in nonlinear dynamics and potential applications to laser dynamics [DBO01], electronic circuits [KYR98], chemical and biological systems [ESH98], and secure communications [KP95]. Synchronization in chaotic oscillators is characterized by the loss of exponential instability in the transverse direction through interaction.…”

“…Synchronization in chaotic oscillators is characterized by the loss of exponential instability in the transverse direction through interaction. In coupled chaotic oscillators, it is known, various types of synchronization are possible to observe, among which are complete synchronization (CS) [FY83,PC90], phase synchronization (PS) [RPK96, ROH98], lag synchronization (LS) [RPK97] and generalized synchronization (GS) [KP96]. One of the noteworthy synchronization phenomena in this regard is PS which is defined by the phase locking between nonidentical chaotic oscillators whose amplitudes remain chaotic and uncorrelated with each other:…”