On the peculiarities of stochastic invariant solutions of a hydrodynamic system accounting for non-local effects. (English) Zbl 0952.34045

Shkil, Mykola (ed.) et al., Symmetry in nonlinear mathematical physics. Proceedings of the second international conference, Kyiv, Ukraine, July 7-13, 1997. Memorial Prof. W. Fushchych conference. Vol. 2. Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine. 409-417 (1997).
The authors investigate the following hydrodynamic-type system of partial differential equations: \[ \begin{aligned} \rho d u/d t+\partial p/\partial x&=\gamma\rho,\\ d\rho/d t +\rho \partial u/\partial x&=0,\\ \tau d p/d t-\chi d\rho/d t&=\kappa \rho-p -h\{d^{2}p/d t^{2}+(2/\rho) (d\rho/d t)^{2}-d^{2}\rho/d t^{2}\},\end{aligned}\tag{1} \] where \(\rho,p,u\) are unknown functions and \(\tau,\kappa,\chi, h\) are some constants. Namely, \(\tau\) is relaxation time, \(\sqrt\kappa\) is proportional to the equilibrium sound velocity, \(h\) is the coefficient of structural relaxation, and \(\chi\) is the volume viscosity coefficient. This system describes pulse load afteraction in active and relaxing media.
The authors further use the invariance of (1) under the Lie symmetry operators \(\partial/\partial t\), \(\partial/\partial x\), \(t \partial/\partial x+\partial/\partial u\), \(\rho \partial/\partial \rho+p \partial/\partial p\), and reduce (1) to a system of ordinary differential equations by means of the ansatz \[ u=D+U(\omega), \quad \rho=\exp(\xi t+S(\omega)),\quad p=\rho Z(\omega), \] where \(\xi\) and \(D\) are arbitrary constants, and \(\omega=x-D t\), built on invariants of the operator \(\partial/\partial t+\partial/\partial x+\xi(\rho \partial/\partial\rho +p \partial/\partial p)\).
In the sequel the authors show that for physically admissible values of parameters thus obtained system of ODEs has only one critical point and study the behavior of the system in question in the vicinity of this point, using both qualitative methods and numerical simulations. In function of values of parameters, this point turns out to be a stable or unstable focus, and the change of its stability is caused by sub- or supercritical Hopf bifurcation. It is further shown that for a certain range of values of parameters the system of ODEs under study exhibits a stochastic behavior and possesses a strange attractor, whose fine structure is studied with usage of the Poincaré sections technique.
As a result of their studies, the authors conclude that, in function of values of parameters, the system of ODEs in question exhibits a variety of regimes, from multiperiodic to stochastic one, and its solutions can be used for the construction of intermediary asymptotics for the solutions to Cauchy and boundary value problems for (1).
For the entire collection see [Zbl 0882.00039].


34D45 Attractors of solutions to ordinary differential equations
76D06 Statistical solutions of Navier-Stokes and related equations
35Q72 Other PDE from mechanics (MSC2000)
34C23 Bifurcation theory for ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
35Q30 Navier-Stokes equations