Cosymmetry, degeneration of solutions of operator equations, and onset of a filtration convection

Yudovich, V. I.

doi:10.1007/bf01142654

Cited by 51 publications

(61 citation statements)

References 2 publications

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“…This fact has 30 been experimentally confirmed in [19], and independently reconfirmed analytically in [20]. As shown in [21], the reason for this unusual first bifurcation transition (loss of stability by the state of rest) is cosymmetry, the theory of which was developed in [21,22,23].…”

Section: Introductionmentioning

confidence: 52%

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Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

Говорухин¹,

Shevchenko²

2013

Fluid Dyn

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We study convection in a two-dimensional container of porous material saturated with fluid and heated from below. This problem belongs to the class of dynamical systems with nontrivial cosymmetry. The cosymmetry gives rise to a hidden parameter in the system and continuous families of infinitely many equilibria, and leads to non-trivial bifurcations. In this article we present our numerical studies that demonstrate nonlinear phenomena resulting from the existence of cosymmetry. We give a comprehensive picture of different bifurcations which occur in cosymmetric dynamical systems and in the convection problem. It includes internal and external (as an invariant set) bifurcations of one-parameter families of equilibria, as well as bifurcations leading to periodic, quasiperiodic and chaotic behaviour. The existence of infinite number of stable steady-state regimes begs the important question as to which of them can realize in physical experiments. In the paper, this question (known as the selection problem) is studied in detail. In particular, we show that the selection scenarios strongly depend on the initial temperature distribution of the fluid. method, and numerical methods used to analyse cosymmetric systems are also described.

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Section: Introductionmentioning

confidence: 52%

“…The cosymmetry of a vector field F on R n (or on a Riemann manifold), or 35 of the autonomous differential equationu = F u is defined as a vector field L which is orthogonal to F in each point, so that (F u, Lu) = 0 for all u [21]. The equilibrium u 0 : F u 0 = 0 is called a noncosymmetric equilibrium if Lu 0 = 0.…”

Section: Introductionmentioning

confidence: 99%

Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

Говорухин¹,

Shevchenko²

2013

Fluid Dyn

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“…Cosymmetry concept was introduced by Yudovich [1,2] and some interesting phenomena were found for both dynamical systems possessing the cosymmetry property.Particularly, it was shown that cosymmetry may be a reason for the existence of the continuous family of regimes of the same type.I f asymmetry group produces a continuous family of identical regimes then it implies the identical spectrum for all points on the family.T he stability spectrum for the cosymmetric system depends on the location of a point, and the family may be formed by stable and unstable regimes.…”

Section: Introductionmentioning

confidence: 99%

“…Following [1], a cosymmetry for a differential equationu = F (u) in a Hilbert space is the operator L(u) which is orthogonal to F at each point of the phase space i.e (F (u),L(u))=0,u∈ R n with an inner product (·, ·). If the equilibrium u 0 is noncosymmetric, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Filtration-Convection Problem: Spectral-Difference Method and Preservation of Cosymmetry

Kantur

Tsybulin

2002

Lecture Notes in Computer Science

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Abstract. For the problem of filtration of viscous fluid in porous medium it was observed that a number of one-parameter families of convective states with the spectrum, which varies along the family. It was shown by V. Yudovich that these families cannot be an orbit of an operation of any symmetry group and as a result the theory of cosymmetry was derived. The combined spectral and finite-difference approach to the planar problem of filtration-convection in porous media with Darcy law is described. The special approximation of nonlinear terms is derived to preserve cosymmetry. The computation of stationary regime transformations is carried out when filtration Rayleigh number varies.

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The cosymmetric bifurcations in the planar filtrational convection problem: numerical results

Говорухин¹,

Shevchenko

2007

Proc Appl Math and Mech

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Here we present survey of our numerical results concerning possible bifurcation in cosymmetric filtrational convection problem. We study bifurcations connected with existence of cycles of equilibria and scenarios for onset of unsteady regimes.Theory of cosymmetry was introduced by V.I. Yudovich to explain a phenomenon of existence of one-parameter family of stationary flows in the planar filtrational convection problem. It was shown [1] that the existence of such a family is due to the cosymmetry of the corresponding differential equations and the branching of cycle of equilibria (CE) is a typical bifurcation in cosymmetric dynamical system [2]. In [3,4,5] and in many other works of these authors the theory of cosymmetric bifurcations was developed.We Here, ψ(x, y, t) is the stream function, θ(x, y, t) is the deviation of the temperature from the (vertical) equilibrium profile, x and y are the Cartesian coordinates in the plane, t is time, and the subscript of a function denotes the corresponding derivative. The bifurcation parameter λ is the seepage Rayleigh number. The problem (1) has cosymmetry and a cycle of stable steadystate regimes branches out from the state of rest as a result of loss of stability. We investigate the problem (1) numerically by Galerkin method (see details in [7,9,10]). The special numerical technique for calculation of cosymmetric CE is described in [8]. The qualitative repetition of bifurcation and consistency of bifurcation parameter values was established by investigation of different dimensions Galerkin's models. We found practically all types of bifurcations studied theoretically in [1]-[5] and some new ones. In Fig. 1 we have plotted the critical curves corresponding to the different bifurcations as the vessel side ratio varies. The specifics of most of the bifurcations mentioned above are attributable to the cosymmetry of the problem considered. Bifurcations of the families of stationary regimesFirstly we study onset of instability on the stable CE (see [6,7,9]). From Fig. 1 we can see that the first loss of stability on a single-parameter family of steady-state convective regimes differs in nature depending on the vessel geometry and may be both monotonic and oscillating and the instability can arise at two, four, six, or eight points. The discontinuities in the critical curves are due to the delay in loss of stability with respect to the parameter λ associated with the bifurcations of the families described below. After the onset of instability and the subsequent increase in the parameter λ the number of unstable regimes increases forming arcs on the family. A further increase in λ can lead to the formation of unstable regimes (both oscillatory and monotonic) on new segments of the family.The series of bifurcations of one-parameter families of equilibria was found out [8,9]: birth of a new family from already existing, intersection and join of equilibrium curve, origin of a cycle of equilibria 'from an air' etc. The bifurcations of families are connected with different equilibr...

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Cosymmetry, degeneration of solutions of operator equations, and onset of a filtration convection

Cited by 51 publications

References 2 publications

Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

Filtration-Convection Problem: Spectral-Difference Method and Preservation of Cosymmetry

The cosymmetric bifurcations in the planar filtrational convection problem: numerical results

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