Some unsolved problems in the theory of differential equations and mathematical physics

Arnold, Vladimir I.; Vishik, M. I.; Ilyashenko, Yu.; Kalashnikov, A. S.; Кондратьев, В А; Kruzhkov, S. N.; Landis, Eugene M.; Millionshchikov, V. M.; Oleinik, O A; Filippov, A. F.; Shubin, M. A.

doi:10.1070/rm1989v044n04abeh002139

Cited by 42 publications

(35 citation statements)

References 9 publications

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“…Therefore the cyclicity of an open period annulus is bounded by the maximal number of zeros which an Abelian integral I(h) on a suitable interval can have (this holds true even for the closed period annulus [34]). Based on this Arnold [4, p. 313] formulated a weakened (or rather infinitesimal) version of the 16th Hilbert problem, which asks for the maximal number of zeros of Abelian integrals of the form (45) (see also [5]). It should be stressed, however, that if the polynomial H(x, y) is not generic, or the degree of the polynomial oneform ω is strictly greater than deg(H ) − 1, the problem of finding the limit cycles of d H + ε ω = 0 is not equivalent to a problem on the zeros of Abelian integrals.…”

Section: Discussionmentioning

confidence: 99%

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The infinitesimal 16th Hilbert problem in the quadratic case

Gavrilov¹

2001

Invent. math.

103

102

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Section: Discussionmentioning

confidence: 99%

“…The precise description of l i ,l ∞ , ∆ is given in Corollary 3,5,6. To prove the above Theorem we shall study each of the sets l ab i , l ab ∞ , ∆ separately. …”

Section: The Bifurcation Set Of the Zeros Of The Abelian Integral D 2mentioning

confidence: 99%

The infinitesimal 16th Hilbert problem in the quadratic case

Gavrilov¹

2001

Invent. math.

103

102

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“…Since v 2 between v 3 , v 4 does not block v 4 ; v 3 , blocking conditions (2.7), (2.8) must both be violated for the triple v 2 , v 3 , v 4 . Therefore…”

Section: Sturm Attractors Hamiltonian Paths and Sturm Permutationsmentioning

confidence: 98%

“…Following Arnol'd [4], we call π ∈ S N a meander permutation, if the following property holds. Whenever π −1 (j ) is between π −1 (j ) and π −1 (j + 1), and j, j have the same parity (−1) j = (−1) j , then π −1 (j + 1) is also between π −1 (j ) and π −1 (j + 1).…”

Section: Sturm Attractors Hamiltonian Paths and Sturm Permutationsmentioning

confidence: 99%

Connectivity and design of planar global attractors of Sturm type. II: Connection graphs

Fiedler

Rocha

2008

Journal of Differential Equations

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Based on a Morse-Smale structure we study planar global attractors A f of the scalar reaction-advectiondiffusion equation u t = u xx + f (x, u, u x ) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f , and hyperbolicity of equilibria. We call A f Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type.The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits. The unique heteroclinic orbits between adjacent Morse levels define a plane graph C f which we call the connection graph. Its 1-skeleton C 1 f consists of the unstable manifolds (separatrices) of the index-1 Morse saddles.We present two results which completely characterize the connection graphs C f and their 1-skeletons C 1 f in purely graph theoretical terms. Connection graphs are characterized by the existence of pairs of Hamiltonian paths with certain chiral restrictions on face passages. Their 1-skeletons are characterized by the existence of cycle-free orientations with certain restrictions on their criticality. Such orientations are called bipolar in [H. de Fraysseix, P.O. de Mendez, P. Rosenstiehl, Bipolar orientations revisited, Discrete Appl. Math. 56 (1995) 157-179]. In [B. Fiedler, C. Rocha, Connectivity and design of planar global attractors of Sturm type. I: Orientations and Hamiltonian paths, Crelle J. Reine Angew. Math. ( 2007), in press] we have shown the equivalence of the two characterizations. Moreover we have established that connection graphs of Sturm attractors indeed satisfy the required properties. In the present paper we show, conversely, how to design a planar Sturm attractor with prescribed plane connection graph or 1-skeleton of the required properties. In [B. Fiedler, C. Rocha, Connectivity and design of planar global attractors of Sturm type. III: Small and Platonic exam-* Corresponding author. publication] we describe all planar Sturm attractors with up to 11 equilibria. We also design planar Sturm attractors with prescribed Platonic 1-skeletons.

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“…For this reason the problem of finding the zeros of I(h) in terms of the degrees of H, f, g was called by Arnold [5, p. 313] the "weakened 16th Hilbert problem" (compare to Hilbert [19]; see also Arnold [6], [7], [8]). Note that the level sets {H = h} will contain in general several continuous families of ovals which need be considered separately.…”

Section: Take Real Polynomials H F G ∈ R[x Y] and Let δ(H) ⊂ {(X mentioning

confidence: 99%