On Structural Stability

Peixoto, M. M.

doi:10.2307/1970100

Cited by 133 publications

(64 citation statements)

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“…Also Poincaré [89], Birkhoff [16], and M. Morse [68] all had some parts of this global picture. Since this earlier work, Andronov and Pontrjagin [6], Elsgolts [30 ], Peixoto [83], Reeb [94], and Thorn [124] among others had made contributions toward the picture given in this section.…”

Section: Figurementioning

confidence: 99%

Differentiable dynamical systems

Smale¹

1967

Bull. Amer. Math. Soc.

2,943

1,301

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1.1. Introduction to conjugacy problems for diffeomorphisms. This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G->Diff(M) such that the induced map GXM->M is differentiable. Here Diff(M) is the group of all diffeomorphisms of Mand a diffeomorphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C 00 or C r point of view. All manifolds maps, etc. will be differentiable (C r , l^r^co) unless stated otherwise.In the beginning we will be restricted to the discrete case, G = Z. Here Z denotes the integers, Z + the positive integers. By taking a generator ƒ G Diff(ikT), this amounts to studying diffeomorphisms on a manifold from the point of view of orbit structure. The orbit of %(EM, relative to/, is the subset {f m (x) \m^Z} of M or else the map Z->M which sends m into f m (x). The finite orbits are called periodic orbits and their points, periodic points. Thus xÇ£M is a periodic point if f m (x)=x for some w£Z + , Here m is called a period of x and if m = l, x is Si fixed point. Our problem is to study the global orbit structure, i.e,, all of the orbits on M.The main motivation for this problem comes from ordinary differential equations, which essentially corresponds to G~R> R the reals acting on M. There are two reasons for this leading to the diffeomorphism problem. One is that certain differential equations have cross-sections (see, e.g., [114]) and in this case the qualitative study of the differential equation reduces to the study of an associated diffeomorphism of the cross-section. This is the reason why Poincaré [90] and Birkhoff [19] studied diffeomorphisms of surfaces, I believe there is a second and more important reason for studying the diffeomorphism problem (besides its great natural beauty). That is, the same phenomena and problems of the qualitative theory of ordinary differential equations are present in their simplest form in the diffeomorphism problem. Having first found theorems in the dif-

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Section: Figurementioning

confidence: 99%

Differentiable dynamical systems

Smale¹

1967

Bull. Amer. Math. Soc.

2,943

1,301

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“…Dynamical Systems not having Morse-Smale structure in 2-D are topologically/structurally unstable and will not occur in nature, as any small perturbation will "destroy" such systems (see, e.g. Hornig and Schindler, 1996;Bruce andGiblin, 1992 or Reitmann, 1990; and for a mathematical proof Peixoto, 1959). Solutions, depending on singular perturbation theory, are therefore exceptions and thus, in a general sense, "unphysical".…”

Section: The Topological and Geometrical Structure Of The Magnetic Fieldmentioning

confidence: 99%

Topological skeleton of the 2-D slightly non-ideal MHD system close to X-type magnetic null points – an analysis of the general solution for the generic case

2012

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Abstract. The appearance of eruptive space plasma processes, e.g. in eruptive flares as observed in the solar atmosphere, is usually assumed to be caused by magnetic reconnection, often connected with singular points of the magnetic field.We are interested in the general relation between the eigenvalues of the Jacobians of the plasma velocity and the magnetic field and their relation to the shape of a spatially varying, localized non-idealness or resistivity, i.e. we are searching for the general solution. We perform a local analysis of almost all regular, generic, structurally stable non-ideal or resistive MHD solutions. Therefore we use Taylor expansions of the magnetic field, the velocity field and all other physical quantities, including the non-idealness, and with the method of comparison of coefficients, the non-linear resistive MHD system is solved analytically, locally in a close vicinity of the null point.We get a parameterised general solution that provides us with the topological and geometrical skeleton of the resistive MHD fields. These local solutions provide us with the "roots" of field and streamlines around the null points of basically all possible 2-D reconnection solutions. We prove mathematically that necessarily, the flow close to the magnetic X-point must also be of X-point type to guarantee positive dissipation of energy and annihilation of magnetic flux. We also prove that, if the non-idealness has only a onedimensional, sheet-like structure, only one separatrix line can be crossed by the plasma flow, similar to known reconnective annihilation solutions.

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“…It is known from Peixoto's work [35,36] that structurally stable flows on surfaces form a dense open set. A few years later Palis formulated the following conjecture for general dynamical systems defined on a closed manifold (flows, diffeomorphisms, or even more general transformations).…”

Section: On Palis' Conjecturementioning

confidence: 99%