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Jose Manuel Rodriguez Sanjurjo - One of the best experts on this subject based on the ideXlab platform.
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Unstable Manifold conley index and fixed points of flows
Journal of Mathematical Analysis and Applications, 2014Co-Authors: Hector Barge, Jose Manuel Rodriguez SanjurjoAbstract:We study dynamical and topological properties of the Unstable Manifold of isolated invariant compacta of flows. We show that some parts of the Unstable Manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the Unstable Manifold and, as a consequence, a relation between the Brouwer degree and the Unstable Manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.
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Unstable Manifold, Conley index and xed points of ows I
2014Co-Authors: Hector Barge, Jose Manuel Rodriguez SanjurjoAbstract:We study dynamical and topological properties of the Unstable Manifold of isolated invariant compacta of flows. We show that some parts of the Unstable Manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the Unstable Manifold and, as a consequence, a relation between the Brouwer degree and the Unstable Manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.
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shape and conley index of attractors and isolated invariant sets
2007Co-Authors: Jose Manuel Rodriguez SanjurjoAbstract:This article is an exposition of several results concerning the theory of continuous dynamical systems, in which Topology plays a key role. We study homological and homotopical properties of attractors and isolated invariant compacta as well as properties of their Unstable Manifolds endowed with the intrinsic topology. We also provide a dynamical framework to express properties which are studied in Topology under the name of Hopf duality. Finally we see how the use of the intrinsic topology makes it possible to calculate the Conley-Zehnder equations of a Morse decomposition of an isolated invariant compactum, provided we have enough information about its Unstable Manifold.
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Morse equations and Unstable Manifolds of isolated invariant sets
Nonlinearity, 2003Co-Authors: Jose Manuel Rodriguez SanjurjoAbstract:We describe a new way of obtaining the Morse equations of a Morse decomposition of an isolated invariant set. This is achieved through a filtration of truncated Unstable Manifolds associated with the decomposition. The results in the paper make it possible to calculate the Morse equations (and also the Conley index) in many interesting situations without using index pairs. We also study the intrinsic topology of the Unstable Manifold and obtain new duality properties of the cohomological Conley index.
Julien Barré - One of the best experts on this subject based on the ideXlab platform.
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vlasov fokker planck equation stochastic stability of resonances and Unstable Manifold expansion
Nonlinearity, 2018Co-Authors: Julien Barré, David MétivierAbstract:We investigate the dynamics close to a homogeneous stationary state of the Vlasov equation in one dimension, in presence of a small dissipation modeled by a Fokker–Planck operator. When the stationary state is stable, we show the stochastic stability of Landau poles. When the stationary state is Unstable, depending on the relative size of the dissipation and the Unstable eigenvalue, we find three distinct nonlinear regimes: for a very small dissipation, the system behaves as the pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. The nonlinear analysis relies on an Unstable Manifold expansion, performed using Bargmann representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.
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Unstable Manifold expansion for Vlasov-Fokker-Planck equation
arXiv: Mathematical Physics, 2017Co-Authors: Julien Barré, David MétivierAbstract:We investigate the bifurcation of a homogeneous stationary state of Vlasov-Newton equation in one dimension, in presence of a small dissipation mod-eled by a Fokker-Planck operator. Depending on the relative size of the dissipation and the Unstable eigenvalue, we find three different regimes: for a very small dissipa-tion, the system behaves as a pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. This work relies on an Unstable Manifold expansion, performed using Bargman representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.
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vlasov fokker planck equation stochastic stability of resonances and Unstable Manifold expansion
arXiv: Mathematical Physics, 2017Co-Authors: Julien Barré, David MétivierAbstract:We investigate the dynamics close to a homogeneous stationary state of Vlasov equation in one dimension, in presence of a small dissipation modeled by a Fokker-Planck operator. When the stationary state is stable, we show the stochastic stability of Landau poles. When the stationary state is Unstable, depending on the relative size of the dissipation and the Unstable eigenvalue, we find three distinct nonlinear regimes: for a very small dissipation, the system behaves as a pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. The non linear analysis relies on an Unstable Manifold expansion, performed using Bargmann representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.
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Bifurcations and singularities for coupled oscillators with inertia and frustration
2016Co-Authors: Julien Barré, D MétivierAbstract:We prove that any non zero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous, or from discontinuous to continuous. This result is obtained through an Unstable Manifold expansion in the spirit of J.D. Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.
Yoshiyuki Y Yamaguchi - One of the best experts on this subject based on the ideXlab platform.
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Trapping scaling for bifurcations in Vlasov systems
Physical Review E : Statistical Nonlinear and Soft Matter Physics, 2016Co-Authors: J Barré, D Métivier, Yoshiyuki Y YamaguchiAbstract:We study non oscillating bifurcations of non homogeneous steady states of the Vlasov equation, a situation occurring in galactic models, or for Bernstein-Greene-Kruskal modes in plasma physics. Through an Unstable Manifold expansion, we show that in one spatial dimension the dynamics is very sensitive to the initial perturbation: the instability may saturate at small amplitude-generalizing the " trapping scaling " of plasma physics-or may grow to produce a large scale modification of the system. Furthermore, resonances are strongly suppressed, leading to different phenomena with respect to the homogeneous case. These analytical findings are illustrated and extended by direct numerical simulations with a cosine interaction potential.
David Métivier - One of the best experts on this subject based on the ideXlab platform.
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Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results
Journal of Statistical Physics, 2019Co-Authors: David Métivier, Shamik GuptaAbstract:In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an Unstable Manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott–Antonsen (OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the Unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the Unstable Manifold derived from the kinetic equation has the same form as the OA Manifold, implying thereby that the OA-ansatz form follows also as a result of the Unstable Manifold expansion. We illustrate our results by showing how delay can affect dramatically the bifurcation of a bimodal distribution.
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vlasov fokker planck equation stochastic stability of resonances and Unstable Manifold expansion
Nonlinearity, 2018Co-Authors: Julien Barré, David MétivierAbstract:We investigate the dynamics close to a homogeneous stationary state of the Vlasov equation in one dimension, in presence of a small dissipation modeled by a Fokker–Planck operator. When the stationary state is stable, we show the stochastic stability of Landau poles. When the stationary state is Unstable, depending on the relative size of the dissipation and the Unstable eigenvalue, we find three distinct nonlinear regimes: for a very small dissipation, the system behaves as the pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. The nonlinear analysis relies on an Unstable Manifold expansion, performed using Bargmann representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.
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Unstable Manifold expansion for Vlasov-Fokker-Planck equation
arXiv: Mathematical Physics, 2017Co-Authors: Julien Barré, David MétivierAbstract:We investigate the bifurcation of a homogeneous stationary state of Vlasov-Newton equation in one dimension, in presence of a small dissipation mod-eled by a Fokker-Planck operator. Depending on the relative size of the dissipation and the Unstable eigenvalue, we find three different regimes: for a very small dissipa-tion, the system behaves as a pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. This work relies on an Unstable Manifold expansion, performed using Bargman representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.
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vlasov fokker planck equation stochastic stability of resonances and Unstable Manifold expansion
arXiv: Mathematical Physics, 2017Co-Authors: Julien Barré, David MétivierAbstract:We investigate the dynamics close to a homogeneous stationary state of Vlasov equation in one dimension, in presence of a small dissipation modeled by a Fokker-Planck operator. When the stationary state is stable, we show the stochastic stability of Landau poles. When the stationary state is Unstable, depending on the relative size of the dissipation and the Unstable eigenvalue, we find three distinct nonlinear regimes: for a very small dissipation, the system behaves as a pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. The non linear analysis relies on an Unstable Manifold expansion, performed using Bargmann representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.
Brian Dennis - One of the best experts on this subject based on the ideXlab platform.
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moving toward an Unstable equilibrium saddle nodes in population systems
Journal of Animal Ecology, 1998Co-Authors: Brian DennisAbstract:1. We identify an Unstable equilibrium with a two-dimensional stable Manifold and a one-dimensional Unstable Manifold in a three-state variable (larva, pupa, adult) insect population growth model. 2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the Unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not. 3. Using two quantitative indices, we examine time series from a Tribolium experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment.