Geometric Theory of Dynamical Systems

Palis, Jacob; Melo, Welington de

doi:10.1007/978-1-4612-5703-5

Cited by 970 publications

(766 citation statements)

References 56 publications

(85 reference statements)

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“…The foliation (2.25) will be called stable, because it is generated by the stable manifold S O (that is associated with the stable eigenvalues of the matrix F in (2.23)). This terminology is in accordance with that given in Palis & de Melo [9]. (Note that a stable distribution is not just the distribution associated with a stable foliation (cf.…”

Section: The Local Disturbance Decoupling Problem With Stabili~supporting

confidence: 59%

3. The local disturbance decoupling problem with stability for nonlinear systems 1

1991

Local Disturbance Decoupling With Stability for Nonlinear Systems

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Abstract:In this paper the Disturbance Decoupling Problem with Stability (DDPS) for nonlinear systems is considered. The DDPS is the problem of finding a feedback such that after applying this feedback the disturbances do not influence the output anymore and x = 0 is an exponentially stable equilibrium point of the feedback system. For systems that can be decoupled by static state feedback it is possible to define (under fairly mild assumptions) a distribution A* which is the nonlinear analogue of the linear ;¢5~*, the largest stabilizable controlled invariant subspace in the kernel of the output mapping, and to prove that the DDPS is locally solvable if and only if the disturbance vector fields are contained in /~.

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Section: The Local Disturbance Decoupling Problem With Stabili~supporting

confidence: 59%

3. The local disturbance decoupling problem with stability for nonlinear systems 1

1991

Local Disturbance Decoupling With Stability for Nonlinear Systems

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“…Therefore, by application of perturbation theory the bifurcations SN 5 ± , H 5 ± , and HSN 5 ± in Theorem 5 persist for the map G 5 . By perturbation theory we mean the implicit function theorem, the theory of persistence of normally hyperbolic invariant manifolds [33,37], the theory of persistence of non-degenerate bifurcations [3,36,44,49,51,52], including quasi-periodic bifurcations [9,10,22,23], and KAM theory [2,3,9,10].…”

Section: The Cylindrical Coordinates Of Pmentioning

confidence: 99%

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Broer

Simó

Vitolo

2008

Physica D: Nonlinear Phenomena

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The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf-saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.

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“…For now on, we assume that λ > 0. In this case the hyperbolic equilibrium 0 has 2-dimensional stable and unstable manifolds W ± loc [20]. Since W ± loc are Lagrangian manifolds and project diffeomorphically to R 2 {x}, they are defined by C 4 generating functions s ± on a small ball U with centre 0 ∈ R 2 .…”

Section: Regularization Of Collisionsmentioning

confidence: 99%

Interaction of two charges in a uniform magnetic field: I. Planar problem

Pinheiro

MacKay

2006

Nonlinearity

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Abstract. The interaction of two charges moving in R 2 in a magnetic field B can be formulated as a Hamiltonian system with 4 degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with 2 degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non-integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion.

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Geometric Theory of Dynamical Systems

Cited by 970 publications

References 56 publications

3. The local disturbance decoupling problem with stability for nonlinear systems 1

3. The local disturbance decoupling problem with stability for nonlinear systems 1

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Interaction of two charges in a uniform magnetic field: I. Planar problem

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