Instituto de Ciência e Tecnologia (ICT)

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Navegando Instituto de Ciência e Tecnologia (ICT) por Orientador(es) "Antoneli Jr, Fernando Martins [UNIFESP]"

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    Redes de sistemas dinâmicos acoplados com estrutura gradiente ou hamiltoniana
    (Universidade Federal de São Paulo (UNIFESP), 2020-03-03) Melo Junior, Antonio Edimar De [UNIFESP]; Antoneli Jr, Fernando Martins [UNIFESP]; Universidade Federal de São Paulo
    A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the digraph of a network, the “input sets”. A symmetry group becomes a groupoid and this formalism makes it possible to extend group theoretic methods to more general networks, and in particular it leads to a classification of patterns of synchrony in terms of the structure of the network. A network of dynamical systems is equipped with a canonical set of observables for the states of its individual nodes. Moreover, the form of the underlying ODE is constrained by the network topology and how those equations relate to each other. For the coupled systems associated with a network, there can be flow-invariant spaces (synchrony subspaces where some subsystems evolve synchronously), whose existence is independent of the systems equations and depends only on the network topology. Furthermore, any coupled system on the network, when restricted to such a synchrony subspace, determines a new coupled system associated with a smaller network (quotient). A regular network is a network with one kind of node and one kind of coupling. We show conditions for a codimension one bifurcation from a synchronous equilibrium in a regular network at linear level be isomorphic to a generalized eigenspace of the adjacency matrix of the network. We then focus on coupled cell systems in which individual cells are also gradient or Hamiltonian. In broad terms, we prove that only systems with bidirectionally coupled digraphs can be gradient or Hamiltonian. We characterize the conditions for the coupled systems property of being gradient or Hamiltonian to be preserved by the lift and quotient coupled systems. Aside from the topological criteria, we also study the linear theory of regular gradient (Hamiltonian) coupled cell systems. We then prove results on steady-state bifurcations and a version of the Equivariant Branching Lemma and the Equivariant Hopf Theorem. We illustrate a neural network given by two sets of neurons that are mutually coupled through either excitatory or inhibitory synapses, which is modelled by a coupled system exhibiting both gradient and Hamiltonian structures, and how periodic solutions from equilibrium appear in the Restricted Three Body Problem.