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- Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic OscillatorsPublication . Pinto, Carla M.A.We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?
- Strange patterns in one ring of Chen oscillators coupled to a ‘buffer’ cellPublication . Pinto, Carla M.A.; Carvalho, Ana R.M.We study curious dynamical patterns appearing in networks of one ring of cells coupled to a ‘buffer’ cell. Depending on how the cells in the ring are coupled to the ‘buffer’ cell, the full network may have a nontrivial group of symmetries or a nontrivial group of ‘interior’ symmetries. This group is Z3 in the unidirectional case and D3 in the bidirectional case. We simulate the coupled cell systems associated with the networks and obtain steady states, rotating waves, quasiperiodic behavior, and chaos. The different patterns seem to arise through a sequence of Hopf, period-doubling, and period-halving bifurcations. The behavior of the systems with exact symmetry are similar to the ones with ‘interior’ symmetry. The network architecture appears to explain some features, whereas the properties of the Chen oscillator, used to model cells’ internal dynamics, may explain others. We use XPPAUT and MATLAB to numerically compute the relevant states.
- Exciting dynamical behavior in a network of two coupled rings of Chen oscillatorsPublication . Pinto, Carla M.A.We study exotic patterns appearing in a network of coupled Chen oscillators. Namely, we consider a network of two rings coupled through a “buffer” cell, with Z3×Z5 symmetry group. Numerical simulations of the network reveal steady states, rotating waves in one ring and quasiperiodic behavior in the other, and chaotic states in the two rings, to name a few. The different patterns seem to arise through a sequence of Hopf bifurcations, period-doubling, and halving-period bifurcations. The network architecture seems to explain certain observed features, such as equilibria and the rotating waves, whereas the properties of the chaotic oscillator may explain others, such as the quasiperiodic and chaotic states. We use XPPAUT and MATLAB to compute numerically the relevant states.