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- Computability of limit sets for two-dimensional flowsPublication . Graça, Daniel; Zhong, NingA classical theorem of Peixoto qualitatively characterizes, on the two-dimensional unit ball, the limit sets of structurally stable flows defined by ordinary differential equations. Peixoto's density theorem further shows that such flows are typical in the sense that structurally stable systems form an open dense set in the space of all continuously differentiable flows. In this note, we discuss the problem of explicitly finding the limit sets of structurally stable planar flows.
- The set of hyperbolic equilibria and of invertible zeros on the unit ball is computablePublication . Graça, Daniel; Zhong, NingIn this note, we construct an algorithm that, on input of a description of a structurally stable planar dynamical flow $f$ defined on the closed unit disk, outputs the exact number of the (hyperbolic) equilibrium points and their locations with arbitrary accuracy. By arbitrary accuracy it is meant that any accuracy required by the input can be achieved. The algorithm can be further extended to a root-finding algorithm that computes the exact number of zeros as well the location of each zero of a continuously differentiable function $f$ defined on the closed unit ball of $\mathbb{R}^{d}$, provided that the Jacobian of $f$ is invertible at each zero of $f$; moreover, the computation is uniform in $f$.
- Computability in planar dynamical systemsPublication . Graça, Daniel; Zhong, NingIn this paper we explore the problem of computing attractors and their respective basins of attraction for continuous-time planar dynamical systems. We consider C1 systems and show that stability is in general necessary (but may not be sufficient) to attain computability. In particular, we show that (a) the problem of determining the number of attractors in a given compact set is in general undecidable, even for analytic systems and (b) the attractors are semi-computable for stable systems. We also show that the basins of attraction are semi-computable if and only if the system is stable.