Robust Approximation Algorithms for the Detection of Attraction Basins in Dynamical Systems

Abstract

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This problem is rather common especially in population dynamics models. Precisely, a particular solution of a dynamical system is completely determined by its initial condition and by the parameters involved in the model. Furthermore, when the omega limit set reduces to a point, the trajectory of the solution evolves towards the steady state. But, in case of multi-stability it is possible that several steady states originate from the same parameter set. Thus, in these cases the importance of accurately reconstruct the attraction basins follows. In this paper we focus on dynamical systems of ordinary differential equations presenting three stable equilibia and we design algorithms for the detection of the points lying on the manifolds determining the basins of attraction and for the reconstruction of such manifolds. The latter are reconstructed by means of the implicit partition of unity method which makes use of radial basis functions (RBFs) as local approximants. Extensive numerical test, carried out with a Matlab package made available to the scientific community, support our findings.

Effects of Viral and Cytokine Delays on Dynamics of Autoimmunity

“…Figure 3c illustrates the bi-stability between S * 2 and S * 6 in terms of their basins of attraction. It is worth noting that recently significant research in approximation theory and meshless interpolation has focused on developing techniques for detection and analysis of attraction basins [63][64][65][66][67][68]. Figure 3c suggests that for very large initial amounts of regulatory T cells, the system converges to the disease-free steady state.…”

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“…So, in this section, we will consider the system (9) with control (18)- (19) and the temporal dependence (20)- (21). If the control (18)- (19) is not applied, then the Wolbachia-infected mosquito population goes to extinction ( Figure 5).…”

Detecting tri‐stability of 3D models with complex attractors via meshfree reconstruction of invariant manifolds of saddle points

“…Therefore, some efforts need to develop a versatile computational framework that could be easily used by researchers of different disciplines who are interested in studying dynamical processes. The first attempts in graphical representation of these surfaces are presented in previous studies . The authors developed a bisection method to detect the separatrix points coupled with a partition of unity meshfree method to reconstruct them.…”

“…The first attempts in graphical representation of these surfaces are presented in previous studies. [11][12][13] The authors developed a bisection method to detect the separatrix points coupled with a partition of unity meshfree method to reconstruct them. Following these ideas in previous studies, [14][15][16] we analyze the Allee effect induced by the pack hunting in a predator-prey model.…”