Abstract
We investigate the robustness with respect to random stimuli of a dynamical system with a plastic self-organising vector field, previously proposed as a conceptual model of a cognitive system and inspired by the self-organised plasticity of the brain. This model of a novel type consists of an ordinary differential equation subjected to the time-dependent “sensory” input, whose time-evolving solution is the vector field of another ordinary differential equation governing the observed behaviour of the system, which in the brain would be neural firings. It is shown that the individual solutions of both these differential equations depend continuously over finite time intervals on the input signals. In addition, under suitable uniformity assumptions, it is shown that the non-autonomous pullback attractor and forward omega limit set of the given two-tier system depend upper semi-continuously on the input signal. The analysis holds for both deterministic and noisy input signals, in the latter case in a pathwise sense.
Highlights
It is well known that when the system parameters are near a critical state, noise can lead to dramatic changes in the observed behaviour, called noise-induced transitions in Horsthemke & Lefever [14]
Noise-induced phenomena have been observed in many models of real systems, such as a climate model [1]
The solution mapping of a non-autonomous ordinary differential equations (ODEs) such as (1) generates a non-autonomous dynamical system on the state space Rd expressed in terms of a 2-parameter semi-group, which is often called a process
Summary
It is important to evaluate the robustness of various devices to random perturbations, such as the stability against noise of the frequency of an electronic clock [3] This is often formulated in terms of the continuous dependence of solutions, and the upper semi-continuous dependence of attractors, on the input signal or a parameter. In this paper we investigate the dependence of the pullback attractor and forward omega limit set on the changes in the input signal η for an appropriate class of input signals This analysis is pathwise, so applies to both deterministic and noisy input signals. 5 the upper semi-continuity of the pullback attractor and of the omega limit set of the ODE (1) in the input signal η is shown to hold These results will be presented in the deterministic setting.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
