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Completely positive trace preserving maps are widely used in quantum information theory. These are mostly studied using the master equation perspective. A central part in this theory is to study whether a given system of dynamical maps…
The evolution of networks of coupled chaotic maps with delayed interactions can be studied in the usual way by analyzing the evolution of the state of elements at each iteration time (the "Simulator" point of view), or it can be analyzed…
We consider a living organism as an observer of the evolution of its environment recording sensory information about the state space X of the environment in real time. Sensory information is sampled and then processed on two levels. On the…
Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps. Our…
Delay-coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time-series outputs. The efficacy of delay-coordinate mapping has long been supported by Takens'…
The celebrated Takens' embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In…
The dynamics of affective decision making is considered for an intelligent network composed of agents with different types of memory: long-term and short-term memory. The consideration is based on probabilistic affective decision theory,…
Recent evidence suggests that Adam performs robustly when its momentum parameters are tied, $\beta_1=\beta_2$, reducing the optimizer to a single remaining parameter. However, how this parameter should be set remains poorly understood. We…
Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals. Unfortunately, continuous attractors suffer from severe structural instability…
If the dynamics of an open quantum systems is non-Markovian, its {asymptotic} state strongly depends on the initial conditions, even if the dynamics possesses an {invariant} state. This is the very essence of memory effects. In particular,…
We investigate the lifetime of dynamical regimes under the impact of noise motivated by low-dimensional models of the atmosphere. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state…
We present a simple dynamical systems model for the effect of invisible space dimensions on the visible ones. There are three premises. A: Orbits consist of flows of probabilities [P].which is the case in the setting of quantum mechanics.…
Habituation - a phenomenon in which a dynamical system exhibits a diminishing response to repeated stimulations that eventually recovers when the stimulus is withheld - is universally observed in living systems from animals to unicellular…
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…
We study a random dynamical system such that one transformation is randomly selected from a family of transformations and then applied on each iteration. For such random dynamical systems, we consider estimates of absolutely continuous…
We introduce a two-player model of reinforcement learning with memory. Past actions of an iterated game are stored in a memory and used to determine player's next action. To examine the behaviour of the model some approximate methods are…
A coupled computational approach to simultaneously learn a vector field and the region of attraction of an equilibrium point from generated trajectories of the system is proposed. The nonlinear identification leverages the local stability…
We examine the one-humped map at the period-doubling transition to chaos, and ask whether its long-term memory is stack-like (last-in, first-out) or queue-like (first-in, first-out). We show that it can be recognized by a real-time…
We consider the dynamical system described by the area--preserving standard mapping. It is known for this system that $P(t)$, the normalized number of recurrences staying in some given domain of the phase space at time $t$ (so-clled…
A succesful method to describe the asymptotic behavior of a discrete time stochastic process governed by some recursive formula is to relate it to the limit sets of a well chosen mean differential equation. Under an attainability condition,…