Related papers: Statistical and Deterministic Dynamics of Maps wit…
We explore how different types and uses of memory can aid spatial navigation in changing uncertain environments. In the simple foraging task we study, every day, our agent has to find its way from its home, through barriers, to food.…
Active systems across scales, ranging from molecular machines to human crowds, are usually modeled as assemblies of self-propelled particles driven by internally generated forces. However, these models often assume memoryless dynamics and…
We extract a two-dimensional dynamical system from the theorems of Pappus and Steiner in classical projective geometry. We calculate an explicit formula for this system, and study its elementary geometric properties. Then we use Artin…
A fundamental problem in neuroscience is understanding how working memory -- the ability to store information at intermediate timescales, like 10s of seconds -- is implemented in realistic neuronal networks. The most likely candidate…
Let $I=[0,1]$ and consider disjoint closed regions $G_{1},....,G_{n}$ in $% I\times I$ and subintervals $I_{1},......,I_{n},$ such that $G_{i}$ projects onto $I_{i.}$ We define the lower and upper maps $\tau_{1},$ $\tau_{2}$ by the lower…
We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we…
We show how random feature maps can be used to forecast dynamical systems with excellent forecasting skill. We consider the tanh activation function and judiciously choose the internal weights in a data-driven manner such that the resulting…
In this paper we discuss the dynamical structure of the rational family $(f_t)$ given by $$f_t(z)=tz^{m}\Big(\frac{1-z}{1+z}\Big)^{n}\quad(m\ge 2,~t\ne 0).$$ Each map $f_t$ has two super-attracting immediate basins and two free critical…
In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…
Dynamical models underpin our ability to understand and predict the behavior of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state…
We consider dynamical systems $(X,T,\mu)$ which have exponential decay of correlations for either H\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\infty$, we give sufficient conditions…
This paper proposes a novel perspective on learning, positing it as the pursuit of dynamical invariants -- data combinations that remain constant or exhibit minimal change over time as a system evolves. This concept is underpinned by both…
Dynamic decisions are pivotal to economic policy making. We show how existing evidence from randomized control trials can be utilized to guide personalized decisions in challenging dynamic environments with budget and capacity constraints.…
Attractor dynamics are a fundamental computational motif in neural circuits, supporting diverse cognitive functions through stable, self-sustaining patterns of neural activity. In these lecture notes, we review four key examples that…
We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity…
Natural memories are associative, declarative and distributed. Symbolic computing memories resemble natural memories in their declarative character, and information can be stored and recovered explicitly; however, they lack the associative…
A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains non-zero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection…
Working memory is a cognitive function involving the storage and manipulation of latent information over brief intervals of time, thus making it crucial for context-dependent computation. Here, we use a top-down modeling approach to examine…
A discrete dynamical system known as Arnold's Discrete Cat Map (Arnold's DCM) is given by (x_t+1, y_t+1) = (x_t + y_t, x_t + 2y_t) mod N; which acts on a two-dimensional square coordinate grid of size Nx?N. The defining characteristic of…
We consider a reaction-diffusion process with retardation. The particles, immersed in traps initially, remain inactive until another particle is annihilated spontaneously with a rate $\lambda$ at a certain point $\vec x$. In that case the…