Related papers: Finite switching near heteroclinic networks
Some aspects of nonlocal dynamics on directed and undirected networks for an initial value problem whose Jacobian matrix is a variable-order fractional power of a Laplacian matrix are discussed here. This is a new extension to…
Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown,…
We classify simple heteroclinic networks for a $\Gamma$-equivariant system in ${\mathbb R}^4$ with finite $\Gamma \subset {\rm O}(4)$, proceeding as follows: we define a graph associated with a given $\Gamma \subset {\rm O}(n)$ and identify…
We discuss the complex dynamics of a non-linear random networks model, as a function of the connectivity k between the elements of the network. We show that this class of networks exhibit an order-chaos phase transition for a critical…
The Melnikov method is applied to periodically perturbed open systems modeled by an inverse--square--law attraction center plus a quadrupolelike term. A compactification approach that regularizes periodic orbits at infinity is introduced.…
In this work, we study a class of hybrid dynamical systems called hybrid gene regulatory networks (HGRNs) which was proposed to model gene regulatory networks. In HGRNs, there exist well-behaved trajectories that reach a fixed point or…
In multiplex networks with a large number of layers, the nodes can have different activities, indicating the total number of layers in which the nodes are present. Here we model multiplex networks with heterogeneous activity of the nodes…
Complex network theory aims to model and analyze complex systems that consist of multiple and interdependent components. Among all studies on complex networks, topological structure analysis is of the most fundamental importance, as it…
We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated…
We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of…
Controlling entanglement and coherence is central to quantum information, yet the two resources often exhibit antagonistic trends and are difficult to optimize within a single platform. Here we show that chaos enables switchable eigenstate…
The interaction of distinct units in physical, social, biological and technological systems naturally gives rise to complex network structures. Networks have constantly been in the focus of research for the last decade, with considerable…
The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose…
In this paper we deal with the notion of the effective impedance of AC networks consisting of resistances, coils and capacitors. Mathematically such a network is a locally finite graph whose edges are endowed with complex-valued weights…
Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification…
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph). Although it is known that every connected $2k$-valent infinite circulant graph has a two-way-infinite…
The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete laplacian is under consideration. The rate of stabilization for the the matrix entries which provides finiteness of the discrete spectrum and is…
Many complex networks are known to exhibit sudden transitions between alternative steady states with contrasting properties. Such a sudden transition demonstrates a network's resilience, which is the ability of a system to persist in the…
We analytically determine the number and distribution of fixed points in a canonical model of a chaotic neural network. This distribution reveals that fixed points and dynamics are confined to separate shells in phase space. Furthermore,…
The concept of limiting step gives the limit simplification: the whole network behaves as a single step. However, in its simplest form this idea is applicable only to the simplest linear cycles in steady states. For such the simplest cycles…