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A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph…
Classifying perturbative fixed points near upper critical dimensions plays an important role in understanding the space of conformal field theories and critical phases of matter. In this work, we consider perturbative fixed points of $N=5$…
Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in…
In four-dimensional symplectic maps complex instability of periodic orbits is possible, which cannot occur in the two-dimensional case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter…
A periodic perturbation generates a complicated dynamics close to separatrices and saddle points. We construct an asymptotic solution which is close to the separatrix for the unperturbed Duffing's oscillator over a long time. This solution…
We present a phenomenological description of the critical slowing down associated with period-doubling bifurcations in discrete dynamical systems. Starting from a local Taylor expansion around the fixed point and the bifurcation parameter,…
We consider the problem of numerically computing a critical point of a functional $J\colon M\rightarrow R$ where $M$ is a Riemannian manifold. Due to local quadratic convergence a popular choice to solve this problem is the geometric Newton…
The aim of this paper is to study global bifurcations of non-constant solutions of some nonlinear elliptic systems, namely the system on a sphere and the Neumann problem on a ball. We study the bifurcation phenomenon from families of…
Saddle-point configurations, such as the Euclidean bounce and sphalerons, are known to be difficult to find numerically. In this Letter we study a new method, Quartic Gradient Flow, to search for such configurations. The central idea is to…
This paper develops a fixed-point iteration to solve the steady-state water flow equations in an urban water distribution network. The fixed-point iteration is derived upon the assumption of turbulent flow solutions and the validity of the…
Let {f_t} be any algebraic family of rational maps of a fixed degree, with a marked critical point c(t). We first prove that the hypersurfaces of parameters for which c(t) is periodic converge as a sequence of positive closed (1,1) currents…
We show the direct applicability of the Brouwer fixed point theorem for the existence of equilibrium points and periodic solutions for differential systems on general domains satisfying geometric conditions at the boundary. We develop a…
We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…
Bifurcation equations, non-degeneracy and transversality conditions are obtained for the fold, transcritical, pitchfork and flip bifurcations for periodic points of one dimensional implicitly defined discrete dynamical systems. The backward…
In this paper we consider a one quartic operator on the $\mathbb{R}^2$ with positive coefficients. Positive fixed points for a quartic operator, were investigated. Theorems on number of positive fixed points of the quartic operator, are…
In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form $$g(z,\varepsilon)=g_0(z)+\sum_{i=1}^k \varepsilon^i g_i(z)+\mathcal{O}(\varepsilon^{k+1}),$$ for $|\varepsilon|\neq0$…
Many natural and social systems possess power-law memory, and their mathematical modeling requires the application of discrete and continuous fractional calculus. Most of these systems are nonlinear and demonstrate regular and chaotic…
In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator…
The existence of cycles of the matrix maps in Fibonacci class of lattices is well established. We show that such cycles are intimately connected with the presence of interesting positional correlations among the constituent `atoms' in a one…
We study the existence of periodic solutions in a class of planar Filippov systems obtained from non-autonomous periodic perturbations of reversible piecewise smooth differential systems. It is assumed that the unperturbed system presents a…