arXiv subjects

Sajjad Bakrani

Publications and source records attributed to Sajjad Bakrani.

3 recordsLinked to original sources

arXiv2024

Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry

We consider a $\mathbb{Z}_{2}$-equivariant 4-dimensional system of ODEs with a smooth first integral $H$ and a saddle equilibrium state $O$. We assume that there exists a transverse homoclinic orbit $Γ$ to $O$ that approaches $O$ along the nonleading directions. Suppose $H(O) = c$. In \cite{Bakrani2022JDE}, the dynamics near $Γ$ in the level set $H^{-1}(c)$ was described. In particular, some criteria for the existence of the stable and unstable invariant manifolds of $Γ$ were given. In the current paper, we describe the dynamics near $Γ$ in the level set $H^{-1}(h)$ for $h\neq c$ close to $c$. We prove that when $h < c$, there exists a unique saddle periodic orbit in each level set $H^{-1}(h)$, and the forward (resp. backward) orbit of any point off the stable (resp. unstable) invariant manifold of this periodic orbit leaves a small neighborhood of $Γ$. We further show that when $h > c$, the forward and backward orbits of any point in $H^{-1}(h)$ near $Γ$ leave a small neighborhood of $Γ$. We also prove analogous results for the scenario where two transverse homoclinics to $O$ (homoclinic figure-eight) exist. The results of this paper, together with \cite{Bakrani2022JDE}, give a full description of the dynamics in a small open neighborhood of $Γ$ (and a small open neighborhood of a homoclinic figure-eight).

math.DS↗

arXiv2024

Cycle-Star Motifs: Network Response to Link Modifications

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master-slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as Rössler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.

nlin.CD↗

arXiv2022

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in (mathbb{R}^4) with (mathbb{Z}_2)-symmetry and integral of motion

We consider a (mathbb{Z}_2)-equivariant flow in (mathbb{R}^{4}) with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit (Gamma). We provide criteria for the existence of stable and unstable invariant manifolds of (Gamma). We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

math.DS↗