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Mohamed Karim Hamdani

Publications and source records attributed to Mohamed Karim Hamdani.

8 recordsLinked to original sources
arXiv2026

A Novel Hyperchaotic System Derived from Asymmetric Bidirectional Coupling of Two Lorenz Oscillators: Synchronization Analysis and Cryptographic Applications

This paper investigates a six-dimensional coupled Lorenz system obtained through bidirectional asymmetric diffusive coupling of two identical Lorenz oscillators. The proposed model is analyzed from both dynamical and synchronization perspectives. Dissipativity conditions, equilibrium points, local stability properties, and Lyapunov spectra are systematically examined. The results reveal wide regions of chaotic and hyperchaotic behavior, with up to three positive Lyapunov exponents depending on the coupling parameters. These findings provide a practical framework for selecting coupling configurations that guarantee hyperchaotic dynamics while preserving the simplicity of the classical Lorenz structure. \\ A Lyapunov-based synchronization analysis is then developed. By exploiting effective attractor bounds obtained from the actual system trajectories rather than conservative worst-case estimates, a less restrictive sufficient synchronization condition is derived. Under the classical Lorenz parameters, synchronization is guaranteed for $α_1+α_2>15.1$. \\ The synchronization performance is further evaluated through a parametric analysis of the synchronization time. Numerical results show that increasing the coupling strength significantly accelerates convergence toward the synchronization manifold. For a broad range of admissible coupling values, synchronization times below $0.5$ second are achieved, whereas previously reported Lorenz-based synchronization schemes typically require approximately $5$ seconds.

math.DS↗
arXiv2023

A new class of multiple nonlocal problems with two parameters and variable-order fractional $p(\cdot)$-Laplacian

In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the $p(x)$-fractional Laplacian equations of variable order. The problem is stated as follows: \begin{eqnarray*} \left\{ \begin{array}{ll} M\Big(σ_{p(x,y)}(u)\Big)(-Δ)^{s(\cdot)}_{p(\cdot)}u(x) =λ|u|^{q(x)-2}u\left(\int_Ø\frac{1}{q(x)} |u|^{q(x)}dx \right)^{k_1}+β|u|^{r(x)-2}u\left(\int_Ø\frac{1}{r(x)} |u|^{r(x)}dx \right)^{k_2} \quad \mbox{in }Ω, \\ u=0 \quad \mbox{on }\partialΩ, \end{array} \right. \end{eqnarray*} where the nonlocal term is defined as $$ σ_{p(x,y)}(u)=\int_{Ω\times Ω}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}} \,dx\,dy. $$ Here, $Ω\subset\mathbb{R}^{N}$ represents a bounded smooth domain with at least $N\geq2$. The function $M(s)$ is given by $M(s) = a - bs^γ$, where $a\geq 0$, $b>0$, and $γ>0$. The parameters $k_1$, $k_2$, $λ$ and $β$ are real parameters, while the variables $p(x)$, $s(\cdot)$, $q(x)$, and $r(x)$ are continuous and can change with respect to $x$. To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when $a>0$ and when $a=0$. To the best of our knowledge, these results are the first contributions to research on the variable-order $p(x)$-fractional Laplacian operator.

math.AP↗
arXiv2023

A degenerate Kirchhoff-type problem involving variable $s(\cdot)$-order fractional $p(\cdot)$-Laplacian with weights

This paper deals with a class of nonlocal variable $s(.)$-order fractional $p(.)$-Kirchhoff type equations: \begin{eqnarray*} \left\{ \begin{array}{ll} \mathcal{K}\left(\int_{\mathbb{R}^{2N}}\frac{1}{p(x,y)}\frac{|φ(x)-φ(y)|^{p(x,y)}}{|x-y|^{N+s(x,y){p(x,y)}}} \,dx\,dy\right)(-Δ)^{s(\cdot)}_{p(\cdot)}φ(x) =f(x,φ) \quad \mbox{in }Ω, \\ φ=0 \quad \mbox{on }\mathbb{R}^N\backslashΩ. \end{array} \right. \end{eqnarray*} Under some suitable conditions on the functions $p,s, \mathcal{K}$ and $f$, the existence and multiplicity of nontrivial solutions for the above problem are obtained. Our results cover the degenerate case in the $p(\cdot)$ fractional setting.

math.AP↗
arXiv2021

Existence and multiplicity of solutions for $m(x)-$polyharmonic elliptic Kirchhoff type equations without Ambrosetti-Rabinowitz conditions

In this paper, we prove the existence of infinitely many solutions for a class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the nonlinear function has a quasicritical growth at infinity and without assuming the Ambrosetti and Rabinowitz type condition. The new aspect consists in employing the notion of a Schauder basis to verify the geometry of the symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution, and also to discuss the sublinear case under large growth conditions at infinity and at zero. Our results are an improvement and generalization of the corresponding results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and Applications, $2011$) and Bae-Kim (Mathematical Methods in the Applied Sciences, $2020$).

math.AP↗
arXiv2021

A class of fractional p()-Kirchhoff type systems

This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional $p(x)$-operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.

math.AP↗
arXiv2019

Multiple solutions for Grushin operator without odd nonlinearity

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: \begin{eqnarray*} (P_g)\quad - Δ_λ u + V(x) u = f(x,u)+g(x),\;\mbox{ in } \R^N,\; \end{eqnarray*} and \begin{eqnarray*} (P_0)\quad - Δ_λ u + V(x) u = K(x)f(x,u),\;\mbox{ in } \R^N,\; \end{eqnarray*} where $Δ_λ$ is the strongly degenerate operator, $V(x)$ is allowed to be sign-changing, $K\in C(\R^N,\R)$, $g:\R^N\to\R$ is a perturbation and the nonlinearity $f(x,u)$ is a continuous function does not satisfy the Ambrosetti-Rabinowitz superquadratic condition ($(AR)$ for short). First, via the mountain pass theorem and the Ekeland's variational principle, existence of two different solutions for $(P_g)$ are obtained when $f$ satisfies superlinear growth condition. Moreover, we prove the existence of infinitely many solutions for $(P_0)$ if $f$ is odd in $u$ thanks an extension of Clark's theorem near the origin. So, our main results considerably improve results appearing in the literature.

math.AP↗
arXiv2019

High-order Kirchhoff problems in bounded and unbounded domains

Consider the following $m-$polyharmonic Kirchhoff problem: \begin{eqnarray} \label{ea} \begin{cases} M\left(\int_Ø|D_r u|^{m} +a|u|^m\right)[Δ^r_m u +a|u|^{m-2}u]= K(x)f(u) &\mbox{in}\quad Ω, \\ u=\left(\frac{\partial}{\partial ν}\right)^k u=0, \quad &\mbox{on}\quad \partialΩ, \quad k=1, 2,..... , r-1, \end{cases} \end{eqnarray} where $r \in \N^*$, $m >1$, $N\geq rm+1$, $a\geq 0$, $K\in L^{\infty}(Ø)$ is a positive weight function, $M \in C([0,+\infty))$ and $f\in C(\mathbb{R})$ which will be specified later. We will study problem \eqref{ea} in the following different type of domains: \begin{enumerate} \item $a=0$ and $K\in L^{\infty}(Ø)$ is a positive weight function if $Ω$ is a smooth bounded domain of $\R^N$. \item $a>0$ and $K\in L^{\infty}(Ø)\cap L^{p}(Ø)$, $p \geq 1$ if $Ω$ is an unbounded smooth domain. \item $Ø=\R^N$ and $a=0$ (which called the $mγ$-zero mass case). \end{enumerate} We prove the existence of infinitely many solutions of \eqref{ea} for some odd functions $f$ in $u$ satisfying subcritical growth conditions at infinity which are weaker than the analogue of the Ambrosetti-Rabinowitz condition and the standard subcritical polynomial growth. The new aspect consists in employing the Schauder basis of $W_0^{r,m}(Ø)$ to verify the geometry of the symmetric mountain pass theorem without any control on $f$ near $0$ if $Ω$ is a bounded domain and under a suitable condition at $0$ if $Ω$ is a unbounded domain allowing only to derive the variational setting of \eqref{ea}. Moreover, we introduce a positive quantity $λ_M$ similar to the first eigenvalue of the $m$-polyharmonic operator to find a mountain pass solution.

math.AP↗