arXiv subjects

Zeraoulia Rafik

Publications and source records attributed to Zeraoulia Rafik.

16 recordsLinked to original sources

arXiv2025

Chaos Analysis in the Hybrid Quintic Duffing-Riemann Zeta System via Decomposition

This paper presents a comprehensive analysis of the driven cubic-quintic Duffing oscillator \[ \ddotϕ+\frac{1}{q}\dotϕ+ϕ^3+ϕ^5=A\cos(ωt), \] advancing both analytical and numerical chaos theory. Using Melnikov analysis on explicit homoclinic orbits \[ ϕ_0(t) = 1-\tanh(t)-\tanh^2(t) \quad \text{and} \quad ϕ_0(t) = {\rm sech}_{\rm RZ}(t) -{\rm sech}_{\rm RZ}^2(t),\] we rigorously predict transverse homoclinic intersections and limit cycle bifurcations surrounding the hyperbolic saddle $(0,0)$, establishing chaos onset at $A_\mathrm{chaos}\approx0.34$. A groundbreaking contribution introduces the hybrid quintic Duffing-Riemann zeta system $\ddotϕ+ϕ^3+ϕ^5=A\cos(ωt)+\Re[ζ(s)]$, where $ζ(s)=X(s)-Y(s)$ via C-transformation decomposition. Bifurcation portraits reveal zeta perturbation delays chaos by $24\%$ ($A_\text{chaos}\approx0.42$) while enhancing Lyapunov exponents by $27\%$ ($λ_\text{max}=0.14>0.11$). Nontrivial zeros $s_k=1/2+it_k$ emerge as chaos suppressors through entropy-matching $|X(s_k,n)|^2=|Y(s_k,n)|^2$. We prove nontrivial zeros manifest as global Lyapunov minimizers $λ(s_k)=\min_{σ\in[0,1]}λ(σ+it_k)$, reformulating the Riemann Hypothesis as a verifiable bifurcation prediction. The unperturbed Hamiltonian $H=\frac{1}{2}\dotϕ^2+\frac{1}{4}ϕ^4+\frac{1}{6}ϕ^6$ and stochastic extensions for biomedical applications are analyzed, positioning number-theoretic chaos control as a novel paradigm bridging nonlinear dynamics and analytic number theory.

math.GM↗

arXiv2025

Analyzing Dynamical Systems Inspired by Montgomery's Conjecture: Insights into Zeta Function Zeros and Chaos in Number Theory

In this study, we analyze a novel dynamical system inspired by Montgomery's pair correlation conjecture, modeling the spacings between nontrivial zeros of the Riemann zeta function via the GUE kernel $g(u) = 1 - \left( \frac{\sin(πu)}{πu} \right)^2 + δ(u)$. The recurrence $x_{n+1} = 1 - \left( \frac{\sin(π/x_n)}{π/x_n} \right)^2 + \frac{1}{x_n}$ emulates eigenvalue repulsion as a quantum operator analogue realizing the Pólya-Hilbert conjecture. Bifurcation analysis and Lyapunov exponents reveal quantum-like chaos: near $x=0$, linearized dynamics $f(x) = 1 - π^2 x^2$ yield Gaussian Lyapunov function $V(x) = C_1 e^{-π^2 x^3/3}$ with LaSalle invariance bounding zeros in $[0,1]$; large $x$ exhibit exponential growth $λ_n \to \ln(π^2/6)$. Entropy analysis confirms GUE level repulsion with zero entropy for small initial conditions. Comparative validation against actual $γ_n$ achieves errors $<10^{-100}$, while spectral density $ρ(E) \sim \frac{\log E}{2π}$ matches zeta zero statistics. This bridges Montgomery pair correlation to quantum chaos, providing computational evidence for Riemann zero spacing distributions and supporting the quantum operator hypothesis for $ζ(1/2+it)$.

math.GM↗

arXiv2025

Iterating sum of power divisor function and New equivalence to the Riemann hypothesis

This paper investigates the dynamics of the iterated sum-of-divisors function $σ_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $σ_k(m) \equiv 0 \pmod m$. Perfect and multiperfect numbers remain extremely rare; odd perfect numbers are still unknown and must be astronomically large. Here, the emphasis is on the dynamical and statistical structure of the iterates rather than on isolated examples. Three main results are obtained. First, it is proved that no integer $m>1$ can satisfy $σ_k(m) \equiv 0 \pmod m$ for all $k \ge 0$, thereby ruling out the existence of "metaperfect" numbers and showing that the iteration of $σ$ cannot remain permanently trapped in the residue class $0$ modulo $m$. Second, for certain explicit integers such as $m=6,12,24$, the sequence $σ_k(m) \bmod m$ is strictly periodic with small period dividing $L=\mathrm{lcm}(e_i+1)$, where the $e_i$ are the prime exponents of $m$. Bifurcation plots and distributional analysis reveal a transition from rigid two-cycle structure to more complex residue dynamics as $m$ increases. Third, a new equivalence with the Riemann Hypothesis is established: RH holds if and only if, for every even non-squarefree $m \ge 5041$ containing a prime fifth power, \[ \frac{σ_k(m)}{σ_{k-1}(m)\log\logσ_{k-1}(m)} \le e^γ, \] and the sequence $σ_k(m) \bmod m$ is eventually periodic, uniformly in $k \ge 0$. Extensive computations support these periodicity phenomena, yield non-normal discrete distribution models for the residues, and suggest a connection with a newly proposed Schrodinger-type "Caceres" operator whose spectrum numerically reproduces key statistical features of the nontrivial zeros of the Riemann zeta function.

math.GM↗

arXiv2025

On Congruences for Iterates of the Sum--Power Divisor Function and Conditional Implications for the Riemann Hypothesis

Inspired by Cohen and te Riele~\cite{Cohen1996}, who computationally verified that for every $n \leq 400$ there exists $k$ such that $σ^k(n) \equiv 0 \pmod{n}$ (where $σ^k$ denotes the $k$-fold iteration of the sum-of-divisors function), this paper resolves their reverse question negatively: no integer $n > 1$ satisfies $σ^k(n) \equiv 0 \pmod{n}$ for \emph{all} $k \geq 1$. The proof eliminates prior gaps via Lenstra's density-zero bounds $σ_k(m) \ll m / \log\log m$ combined with Robin's RH-equivalent criterion $σ(n) < e^γn \log\log n + 0.6483 n / \log\log n$ ($n \geq 5041$), showing universal metaperfect divisibility implies RH-violating $σ$ growth or low-lying zeta zeros near $s=1$. Among multiperfect $n$ with prime $L = \mathrm{lcm}(1+e_p : p \mid n)$, only $n=6$ satisfies the congruence for all odd $k$, with Shannon entropy $H(σ^k(6) \mod 6) \to \log 2$ reflecting periodic order. We analyze bifurcation phenomena in the dynamics $σ^k(n) \mod n$, where high-entropy chaotic residues for other $n$ mirror GUE statistics of zeta zeros ($\sim \log T / 2π$ near $s=1/2$, $>41\%$ verified on critical line), contrasting the ordered $n=6$ case. Zero rates near $s=1$ (simple pole) and $s=1/2$ bound iterated $σ$ distributions, linking to RH via divisor sums and dynamical bifurcations; we conjecture $n=6$ uniquely achieves odd-$k$ divisibility with small period dividing $L$.

math.NT↗

arXiv2025

A note on an open conjecture in rational dynamical systems

Recently ,mathematicians have been interested in studying the theory of discrete dynamical system, specifically difference equation, such that considerable works about discussing the behavior properties of its solutions (boundedness and unboundedness) are discussed and published in many areas of mathematics which involves several interesting results and applications in applied mathematics and physics ,One of the most important discrete dynamics which is became of interest for researchers in the field is the rational dynamical system .In this paper we give a negative answer to the eight open conjecture in rational dynamical system proposed by G.Ladas and Palladino many years ago which states : Assume $α,β, λ\in [0,\infty)$. Then every positive solution of the difference equation \\: \begin{align*} z_{n+1}=\frac{α+z_{n}β+z_{n-1}λ}{z_{n-2}},\quad n=0,1,\ldots \end{align*} is bounded if and only if $β=λ$. We will use a construction of subenergy function and some properties of Todd's difference equation to disprove that conjecture in general.Some new results (Chebychev approximation) and analysis regarding that open conjecture are presented.

math.DS↗

arXiv2025

If our chaotic operator is derived correctly, then the Riemann hypothesis holds true

This work develops an operator-theoretic and dynamical framework inspired by the Riemann--von Mangoldt formula, chaotic dynamics, and random-matrix models for the Riemann zeta function, without attempting to prove the Riemann Hypothesis. Starting from the explicit zero-counting function $N(T)$, we construct a discrete map on the critical line and analyse its Lyapunov exponents and bifurcation diagrams, showing that the smooth von Mangoldt term generates a strongly unstable flow that captures the global growth of the zero density. Motivated by this dynamics, we define a self-adjoint ``chaotic'' operator $\mathcal{O}_α$ on a weighted Hilbert space with weight $\mathrm{d}N/\mathrm{d}T$, prove its unboundedness and essential self-adjointness, and describe its spectral resolution via the spectral theorem. Finite-dimensional truncations of $\mathcal{O}_α$ yield Hermitian random matrices whose eigenvalue statistics agree numerically with Gaussian unitary ensemble predictions and show qualitative similarities to both Odlyzko's zeta zeros and the hydrogen-atom spectrum, suggesting that $\mathcal{O}_α$ lies in the same universality class as the nontrivial zeros and providing a concrete Hilbert--Pólya--type framework rather than a proof of the conjecture.

math.GM↗

arXiv2025

Chaotic Dynamics Derived from the Montgomery Conjecture: Application to Electrical Systems

Here, we introduce a novel method for obtaining chaotic dynamics based on the Montgomery conjecture for the pair correlation of zeros of the Riemann zeta function. Motivated by the conjecture, we present a recursive relation that reveals chaotic behavior. Notably, we provide insights into the possible uses of this derived chaotic dynamics in electrical engineering by interpreting it as a unique representation of an electrical system. Furthermore, we investigate the relevance of entropy, bifurcation analysis, and chaos theory in this framework for electrical systems. We look into its applicability to signal processing, stability analysis through bifurcation, and how entropy measures the predictability or unpredictability of electrical signals. Additionally, we discuss the system's strange attractor and its transition to voltage collapse, highlighting the interplay between chaotic dynamics and stability in electrical systems. Furthermore, we analyze the system's energy distribution, taking into account how chaotic dynamics may affect energy allocation or dissipation. Furthermore, we compare the chaotification and Hermiticity of the resulting operators between Yitang dynamics and Montgomery dynamics. To have a better grasp of the spectrum features of each operator, we calculate the eigenvalues for each one obtained from the corresponding dynamics. Our results provide fresh insights into number-theoretic chaotic dynamics and how they might be applied in real-world electrical engineering applications. This work provides encouraging opportunities for further research and technology developments by laying the foundation for creative investigations in system dynamics.

math.GM↗

arXiv2025

A New Special Function and Its Application in Probability

In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we provide its series representation using Padé approximant. We show convincing numerical evidence of an accuracy of $10^{-6}$ for the approximants in the sense of the quadratic mean norm. A similar approach may be applied to other probability distributions, for example, the Maxwell--Boltzmann distribution and the normal distribution, and we show its application using both of those distributions.

math.GM↗

arXiv2025

Chaotic Dynamics and Zero Distribution: Implications and Applications in Control Theory for Yitang Zhang's Landau Siegel Zero Theorem

This study delves into the realm of chaotic dynamics derived from Dirichlet L-functions, drawing inspiration from Yitang Zhang's groundbreaking work on Landau-Siegel zeros. The dynamic behavior reveals profound chaos, corroborated by the calculated Lyapunov exponents and entropy, attesting to the system's inherent unpredictability. Furthermore, we establish a novel connection between Fractal geometry and Quantum chaos, predicting the distributions of zeros for both Yitang dynamics and Riemann dynamics. These findings offer indirect support for Zhang's groundbreaking theorem concerning Landau-Siegel zeros and suggest that these chaotic dynamics could find application in engineering and control systems, demonstrating the potential to harness chaos for beneficial purposes. The exploration of stability within electrical systems further uncovers the instability of fixed points, highlighting both the challenges and opportunities for harnessing chaotic behavior to achieve specific control objectives. This study not only contributes to our understanding of chaotic dynamics but also opens new avenues for exploring the potential applications of Yitang dynamics in the field of electrical control systems. It paves the way for innovative approaches to address real-world engineering challenges and may be considered as a new consequence for the generalized Riemann hypothesis.

math.DS↗

arXiv2025

A Hybrid Deep Learning and Anomaly Detection Framework for Real-Time Malicious URL Classification

Malicious URLs remain a primary vector for phishing, malware, and cyberthreats. This study proposes a hybrid deep learning framework combining \texttt{HashingVectorizer} n-gram analysis, SMOTE balancing, Isolation Forest anomaly filtering, and a lightweight neural network classifier for real-time URL classification. The multi-stage pipeline processes URLs from open-source repositories with statistical features (length, dot count, entropy), achieving $O(NL + EBdh)$ training complexity and a 20\,ms prediction latency. Empirical evaluation yields 96.4\% accuracy, 95.4\% F1-score, and 97.3\% ROC-AUC, outperforming CNN (94.8\%) and SVM baselines with a $50\!\times$--$100\!\times$ speedup (Table~\ref{tab:comp-complexity}). A multilingual Tkinter GUI (Arabic/English/French) enables real-time threat assessment with clipboard integration. The framework demonstrates superior scalability and resilience against obfuscated URL patterns.

cs.CR↗

arXiv2024

Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

This paper explores a novel approach to modeling the positional dynamics of stars using discrete dynamical systems. We define star evolution through discrete-time update rules based on right ascension, declination, and distance, incorporating chaotic behavior via nonlinear functions and external perturbations. By applying Ricci flow and Riemannian metrics, we provide new insights into the positional dynamics of stars. Theoretical computations of Perelman entropy are used to assess system complexity, with high-precision Runge-Kutta methods ensuring accurate solutions for our chaotic model. We quantify chaos using Lyapunov exponents and perform bifurcation analysis to study how parameter variations affect the dynamics. Comparing our model to the Lorenz attractor reveals both similarities and unique characteristics in stellar dynamics. Our results show that entropy increases exponentially, indicating that predicting star positions with precision becomes increasingly challenging over time. This study advances the understanding of chaos in celestial systems and contributes to dynamical systems theory by integrating chaos theory with astronomical modeling.

math.DS↗

arXiv2024

Does the Zeraoulia Sequences Converges?

In this note we present a collection of attempts of some researchers to prove or disprove whether the Zeraoulia sequences convergent. Even nowdays convergence of Zeraoulia sequences still open.

math.GM↗

arXiv2023

New Chaotic dynamics for Yitang Zhang latest results on Landau-Siegel zero

The first part of this paper is about Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zero posted by some mathematicians in Mathoverflow ,For the second part we are able to derive new Chaotic dynamics for Yitang Zhang on Landau-Siegel zero such that the behavior of the new dynamics has been discussed ,Lyaponove Exponents has been computed and bifurcation diagram has been achieved ,The number of limit cycle and orbits are predicted.The behavior of this new dynamics roughly proves the validity of Yitang latest results .

math.GM↗

arXiv2022

Amazing behavior and transition to chaos of some sequences using Collatz like problems and Quibic duffing

In this paper we shall show amazing behavior of some discrete maps using Collatze like problems and some advanced theories in analytic number theory and dynamical system,we have investigated the driven cubic-quintic Duffing equation such that , We were able to predict the number of limit cycles around the equilibrium and to develop a theoretical approach to chaos suppression in damped driven systems using Collatze like problem sequences , some new results regarding behavior of that sequence are presented.

math.GM↗

arXiv2022

On the Boundedness solutions of the difference equation $x_{n+1}=a x^α_{n}+bx^α_{n-1},0<α\leq2$ and its application in medicine

math.DS↗

arXiv2022

Smoothness and analyticity of $f'=\exp({f^{-1}})$

This paper considers some analytical and numerical aspects of the problem defined by an equation of the type $f'=\exp({f^{-1}})$ with $f^{-1}$ is a composional inverse of $f$,some new analytical and numerical results are presented using RK4 and Explicit Runge kuta methods.

math.GM↗