\begin{abstract}

\textbf{Introduction:}
Gradient-based optimization methods are widely used due to their simplicity and fast convergence. However, when applied to highly nonlinear and multimodal objective functions, these methods frequently converge to local optima because they rely exclusively on local gradient information. Chaotic optimization techniques provide an alternative mechanism for global exploration, yet they generally lack directional guidance and therefore may exhibit slow convergence during local refinement.

\textbf{Methods:}
To address these limitations, this paper proposes a novel Chaotic Gradient Method (CGM) that integrates gradient descent with density-controlled chaotic perturbations within a unified optimization framework. In contrast to existing hybrid chaotic approaches, where chaos is mainly used for initialization or stochastic exploration, the proposed method incorporates chaos directly into the gradient update rule. The key novelty lies in the introduction of nonlinear density transformation functions applied to chaotic sequences, which reshape their probability distribution and allow controlled perturbations around the gradient direction. This mechanism enables the algorithm to dynamically guide chaotic exploration toward promising regions of the search space while preserving the descent property of gradient-based optimization.

\textbf{Results:}
The performance of the proposed CGM is evaluated on several challenging multimodal benchmark functions, including Styblinski–Tang, Goldstein–Price, and Bukin functions. Numerical experiments demonstrate that the proposed approach improves the robustness of gradient-based optimization by increasing the success rate of reaching global optima and accelerating convergence compared with conventional gradient descent methods.

\textbf{Conclusions:}
The results confirm that density-controlled chaotic perturbations provide an effective mechanism for enhancing gradient-based optimization in complex multimodal landscapes. The proposed framework offers a new perspective for combining deterministic search with controlled chaotic dynamics. Future work will focus on adaptive selection of transformation functions and the application of the method to real-world optimization problems.

\end{abstract}