In recent years, with the development of novel hardware such as memristors, integrating chaotic systems with hardware implementations has enabled efficient and controllable generation of chaotic signals, providing new avenues for both theoretical research and engineering applications. In this work, we propose a novel memristor-based chaotic system, named the three-dimensional discrete echo-memristor map (3D-DEMM). The 3D-DEMM is capable of generating complex dynamic behaviors with self-similar attractor structures; specifically, under different parameters and initial conditions, the system produces similar attractor shapes at different amplitudes, which we refer to as an echo chaotic map. By incorporating the discrete nonlinear characteristics of memristors, the 3D-DEMM is systematically designed. We first conduct a thorough dynamic analysis of the 3D-DEMM, including attractor visualization, Lyapunov exponents, and NIST tests, to verify its chaoticity and self-similarity. Subsequently, the attractors of the 3D-DEMM are captured on a DSP platform, demonstrating discrete-time hardware simulation and real-time operation. Experimental results show that the proposed system not only exhibits highly controllable chaotic behavior but also demonstrates strong robustness in maintaining amplitude-invariant attractor shapes, providing a new theoretical and practical approach for memristor-based chaotic signal generation and applications in information security.

Time series forecasting has long been dominated by advances in model architecture, with recent progress driven by deep learning and hybrid statistical techniques. However, as forecasting models approach diminishing returns in accuracy, a critical yet underexplored opportunity emerges: the strategic use of post-processing. In this paper, we address the last-mile gap in time-series forecasting, which is to improve accuracy and uncertainty without retraining or modifying a deployed backbone. We propose $δ$-Adapter, a lightweight, architecture-agnostic way to boost deployed time series forecasters without retraining. $δ$-Adapter learns tiny, bounded modules at two interfaces: input nudging (soft edits to covariates) and output residual correction. We provide local descent guarantees, $O(δ)$ drift bounds, and compositional stability for combined adapters. Meanwhile, it can act as a feature selector by learning a sparse, horizon-aware mask over inputs to select important features, thereby improving interpretability. In addition, it can also be used as a distribution calibrator to measure uncertainty. Thus, we introduce a Quantile Calibrator and a Conformal Corrector that together deliver calibrated, personalized intervals with finite-sample coverage. Our experiments across diverse backbones and datasets show that $δ$-Adapter improves accuracy and calibration with negligible compute and no interface changes.