Dynamical systems exhibit rich and intricate behaviors that can be harnessed for physical computation. Physical computing draws inspiration from complex systems that continuously adapt, self-organize, and minimize energy as they evolve toward stable configurations, naturally enabling parallel processing. These characteristics show promise for tackling difficult scientific challenges, including NP-hard combinatorial optimization problems. However, designing dynamical systems for computation remai

Dynamical systems exhibit rich and intricate behaviors that can be harnessed for physical computation. Physical computing draws inspiration from complex systems that continuously adapt, self-organize, and minimize energy as they evolve toward stable configurations, naturally enabling parallel processing. These characteristics show promise for tackling difficult scientific challenges, including NP-hard combinatorial optimization problems. However, designing dynamical systems for computation remains challenging, particularly in choosing appropriate technologies and developing scalable circuit implementations. This invited talk will provide an overview of circuit-level implementations of physical computing using coupled oscillatory neural networks (ONNs).

ABSTRACT This work presents a unified fractal law that explains how natural systems evolve across all scales—from geology, climate, and biology on Earth to stars, galaxies, and cosmic structures. The study demonstrates that diverse processes follow the same numerical architecture: fractal scaling sequences (1–3–7–10–30–70–...), threshold nodes (7 × 10ⁿ), and golden-ratio stability and collapse boundaries (1.618 and 2.618). A central result of this research is the formulation of the Orymbetov Law of Transitional Chaos: “Chaos in natural systems is not destruction, but the mandatory reorganization phase between two stable fractal regimes.” This principle reframes chaos as a structured transition rather than a breakdown. Nature consistently reorganizes systems through controlled instability before re-establishing a new, more efficient configuration. This mechanism is visible in geological cycles, climate shifts, ecosystem restructuring, biological development, neural degradation, stellar evolution, and galactic dynamics. Even simple physical processes—such as ice melting into water and reorganizing into a flowing river—illustrate the same universal pattern: stable structure → transitional chaos → new stable flow. The study further emphasizes the Optimization Principle of Nature: systems naturally evolve toward configurations that minimize energy, maximize stability, and maintain coherence across fractal scales. This principle explains why similar geometric patterns appear in pro

Energy minimizing maps (E.M.M.s) play a central role in the calculus of variations, partial differential equations (PDEs), and geometric analysis. These maps are often embedded into $C^\infty$ Riemannian manifolds to minimize the Dirichlet Energy functional under certain prescribed conditions. For understanding physical phenomena where systems naturally evolve to states of minimal energy, the geometric analysis of these maps has provided elucidating insights. This paper explores the geometric and analytic properties of energy minimizing maps, tangent maps, and the singular set (sing(u)). We begin by establishing key concepts from analysis, including the Sobolev Space $W^{1,2}$ harmonic functions, and Hausdorff dimension. Significant results about the density function, its upper semi-continuity, and the compactness theorem for tangent maps, and theorems for homogeneous degree zero minimizers are presented. Also analyzed in detail is the singular set (sing(u)), its Hausdorff dimension, and geometric structure. We conclude with open problems that are rich in research potential, and the far-reaching implications if these problems are to be solved.