Mathematical Sciences: Chaos-Integrability Transition in Nonlinear Dynamical Systems: Exponental Asymptotics Approach

9500644 Tovbis该项目的目标是研究扰动非线性系统中从可积（可预测）到不可积（混乱）动力学的转变。这种转变在各种各样的物理、生物、化学等问题（不可压缩流体中的湍流、原子从金属表面的散射、种群动力学中的宿主-寄生虫模型、原子-硅藻碰撞中的复合物形成等）中起着重要作用。它也是理解计算问题中数值不稳定性（数值混沌）的本质的关键。该方法结合了经典的分析和现代渐近技术（指数渐近）的问题。 在我们的初步研究中，模型的例子显示了一个很好的协议之间的“理论”结果和数值模拟。 该项目的目标是研究奇异摄动非线性系统中从可积到不可积（混沌）动力学的过渡。这种转变在许多物理、生物、化学等问题中起着重要的作用。这也是至关重要的理解数值不稳定性（数值混沌）il计算问题的性质。我们特别感兴趣的情况下，扰动的影响是指数小的小参数的问题。我们建议应用指数渐近的技术研究扰动可积系统中混沌的发生的分析机制，并近似扰动系统的动力学。 这种方法在我们的初步调查中证明是卓有成效的。 ***

9500644 Tovbis The objective of the project is to study the transition from integrable (predictable) to nonintegrable (chaotic) dynamics in perturbed nonlinear systems. This transition plays a fundamental role in a very wide variety of physical, biological, chemical, etc. problems (turbulence in incompressible fluids, scattering of atoms from metal surfaces, host - parasitoid models in population dynamics, complex formation in atom-diatom collisions, etc.). It is also crucial in understanding the nature of numerical instabilities (numerical chaos) in computational problems. The proposed approach to the problem combines classical analytic and modern asymptotic technique (exponential asymptotics). In our initial studies the model example shows an excellent agreement between the "theoretical" results and numerical simulations. %%% The objective of the project is to study the transition from integrable to nonintegrable (chaotic) dynamics in singularly perturbed nonlinear systems. This transition plays a fundamental role in a very wide variety of physical, biological, chemical, etc. problems. It is also crucial in understanding the nature of numerical instabilities (numerical chaos) il computational problems. Of our particular interest are situations when the effect of perturbation is exponentially small in the small parameter of the problem. We propose to apply the technique of exponential asymptotics to the study of the analytical mechanism of the onset of chaos in perturbed integrable systems and to approximate the dynamics of perturbed systems. This approach has proven fruitful in our initial investigation. ***