Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

ABSTRACT(Chemical) kinetic models in terms of ordinary differential equations correspond to finite-dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the so called reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion and exploitation for deeper insight of unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We newly relate both of them within our variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as novel viewpoints on dimension reduction approaches that might open up new deep perspectives and approaches to the problem from dynamical systems theory.

The "criticality hypothesis," based on observed scale-free correlations in neural activity, posits that the brain operates at a critical point of transition between two phases. However, the validity of this hypothesis is still debated. Here, employing a commonly used model of cortical dynamics, we find that a phase of long-range order (LRO) in neural activity may be induced by memory (time nonlocality) without invoking criticality. The cortical-dynamics model contains fast and slow time scales that govern the neural and resource (memory) dynamics, respectively. When the resource dynamics are sufficiently slow, we observe an LRO phase, which manifests in avalanche size and duration probability distributions that are fit well by power laws. When the slow and fast time scales are no longer sufficiently distinct, LRO is destroyed. Since this LRO phase spans a wide range of parameters, it is robust against perturbations, unlike critical systems.

The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation–oscillations

We examined the role of the cerebellum in patients with tremor-dominant cervical dystonia by measuring the adaptive capacity of rapid reflexive eye movements (saccades). We chose the saccade adaptation paradigm because, unlike other motor learning paradigms, the real-time modification of saccades cannot "wait" for the sensory (visual) feedback. Instead, saccades rely primarily on the internal reafference modulated by the cerebellum. The saccade adaptation happens over fast and slow timescales. The fast timescale has poor retention of learned response, while the slow timescale has strong retention. Cerebellar defects resulting in loss of function affect the fast timescale but the slow timescale of saccade adaptation is retained. In contrast, maladaptive cerebellar disorders feature the absence of both fast and slow timescales. We were able to measure both timescales using noninvasive oculography in 6 normal individuals. In contrast, both timescales were absent in 12 patients with tremor-dominant cervical dystonia. These findings are consistent with maladaptive cerebellar outflow as a putative pathophysiological basis for tremor-dominant cervical dystonia.

This paper presents a novel tabulation strategy for the adaptive numerical integration of chemical kinetics using the computational singular perturbation (CSP) method. The strategy stores and reuses CSP quantities required to filter out fast dissipative processes, resulting in a non-stiff chemical source term. In particular, non-parametric regression on low-dimensional slow invariant manifolds (SIMs) in the chemical state space is used to approximate the CSP vectors spanning the fast chemical subspace and the associated fast chemical time-scales. The relevant manifold and its dimension varies depending on the local number of exhausted modes at every location in the chemical state space. Multiple manifolds are therefore tabulated, corresponding to different numbers of exhausted modes (dimensions) and associated radical species. Non-parametric representations are inherently adaptive, and rely on efficient approximate-nearest-neighbor queries. As the CSP information is only a function of the non-radical species in the system and has relatively small gradients in the chemical state space, tabulation occurs in a lower-dimensional state space and at a relatively coarse level, thereby improving scalability to larger chemical mechanisms. The approach is demonstrated on the simulation of homogeneous constant pressure H2–air and CH4–air ignition, over a range of initial conditions. For CH4–air, results are shown that outperform direct implicit integration of the stiff chemical kinetics while maintaining good accuracy.

Model reduction methods in chemical kinetics are used for simplification of models which involve a number of different time scales. Slow invariant manifolds in chemical composition space are supposed to be identified. A selection of state variables serve for parametrization of these manifolds. Species reconstruction methods are used to compute the values of the remaining variables in dependence of the parameters. We discuss theoretical results and numerical methods for an application of a model reduction method that is developed by D. Lebiedz based on optimization of trajectories. The main focus of this work is an application of the model reduction method to models of chemical combustion. The existence of a solution of the semi-infinite optimization problem, which has to be solved to obtain a local approximation of the slow manifold, is proven. A finite optimization problem for the same purpose is presented which can be solved with a generalized Gauss-Newton method. This method is used with an active set strategy. A filter framework and iterations with second order correction are employed for globalization of convergence. Families of neighboring optimization problems can be solved efficiently in a predictor corrector continuation scheme. The tangent space of the slow manifold can be computed by evaluation of sensitivity equations for the parametric optimization problem. A step size strategy is applied in the continuation scheme for efficient progress along the homotopy path. Results of an application of the presented method are shown and discussed. The test models range from simple test examples to realistic models of syngas combustion in air.

A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identifies the exact slow invariant manifold in the limit of both an infinite time horizon of the variational problem with fixed spectral gap and infinite spectral gap with a fixed finite time horizon. Numerical results for the linear and nonlinear model problems as well as a more realistic higher-dimensional chemical reaction mechanism are presented.

The paper considers the eighth-order proton exchange membrane (PEM) fuel-cell mathematical model and shows that it has a multi-time scale property, indicating that the dynamics of three model state space variables operate in the slow time scale and the dynamics of five state variables operate in the fast time scale. This multi-scale nature allows independent controllers to be designed in slow and fast time scales using only corresponding reduced-order slow (of dimension three) and fast (of dimension five) sub-models. The presented design facilitates the design of hybrid controllers, for example, the linear-quadratic optimal controller for the slow subsystem and the eigenvalue assignment controller for the fast subsystem. The design efficiency and its high accuracy are demonstrated via simulation on the considered PEM fuel cell model.

Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized Relaxation Redistribution Method

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

This work is about a slow-fast data assimilation system under Gaussian noisy fluctuations. First, we obtain its low dimensional reduction via an invariant slow manifold. Second, we prove that the low dimensional filter on the slow manifold approximates the original filter in a suitable metric. Finally, we illustrate this approximate filter numerically in an example.

Ascent Guidance for an Aerospace Plane

HVAC system dynamic management in communities via an aggregation–disaggregation framework

In this paper, we present a review about mathematical models of population and community dynamics incorporating individual behaviour. The considered models consist in coupling a game theoretic model, going on at a fast time scale and that describes the change of tactics among individuals in a population or in a community to a model going on at a slow time scale describing the effects in the long term of individual tactics on reproduction and mortality of the populations. As application, an extension of a stage-structured population with intra-individual behavioural variability is presented. Individuals can adopt three different tactics: hawk, dove and bully. We consider a stage-structured population with immature and mature individuals. A mature individual experiences the different tactics in his life and it is assumed that the game is played at a fast time scale in comparison to population dynamics. This fast part of the model is coupled to a slow part, which describes the growth of each sub-population. By use of aggregation of variables methods, we obtain a reduced system of two differential equations governing immature and total mature sub-populations at the slow time scale. The analysis of the model shows that when the cost of fightings is smaller, the proportion of hawks is larger. The total population size equilibrium shows a maximum with respect to the gain of the game.

Meta-equilibrium state of multi-species ambipolar diffusion and its relevance to polar mesopheric summer echoes