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Controlling chaos is fundamental in many applications, and for this reason, many techniques have been proposed to address this problem. Here, we propose a strategy based on an optimal placement of the sensor and actuator providing global observability of the state space and global controllability to any desired state. The first of these two conditions enables the derivation of a model of the system by using a global modeling technique. In turn, this permits the use of feedback linearization for designing the control law based on the equations of the obtained model and providing a zero-flat system. The procedure is applied to three case studies, including two piecewise linear circuits, namely, the Carroll circuit and the Chua circuit whose governing equations are approximated by a continuous global model. The sensitivity of the procedure to the time constant of the dynamics is also discussed. [ABSTRACT FROM AUTHOR]

The great interest of this system is, when the single parameter value HT ht is varied, to progressively switch from a dissipative to a conservative dynamics ( HT ht ), the so-called I labyrinth walk i . This is based on a pinning error vector whose HT ht th component is given by the difference between the output of the pinner and the output of the HT ht th node. In fact, a pioneering paper by Rössler in 1976 directly demonstrates that simple (bio)chemical reaction systems may show chaotic dynamics.[8] Shortly thereafter, chaos was found experimentally in two (bio)chemical reactions systems,[[99]] and later on in multiple other chemical reactions.[[101]] An article in the Focus Issue discusses chaos and other complex dynamics in the peroxidase-oxidase (PO) reaction,[62] which is the first experimental (bio)chemical reaction system to display chaotic dynamics. [Extracted from the article]

Considering in situ observations, chaos theory was taken as a basis to study the activity of anecic earthworms based on cast production from September 2016 to January 2018 in the Dong Cao watershed (Vietnam). To study this activity, the global modeling technique was used to obtain deterministic models of ordinary differential equations directly from observational time series. The obtained models show that the behavior of earthworms is chaotic; it is coupled to the dynamics of soil water content in a complex (integrative) way and can be interpreted as an habituation/sensitization process. However, this coupling is insufficient to explain the desynchronization of the cast production variations observed at different study sites. The retroaction of earthworm activity on soil is required to explain the spatiotemporal discrepancies. [ABSTRACT FROM AUTHOR]

When a chaotic attractor is produced by a three-dimensional strongly dissipative system, its ultimate characterization is reached when a branched manifold—a template—can be used to describe the relative organization of the unstable periodic orbits around which it is structured. If topological characterization was completed for many chaotic attractors, the case of toroidal chaos—a chaotic regime based on a toroidal structure—is still challenging. We here investigate the topology of toroidal chaos, first by using an inductive approach, starting from the branched manifold for the Rössler attractor. The driven van der Pol system—in Robert Shaw's form—is used as a realization of that branched manifold. Then, using a deductive approach, the branched manifold for the chaotic attractor produced by the Deng toroidal system is extracted from data. [ABSTRACT FROM AUTHOR]

Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed. [ABSTRACT FROM AUTHOR]

The aim of the present work is to investigate the possibility to retrieve the original sets of dynamical equations directly from observational time series when all the system variables are observed. Time series are generated from chosen dynamical systems, and the global modeling technique is applied to obtain optimal models of parsimonious structure from these time series. The obtained models are then compared to the original equations to investigate if the original equations can be retrieved. Twenty-seven systems are considered in the study. The Rössler system is first used to illustrate the procedure and then to test the robustness of the approach under various conditions, varying the initial conditions, time series length, dynamical regimes, subsampling (and resampling), measurement noise, and dynamical perturbations. The other 26 systems (four rational ones included) of various algebraic structures, sizes, and dimensions are then considered to investigate the generality of the approach. [ABSTRACT FROM AUTHOR]

One of the main tasks in network theory is to infer relations among interacting elements. We propose global modeling as a tool to detect links between nodes and their nature. Various situations using small network motifs are investigated under the assumption that the variable to be measured at each node provides full observability when isolated. Such a choice ensures no intrinsic difficulties for getting a global model in the coupled situation. As a first step toward unveiling the coupling function in larger network motifs, we consider three different scenarios involving Rössler systems diffusively coupled, in a couple or embedded in a network, or parametrically forced. We show that the global modeling is able to determine not only the existence of an interaction but also its functional form, to retrieve the dynamics of the whole system, and to extract the equations governing the single node dynamics as if it was isolated. [ABSTRACT FROM AUTHOR]

Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson's group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson's early career in interaction with Otto E. Rössler. [ABSTRACT FROM AUTHOR]

An epidemic of Ebola Virus Disease (EVD) broke out in Guinea in December 2013. It was only identified in March 2014 while it had already spread out in Liberia and Sierra Leone. The spill over of the disease became uncontrollable and the epidemic could not be stopped before 2016. The time evolution of this epidemic is revisited here with the global modeling technique which was designed to obtain the deterministic models from single time series. A generalized formulation of this technique for multivariate time series is introduced. It is applied to the epidemic of EVD in West Africa focusing on the period between March 2014 and January 2015, that is, before any detected signs of weakening. Data gathered by the World Health Organization, based on the official publications of the Ministries of Health of the three main countries involved in this epidemic, are considered in our analysis. Two observed time series are used: the daily numbers of infections and deaths. A four-dimensional model producing a very complex dynamical behavior is obtained. The model is tested in order to investigate its skills and drawbacks. Our global analysis clearly helps to distinguish three main stages during the epidemic. A characterization of the obtained attractor is also performed. In particular, the topology of the chaotic attractor is analyzed and a skeleton is obtained for its structure. [ABSTRACT FROM AUTHOR]

This paper attempts to bridge the gap between tokamak reactor design and plasma physics. The analysis demonstrates that the overall design of a tokamak fusion reactor is determined almost entirely by the constraints imposed by nuclear physics and fusion engineering. Virtually, no plasma physics is required to determine the main design parameters of a reactor: a; R0; B0; Ti; Te; p; n; τE; I. The one exception is the value of the toroidal current I, which depends upon a combination of engineering and plasma physics. This exception, however, ultimately has a major impact on the feasibility of an attractive tokamak reactor. The analysis shows that the engineering/nuclear physics design makes demands on the plasma physics that must be satisfied in order to generate power. These demands are substituted into the well-known operational constraints arising in tokamak physics: the Troyon limit, Greenwald limit, kink stability limit, and bootstrap fraction limit. Unfortunately, a tokamak reactor designed on the basis of standard engineering and nuclear physics constraints does not scale to a reactor. Too much current is required to achieve the necessary confinement time for ignition. The combination of achievable bootstrap current plus current drive is not sufficient to generate the current demanded by the engineering design. Several possible solutions are discussed in detail involving advances in plasma physics or engineering. The main contribution of the present work is to demonstrate that the basic reactor design and its plasma physics consequences can be determined simply and analytically. The analysis thus provides a crisp, compact, logical framework that will hopefully lead to improved physical intuition for connecting plasma physic to tokamak reactor design. [ABSTRACT FROM AUTHOR]

The dynamics underlying cereal crops in the northern region of Morocco is investigated using a global modelling technique applied to a vegetation index time series derived from satellite measurements, namely, the normalized difference vegetation index from 1982 to 2008. Two three-dimensional chaotic global models of reduced size (14-term and 15-term models) are obtained. The model validation is performed by comparing their horizons of predictability with those provided in previous studies. The attractors produced by the two global models have a complex foliated structure--evidenced in a Poincare section--rending a topological characterization difficult to perform. Thus, the Kaplan-Yorke dimension is estimated from the synthetic data produced by our global models. Our results suggest that cereal crops in the northern Morocco are governed by a weakly dissipative three-dimensional chaotic dynamics. [ABSTRACT FROM AUTHOR]

General Lagrangian theory of even and odd fields on an arbitrary smooth manifold is considered. Its nontrivial reducible gauge symmetries and their algebra are defined in this very general setting by means of the inverse second Noether theorem. In contrast with gauge symmetries, nontrivial Noether and higher-stage Noether identities of Lagrangian theory can be intrinsically defined by constructing the exact Koszul–Tate complex. The inverse second Noether theorem that we prove associates with this complex the cochain sequence with the ascent operator whose components define nontrivial gauge and higher-stage gauge symmetries. These gauge symmetries are said to be algebraically closed if the ascent operator can be extended to a nilpotent operator. The necessary conditions for this extension are stated. The characteristic examples of Yang–Mills supergauge theory, topological Chern–Simons theory, gauge gravitation theory, and topological background field (BF) theory are presented. [ABSTRACT FROM AUTHOR]

Geometric (Schrödinger) quantization of nonrelativistic mechanics with respect to different reference frames is considered. In classical nonrelativistic mechanics, a reference frame is represented by a connection on a configuration space fibered over a time axis R. Under quantization, it yields a connection on the quantum algebra of Schrödinger operators. The operators of energy with respect to different reference frames are examined. [ABSTRACT FROM AUTHOR]

A generic degenerate Lagrangian system of even and odd fields is examined in algebraic terms of the Grassmann-graded variational bicomplex. Its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. We show that, if a certain necessary and sufficient condition holds, one can associate to a degenerate Lagrangian system the exact Koszul–Tate complex with the boundary operator whose nilpotency condition restarts all its Noether and higher-stage Noether identities. This complex provides a sufficient analysis of the degeneracy of a Lagrangian system for the purpose of its BV quantization. [ABSTRACT FROM AUTHOR]

We present Noether’s second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noether’s second theorem for Lagrangian systems on fiber bundles Y→X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved. [ABSTRACT FROM AUTHOR]

Given a completely integrable system, we associate to any connection on a fiber bundle in invariant tori over a parameter manifold the classical and quantum holonomy operator (generalized Berry’s phase factor), without any adiabatic approximation. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighborhood of its regular (not necessarily compact) invariant manifold which makes this dynamical system into a partially integrable Hamiltonian system. This Poisson structure is by no means unique. Bi-Hamiltonian partially integrable systems are described in some detail. As an outcome, we state the conditions of quasiperiodic stability (the KAM theorem) for partially integrable Hamiltonian systems. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Quantum systems with adiabatic classical parameters are widely studied, e.g., in the modem holonomic quantum computation. We here provide complete geometric quantization of a Hamiltonian system with time-dependent parameters, without the adiabatic assumption. A Hamiltonian of such a system is affine in the temporal derivative of parameter functions. This leads to the geometric Berry factor phenomena. [ABSTRACT FROM AUTHOR]

We provide geometric quantization of the vertical cotangent bundle V[sup *]Q→Q→R, equipped with the canonical Poisson structure and treated as a momentum phase space of nonrelativistic time-dependent mechanics. We show that this quantization is equivalent to fiberwise quantization of symplectic fibers of V[sup *]Q→R and that the quantum algebra of time-dependent mechanics is an instantwise algebra. Quantization of the classical evolution equation defines a connection on this instantwise algebra and describes quantum evolution in time-dependent mechanics as a parallel transport. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Reports on the constraint reaction force of ideal nonholonomic constraints in time-dependent mechanics. Use of the vertical extension of the Hamiltonian formalism; Use of real, finite dimensional, second countable manifolds; Newtonian systems with nonholonomic constraints.

In the framework of the adjoint forms over the jet spaces of connections and using a canonical jet shift differential, we give a geometrical interpretation of the Yang-Mills equations both in a direct and Lagrangian formulation. [ABSTRACT FROM AUTHOR]

Cohomology of the bicomplex of exterior forms on the infinite-order jet space of a smooth fiber bundle of a field model is computed. This provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory and time-dependent mechanics. In the case of an affine fiber bundle, the outcomes to BRST theory are discussed. We show that there is no topological obstruction to constructing global descent equations in the even field sector of BRST theory. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The key point of the study of constraints in Hamiltonian time-dependent mechanics lies in the fact that a Poisson structure does not provide dynamic equations and a Poisson bracket of constraints with a Hamiltonian is ill-defined. We describe Hamiltonian dynamics in terms of Hamiltonian forms and connections on the vertical cotangent bundle V[sup *]Q→R seen as a momentum phase space. A Poisson bracket {,}[sub V] on V[sup *]Q is induced by the canonical Poisson bracket {,}[sub T] on the cotangent bundle T[sup *]Q. With {,}[sub V], an algebra of first and second class time-dependent constraints is described, but we use the pull-back of the evolution equation onto T[sup *]X and the bracket {,}[sub T] in order to extend the constraint algorithm to time-dependent constraints. The case of Lagrangian constraints of a degenerate almost regular Lagrangian is studied in detail. One can assign to this Lagrangian L a set of Hamiltonian forms (which are not necessarily degenerate) such that any solution of the corresponding Hamilton equations which lives in the Lagrangian constraint space is a solution of the Lagrange equations for L. In the case of an almost regular quadratic Lagrangian, the complete set of global nondegenerate Hamiltonian forms with the above-mentioned properties is described. We construct the Koszul-Tate resolution of the Lagrangian constraints for this Lagrangian in an explicit form. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

It is shown that any second-order dynamic equation on a configuration bundle Q→R of nonrelativistic mechanics is equivalent to a geodesic equation with respect to a (nonlinear) connection on the tangent bundle TQ→Q. The case of quadratic dynamic equations is analyzed in detail. The equation for Jacobi vector fields is constructed and investigated by the geometric methods. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]