Phase Portraits Of Dynamical Systems Research Articles

Stratifications and foliations in phase portraits of gene network models.

Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section "Investigation of functions and plot of their graphs" of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.

Classes of Dynamic Systems with Various Combinations of Multipliers in Their Reciprocal Polynomial Right Parts

A family of differential dynamic systems is considered on a real plane of their phase variables x, y. The main common feature of systems under consideration is: every particular system includes equations with polynomial right parts of the third order in one equation and of the second order in another one. These polynomials are mutually reciprocal, i.e., their decompositions into forms of lower orders do not contain common multipliers. The whole family of dynamic systems has been split into subfamilies according to the numbers of different reciprocal multipliers in the decompositions and depending on an order of sequence of different roots of polynomials. Every subfamily has been studied in a Poincare circle using Poincare mappings. A plan of the investigation for each selected subfamily of dynamic systems includes the following steps. We determine a list of singular points of systems of the fixed subfamily in a Poincare circle. For every singular point in the list, we use the notions of a saddle (S) and node (N) bundles of adjacent to this point semi trajectories, of a separatrix of the singular point, and of a topo dynamical type of the singular point (its TD – type). Further we split the family under consideration to subfamilies of different hierarchical levels with proper numbers. For every chosen subfamily we reveal topo dynamical types of singular points and separatrices of them. We investigate the behavior of separatrices for all singular points of systems belonging to the chosen subfamily. Very important are: a question of a uniqueness of a continuation of every given separatrix from a small neighborhood of a singular point to all the lengths of this separatrix, as well as a question of a mutual arrangement of all separatrices in a Poincare circle Ω. We answer these questions for all subfamilies of studied systems. The presented work is devoted to the original study. The main task of the work is to depict and describe all different in the topological meaning phase portraits in a Poincare circle, possible for the dynamical differential systems belonging to a broad family under consideration, and to its numerical subfamilies of different hierarchical levels. This is a theoretical work, but due to special research methods it may be useful for applied studies of dynamic systems with polynomial right parts. Author hopes that this work may be interesting and useful for researchers as well as for students and postgraduates. As a result, we describe and depict phase portraits of dynamic systems of a taken family and outline the criteria of every portrait appearance.

Measurement of the threshold of community seismic resilience using dynamics-based metrics

Two dynamics-based threshold metrics are developed for characterizing seismic resilience of a community. An overdamped single-degree-of-freedom (SDOF) system is employed as an analogy to the physical modeling of the recovery process of a community. Three major intrinsic recovery mechanisms, i.e. long-term socioeconomic development, short-term socioeconomic assistance, robustness of engineering systems, are further clarified. Phase plane method which can characterize system stability straightforwardly is employed on the basis of some slow variables called the governing parameters. Two threshold metrics, i.e. recovery rapidity and recovery distance, are then proposed respectively, based on the observation about the features of phase portraits of dynamic systems treated in two different ways. Recovery angle which exhibits independence from initial conditions of dynamic systems performs well in representing resilience rapidity. Recovery distance quantitatively identifies a tolerance that can indicate if a community will be completely recovered or not under a certain seismic damage. The threshold metrics are believed to comply with the intrinsic physical definition of resilience more seriously.

Unified description of cosmological and static solutions in affine generalized theories of gravity: Vecton-scalaron duality and its applications

We briefly describe the simplest class of affine theories of gravity in multidimensional space-times with symmetric connections and their reductions to two-dimensional dilaton - vecton gravity field theories (DVG). The distinctive feature of these theories is the presence of an absolutely neutral massive (or tachyonic) vector field (vecton) with essentially nonlinear coupling to the dilaton gravity (DG). We show that in DVG the vecton field can be consistently replaced by an effectively massive scalar field (scalaron) with an unusual coupling to dilaton gravity. With this vecton - scalaron duality, one can use methods and results of the standard DG coupled to usual scalars (DGS) in more complex dilaton - scalaron gravity theories (DSG) equivalent to DVG. We present the DVG models derived by reductions of multidimensional affine theories and obtain one-dimensional dynamical systems simultaneously describing cosmological and static states in any gauge. Our approach is fully applicable to studying static and cosmological solutions in multidimensional theories as well as in general one-dimensional DGS models. We focus on global properties of the models, look for integrals and analyze the structure of the solution spaces. In integrable cases, it can be usefully visualized by drawing a `topological portrait' resembling phase portraits of dynamical systems and simply exposing global properties of static and cosmological solutions, including horizons, singularities, etc. For analytic approximations we also propose an integral equation well suited for iterations.

Elliptic stars in a chaotic night

A recurrent theme in the description of phase portraits of dynamical systems is that of elliptic islands in a chaotic sea. Usually this picture is invoked in the context of smooth twist maps of the annulus or the torus, like the standard map. In this setting, ‘elliptic islands’ refer to the topological disks bounded by periodic smooth curves surrounding elliptic periodic points. Establishing the existence of these curves is one of the many achievements of KAM Theory. The aim of this note is to approach the topic from a different angle, namely from the viewpoint of rotation theory in a purely topological setting. We study homeomorphisms of the two-torus, homotopic to the identity, which have no wandering open sets (as in the area-preserving case) and whose rotation set has a non-empty interior. We define local rotation subsets ρU(F) by restricting Misiurewicz and Ziemian's definition of the rotation set to starting points in a small open disk U. Our main result is the following dichotomy: Either ρU(F) is reduced to a single rational vector and U is contained in a periodic topological open disk which contains a periodic point, or ρU(F) is large, in the sense that its convex hull has a non-empty interior. This allows us to distinguish an ‘elliptic’ and a ‘chaotic’ regime, and as a consequence we obtain that in the chaotic region the dynamics are sensitive with respect to initial conditions.

Geometric Theory of Dynamic Systems with Control

The phase portraits of dynamic systems with control (differential inclusions) form a new class of objects which yield full information on system behavior through the analysis of singular sets of the phase portraits. A dynamic systems with control phase portrait is generated by filling the phase space with a family of trajectory funnel boundaries of the systems. The phase portrait approach permits using advanced mathematical techniques for studies of the dynamic systems with control properties.