Bifurcations and Arnold Tongues of a Multiplier-Accelerator Model

In this paper, we investigate the bifurcations of a multiplier-acceler-ator model with nonlinear investment function in an anti-cyclical fiscal policy rule. Firstly, we give the conditions that the model produces supercritical flip bifurcation and subcritical one respectively. Secondly, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two 2-periodic orbits. Thirdly, it is proved that the model undergoes supercritical Neimark-Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourthly, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark-Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate a attracting 2-periodic orbit, an stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting 7-periodic orbit on the invariant circle.

Topological stability and shadowing of dynamical systems from measure theoretical viewpoint

In this paper it is proved that a topologically stable invariant measure has no sinks or sources in its support; an expansive homeomorphism is topologically stable if it exhibits a topologically stable nonatomic Borel support measure and a continuous map has the shadowing property if there exists an invariant measure with the shadowing property such that each almost periodic point is contained in the support of the invariant measure.

A seasonally invariant deep transform for visual terrain-relative navigation

Visual terrain-relative navigation (VTRN) is a localization method based on registering a source image taken from a robotic vehicle against a georeferenced target image. With high-resolution imagery databases of Earth and other planets now available, VTRN offers accurate, drift-free navigation for air and space robots even in the absence of external positioning signals. Despite its potential for high accuracy, however, VTRN remains extremely fragile to common and predictable seasonal effects, such as lighting, vegetation changes, and snow cover. Engineered registration algorithms are mature and have provable geometric advantages but cannot accommodate the content changes caused by seasonal effects and have poor matching skill. Approaches based on deep learning can accommodate image content changes but produce opaque position estimates that either lack an interpretable uncertainty or require tedious human annotation. In this work, we address these issues with targeted use of deep learning within an image transform architecture, which converts seasonal imagery to a stable, invariant domain that can be used by conventional algorithms without modification. Our transform preserves the geometric structure and uncertainty estimates of legacy approaches and demonstrates superior performance under extreme seasonal changes while also being easy to train and highly generalizable. We show that classical registration methods perform exceptionally well for robotic visual navigation when stabilized with the proposed architecture and are able to consistently anticipate reliable imagery. Gross mismatches were nearly eliminated in challenging and realistic visual navigation tasks that also included topographic and perspective effects.

Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

Stable invariant manifolds with application to control problems

<p style='text-indent:20px;'>In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.</p>

Calculation of Invariant Manifolds of Piecewise-Smooth Maps

Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. Results. A method for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps is developed. The main result is formulated as a statement. The method is based on an original approach to finding the inverse function, the idea of which is to reduce the problem to a nonlinear first-order equation. Conclusion. A numerical method is described for calculating stable invariant manifolds of piecewise smooth maps that simulate impulse automatic control systems. The method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. The method is based on an original approach to finding the inverse function, which consists in reducing the problem to solving a nonlinear first-order equation. This approach eliminates the need to solve systems of nonlinear equations to determine the inverse function and overcome the accompanying computational problems. Examples of studying the global dynamics of piecewise-smooth mappings with multistable behavior are given.

The role of the Manager in the organization's personnel management system (ABC-Electro)

The management style does not characterize the Manager's behavior in General, but rather it is stable, invariant, and constantly manifests itself in various situations. The search for and use of optimal management styles are designed to increase employee achievement and satisfaction, and, as a result, the level of productivity. The management style is characterized by a stable set of traits of the Manager, which are manifested in his relations with subordinates. The management style does not reflect the General behavior of the Manager, but rather a stable, invariant one that constantly manifests itself in various situations. In modern conditions, the success of the case is determined not only by the nature of the relationship between the Manager and subordinate and the degree of freedom that they are granted, but also by a number of other circumstances. This is expressed in "multidimensional" management styles, which represent a set of complementary, intertwining approaches, each of which is independent of the others. The search for and use of optimal management styles are designed to increase employee achievement and satisfaction, and, as a result, the level of productivity. When choosing management methods, you must adhere to the following requirements of the "Golden mean": an effective Manager must be able to balance between different management styles of his team, and then the motivation of employees will be much higher. The leader of the future should be focused on the market and customers, constantly strive for progress, set development directions, be a driving force for effective changes, be talented and develop leadership qualities and teamwork skills of employees. In modern conditions, in their practical activities, the Manager must constantly adjust their management style in accordance with changing internal and external conditions.

Structuralism and contemporary mass art

The subject of this research is the impact of structuralism as a scientific direction upon mass art. Stable invariant structures discovered by the structuralists in multiple artworks can be observed in mass art. Structuring, which initially was a method of research, turned into one of the practical recommendations on reating new works in mass art. The goal consists in the analysis of susceptibility factors of mass culture to the ideas of structuralism and results of using methodology of structuralism in mass artistic production. The initial methodological focus of this work lied in the concept of juxtaposition of craft and art, which goes back to I. Kant and is applied in modern aesthetics. The author leans on the methods of structuring and comparative structural analysis, as well as the elements of functional analysis. The main conclusion of consists in the statement that susceptibility of mass culture to the ideas of structuralism is substantiated by its economic goals, need to possess reliable and scientifically proven tools that would ensure commercial success of the artworks. However, the patterned application of structuring methods in mass art is capable of creating only craft products, rather than actual art.

Bifurcation Analysis of a Discrete-Time Two-Species Model

We study the local dynamics and bifurcation analysis of a discrete-time modified Nicholson–Bailey model in the closed first quadrant R+2. It is proved that model has two boundary equilibria: O0,0,Aζ1−1/ζ2,0, and a unique positive equilibrium Brer/er−1,r under certain parametric conditions. We study the local dynamics along their topological types by imposing method of Linearization. It is proved that fold bifurcation occurs about the boundary equilibria: O0,0,Aζ1−1/ζ2,0. It is also proved that model undergoes a Neimark–Sacker bifurcation in a small neighborhood of the unique positive equilibrium Brer/er−1,r and meanwhile stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the period or quasi-periodic oscillations between host and parasitoid populations. Some simulations are presented to verify theoretical results. Finally, bifurcation diagrams and corresponding maximum Lyapunov exponents are presented for the under consideration model.

Nested Closed Invariant Curves in Piecewise Smooth Maps

The paper describes how several coexisting stable closed invariant curves embedded into each other can arise in a two-dimensional piecewise-linear normal form map. Phenomena of this type have been recently reported for a piecewise smooth map, modeling the behavior of a power electronic DC–DC converter. In the present work, we demonstrate that this type of multistability exists in a more general class of models and show how it may result from the well-known period adding bifurcation structure due to its deformation so that the phase-locking regions start to overlap. We explain how this overlapping structure is related to the appearance of coexisting stable closed invariant curves nested into each other. By means of detailed, numerically calculated phase portraits we hereafter present an example of this type of multistability. We also demonstrate that the basins of attraction of the nested stable invariant curves may be separated from each other not only by repelling closed invariant curves, as previously reported, but also by a chaotic saddle. It is suggested that the considered kind of multistability is a generic phenomenon in piecewise smooth dynamical systems.