scholarly journals Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

2018 ◽  
Author(s):  
Irina Andreeva ◽  
Alexey Andreev
Keyword(s):  
Dynamic Systems ◽  
Phase Portraits

scholarly journals Notes on the behaviour of trajectories of polynomial dynamic systems

2020 ◽  
Vol 313 ◽  
pp. 00014
Author(s):  
Irina Andreeva
Keyword(s):  
Dynamic Systems ◽  
Wide Spectrum ◽  
Modern Science ◽  
Seismic Stability ◽  
Phase Portraits ◽  
Main Task ◽  
Physical Processes ◽  
Science And Engineering ◽  
Research Stage

Dynamic systems play a key role in various directions of modern science and engineering, such as the mathematical modeling of physical processes, the broad spectrum of complicated and pressing problems of civil engineering, for example, in the analysis of seismic stability of constructions and buildings, in the fundamental studies of computing and producing systems, of biological and sociological events. A researcher uses a dynamic system as a mathematical apparatus to study some phenomena and conditions, under which any statistical events are not important and may be disregarded. The main task of the theory of dynamic systems is to study curves, which differential equations of this system define. During such a research, firstly we need to split a dynamic system’s phase space into trajectories. Secondly, we investigate a limit behavior of trajectories. This research stage is to reveal equilibrium positions and make their classification. Also, here we find and investigate sinks and sources of the system’s phase flow. As a result, we obtain a full set of phase portraits, possible for a taken family of differential dynamic systems, which describe a behavior of some process. Namely polynomial dynamic systems often play a role of practical mathematical models hence their investigation has significant interest. This paper represents the original study of a broad family of differential dynamic systems having reciprocal polynomial right parts, and describes especially developed research methods, useful for a wide spectrum of applications.

Kink soliton solutions and bifurcation for a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation in circuitry, chemistry or biology

2019 ◽  
Vol 33 (30) ◽  
pp. 1950372
Author(s):  
Mei-Xia Chu ◽  
Bo Tian ◽  
Hui-Min Yin ◽  
Su-Su Chen ◽  
Ze Zhang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wave Velocity ◽  
Dynamic Systems ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Mutant Gene ◽  
Phase Portraits ◽  
Original Equation ◽  
Kink Soliton

Circuitry and chemistry are applied in such fields as communication engineering and automatic control, environmental protection and material/medicine sciences, respectively. Biology works as the basis of agriculture and medicine. Studied in this paper is a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation for the electronic circuitry, chemical kinetics, population dynamics, neurophysiology, population genetics, mutant gene propagation, nerve impulses transmission or molecular crossbridge property in living muscles. Kink soliton solutions are obtained via the fractional sub-equation method. Change of the fractional order does not affect the amplitudes of the kink solitons. Via the traveling transformation, the original equation is transformed into the ordinary differential equation, while we obtain two equivalent two-dimensional planar dynamic systems of that ordinary differential equation. According to the bifurcation and qualitative considerations of the planar dynamic systems, we display the corresponding phase portraits when the traveling-wave velocity is nonzero or zero. Nonlinear periodic waves of the original equation are obtained when the traveling-wave velocity is zero.

scholarly journals Graph-analytical interpretation and modeling of phase portraits of cellular structures during their phase transformations

2018 ◽  
Vol 170 ◽  
pp. 01135
Author(s):  
Vladimir Kulikov
Keyword(s):  
Dynamical Systems ◽  
Dynamic Systems ◽  
Recursive Algorithm ◽  
Laser Speckle ◽  
Phase Portraits ◽  
Cellular Structures ◽  
Simulation Algorithm ◽  
Systems Models ◽  
Initial States ◽  
Discrete Moments

The article shows the possibility of representing the state of the cellular structures in the form of “phase portraits” of states in discrete moments of time. The generated structures are presented in the form of dynamic systems. Models of states of dynamical systems, is developed, based on the analysis and processing of laser speckle and other structures. The simulation algorithm of the phase portraits given in the paper allows to create information systems casts at given discrete points in time, and to control the process of formation of properties of structures. The transition of the studied systems from their initial states to the final presents a recursive algorithm with deferred calculation of the parameters of designed objects. The initial parameters of the studied systems are reported in the form of a graphic-analytical statistics and are defined further as texture geometric shapes of cellular structures.

EMERGENCIES AS A MANIFESTATION OF THE EFFECT OF BIFURCATION MEMORY IN CONTROLLED UNSTABLE SYSTEMS

2004 ◽  
Vol 14 (07) ◽  
pp. 2439-2447 ◽  
Author(s):  
M. I. FEIGIN ◽  
M. A. KAGAN
Keyword(s):  
Steady State ◽  
Dynamic Systems ◽  
Control Parameter ◽  
Transition Process ◽  
Underwater Vehicles ◽  
Phase Portraits ◽  
Unusual Behavior ◽  
Efficient Operation ◽  
Universal Dependence ◽  
Qualitative Changes

Unusual behavior of dynamic systems with a control parameter is analyzed. Bifurcation induced qualitative changes in phase portraits of such systems are quite routine and ensure efficient operation. This class of systems includes ships with high maneuvering capabilities, aircraft and controlled underwater vehicles designed to be unstable in steady-state motion that are interesting in terms of applications. Bifurcations may generate tracks of bifurcation memory that are "indicators" of regions of reduced controllability referred to as phase spots. The transition process in the phase spot is estimated qualitatively as a universal dependence of the index of loss of controllability on the control parameter. The proposed approach has allowed us to predict and investigate emergency manifestations of the bifurcation memory effect occurring during routine maneuvering.

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

2021 ◽  
Vol 2090 (1) ◽  
pp. 012095
Author(s):  
I A Andreeva
Keyword(s):  
Singular Point ◽  
Dynamic Systems ◽  
Small Neighborhood ◽  
Theoretical Work ◽  
Singular Points ◽  
Phase Portraits ◽  
Mutual Arrangement ◽  
Real Plane ◽  
Main Task ◽  
Hierarchical Levels

Abstract 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.

scholarly journals Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

2017 ◽  
Vol 27 (14) ◽  
pp. 1730051
Author(s):  
Amalia Luque ◽  
Julio Barbancho ◽  
Javier Fernández Cañete ◽  
Antonio Córdoba
Keyword(s):  
Phase Portrait ◽  
Dynamic Systems ◽  
Limit Cycles ◽  
Nonlinear Dynamic ◽  
Equilibrium Points ◽  
Nonlinear Dynamic Systems ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
System Behavior ◽  
Cylindrical Phase

Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360[Formula: see text]. In these cases the use of the phase portrait does not properly depict the system’s evolution. Normalized phase portraits or cylindrical phase portraits have been extensively used to overcome the original phase portrait’s disadvantages. In this research a new graphic representation is introduced: the phase shadow. Its use clearly reveals the system behavior while overcoming the drawback of the existing plots. Through the paper the method to obtain the graphic is stated. Additionally, to show the phase shadow’s expressiveness, a rotating pendulum is considered. The work exposes that the new graph is an enhanced representational tool for systems having equilibrium points, limit cycles, chaotic attractors and/or bifurcations.

Phase behavior of cationic and anionic surfactants at low concentrations

1992 ◽  
Vol 50 (1) ◽  
pp. 694-695
Author(s):  
E. Naranjo
Keyword(s):  
Phase Behavior ◽  
Structural Characterization ◽  
Dynamic Systems ◽  
Anionic Surfactants ◽  
Intermolecular Forces ◽  
Stable Form ◽  
Surfactant Mixtures ◽  
Low Concentrations ◽  
Cationic And Anionic Surfactants ◽  
General Method

Equilibrium vesicles, those which are the stable form of aggregation and form spontaneously on mixing surfactant with water, have never been demonstrated in single component bilayers and only rarely in lipid or surfactant mixtures. Designing a simple and general method for producing spontaneous and stable vesicles depends on a better understanding of the thermodynamics of aggregation, the interplay of intermolecular forces in surfactants, and an efficient way of doing structural characterization in dynamic systems.

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