scholarly journals Dynamical phenomena connected with stability loss of equilibria and periodic trajectories

2021 ◽  
Vol 76 (5) ◽  
pp. 883-926
Author(s):  
A. I. Neishtadt ◽  
D. V. Treschev
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Small Parameter ◽  
Period Doubling ◽  
Stability Loss ◽  
Loss Of Stability ◽  
Soft Or ◽  
Periodic Trajectories ◽  
The Subject ◽  
Dynamic Bifurcation

Abstract This is a study of a dynamical system depending on a parameter . Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on , the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, varies slowly with time (the case of a dynamic bifurcation). In the simplest situation , where is a small parameter. More generally, may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. Bibliography: 88 titles.

scholarly journals Periodic Solutions for the Eccentricity and Inclination First Order Resonance

1992 ◽  
Vol 152 ◽  
pp. 231-232
Author(s):  
Marisa A. Nitto ◽  
Wagner Sessin
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Periodic Solutions ◽  
Reference System ◽  
Resonance Problem ◽  
Order Resonance ◽  
First Order ◽  
Primary Body ◽  
Different Types ◽  
The Subject

For the first order resonance, the problem of the motion of two small masses around a primary body can be of three different types: eccentricity, inclination or eccentricity-inclination. The eccentricity type resonance problem has been the subject of several works since Poincaré(1902). The inclination type resonance problem was studied by Greenberg(1973) who used a particular reference system to obtain an integrable auxiliary system. Sessin and Ferraz-Mello(1984) studied the eccentricity type resonance problem considering the eccentricities of the orbits of the two small masses. Sessin(1991) study the inclination type resonance problem for an arbitrary reference system. In this paper we will study a dynamical system that includes both types of resonance. This study is based in the models developed by Sessin and Ferraz-Mello(1984) and Sessin(1991). The resulting system of differential equation is non-integrable; thus, the families of trivial periodic solutions are studied.

Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories

2008 ◽  
Vol 4 (1) ◽  
Author(s):  
Wonmo Kang ◽  
Phanikrishna Thota ◽  
Bryan Wilcox ◽  
Harry Dankowicz
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Period Doubling ◽  
Hybrid Dynamical Systems ◽  
Hybrid Dynamical System ◽  
Multiple Impacts ◽  
Grazing Bifurcation ◽  
Periodic Trajectories ◽  
Time Dynamics ◽  
Saddle Node

This paper presents the application of a newly developed computational toolbox, TC-HAT(TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of a sample hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, period-doubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.

Period-Increasing Indicates Loss of Stability

2019 ◽  
Vol 29 (12) ◽  
pp. 1950159
Author(s):  
Xiao-Song Yang ◽  
Lei Wang
Keyword(s):  
Dynamical System ◽  
Mathematical Theory ◽  
Regime Shift ◽  
Stability Loss ◽  
Ecological Models ◽  
Loss Of Stability ◽  
Two Dimensional ◽  
Heteroclinic Bifurcation

We first present a mathematical theory on period-increasing as an indicator of stability loss in a dynamical system of any dimension not less than two, and show that in two-dimensional autonomous ODEs, the period-increasing is closely associated with homoclinic or heteroclinic bifurcation. Then we demonstrate that the proposed period-increasing regime shift can occur in some ecological models.

Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories

2007 ◽  
Author(s):  
Wonmo Kang ◽  
Bryan Wilcox ◽  
Harry Dankowicz ◽  
Phanikrishna Thota
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Period Doubling ◽  
Hybrid Dynamical Systems ◽  
Hybrid Dynamical System ◽  
Multiple Impacts ◽  
Grazing Bifurcation ◽  
Periodic Trajectories ◽  
Time Dynamics ◽  
Saddle Node

This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of an example hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, perioddoubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.

RANDOM DIFFERENTIAL EQUATIONS WITH RANDOM DELAYS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 369-388 ◽  
Author(s):  
M. J. GARRIDO-ATIENZA ◽  
A. OGROWSKY ◽  
B. SCHMALFUSS
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Random Dynamical System ◽  
Random Delay ◽  
Random Differential Equation ◽  
Autonomous Case ◽  
Random Differential Equations ◽  
Uniqueness Of A Solution

We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals ALGORITHM AND SOFTWARE FOR CALCULATING THE STABILITY OF THE FORM OF EQUILIBRIUM OF DISCRETE SYSTEMS

2019 ◽  
Vol 13 (3) ◽  
pp. 44-49
Author(s):  
A.A. SHKURUPIY ◽  
A.N. PASCHENKO ◽  
P.B. MYTROFANOV
Keyword(s):  
Discrete Systems ◽  
Stability Loss ◽  
Transcendental Equation ◽  
Loss Of Stability ◽  
Displacement Method ◽  
Software Complex ◽  
Complex Functions ◽  
Transcendental Functions ◽  
The Stability ◽  
Theoretical Mechanics

The paper presents an algorithm for calculating the stability of the form of equilibrium of the first kind of compressed discrete systems by the method of displacements in combination with themethods of iterations and bisection. The use of the displacement method in combination with the iteration and bisection methods makes it possible to effectively determine the minimum critical stress or strain at the first bifurcation and their corresponding form of loss of stability, both for statically determined and statically undetectable systems. This approach, using matrixforms, makes it possible to significantly simplify the calculations of the analytical condition for the loss of stability of compressed discrete systems (the stability loss equation), which has high orders, as well as to construct the form of loss of stability corresponding to a critical load, that is, to solve the problem of loss of stability of equilibrium. The calculation of the compressed discrete system on the stability of the form of equilibrium actually reduces to the solution of the difficultly described nonlinear transcendental equation, which is the equation of loss of stability. The difficulty lies in the absence of an analytical solution of such an equation due to the presence of complex functions of Zhukovsky, which have transcendental functions in their structure. Such solution can be performed only with the use of numerical methods. This algorithm for calculating the loss of equilibrium of the first kind of compressed discrete systems by displacement in combination with the methods of iteration and bisection is implemented in the software complex "Persist" for a PC in Windows OS. The program was approbated and implemented in theeducational process at the Department of Structural and Theoretical Mechanics of the Poltava National Technical Yuri Kondratyuk University during the training of specialists in engineering specialties.

Self-Excited Oscillations in Dynamical Systems Possessing Retarded Actions

1942 ◽  
Vol 9 (2) ◽  
pp. A65-A71 ◽  
Author(s):  
Nicholas Minorsky
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Linear Equation ◽  
Finite Order ◽  
Characteristic Equation ◽  
High Derivative ◽  
Time Lags ◽  
Constant Coefficients ◽  
The Subject

Abstract There exists a variety of dynamical systems, possessing retarded actions, which are not entirely describable in terms of differential equations of a finite order. The differential equations of such systems are sometimes designated as hysterodifferential equations. An important particular case of such equations, encountered in practice, is when the original differential equation for unretarded quantities is a linear equation with constant coefficients and the time lags are constant. The characteristic equation, corresponding to the hysterodifferential equation for retarded quantities in such a case, has a series of subsequent high-derivative terms which generally converge. It is possible to develop a simple graphical interpretation for this equation. Such systems with retarded actions are capable of self-excitation. Self-excited oscillations of this character are generally undesirable in practice and it is to this phase of the subject that the present paper is devoted.

scholarly journals An Elementary Proof of the Theorems of Cauchy and Mayer

1935 ◽  
Vol 4 (3) ◽  
pp. 112-117
Author(s):  
A. J. Macintyre ◽  
R. Wilson
Keyword(s):  
Differential Equation ◽  
Existence Theorem ◽  
Elementary Proof ◽  
Theoretical Discussion ◽  
Practical Advantage ◽  
Total Differential ◽  
Method Of Solution ◽  
The Subject ◽  
Image Position ◽  
Natural Way

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equationThis method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

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