scholarly journals Properties of the solution set of a generalized differential equation

1972 ◽  
Vol 6 (3) ◽  
pp. 379-398 ◽  
Author(s):  
J.L. Davy
Keyword(s):  
Differential Equation ◽  
Additional Assumption ◽  
Initial Conditions ◽  
Upper Semicontinuity ◽  
The State ◽  
Solution Set ◽  
Initial Time ◽  
State Variable ◽  
Generalized Differential Equation ◽  
Generalized Differential

We prove that the solution set of a generalized differential equation is connected and that points on the boundary of the solution funnel are peripherally attainable. This is done without the additional assumption of continuity in the state variable required in previous results. The result on upper semicontinuity of the solution set with respect to initial conditions is extended to include variations of initial time.

scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

Difference Synchronization of Identical and Nonidentical Chaotic and Hyperchaotic Systems of Different Orders Using Active Backstepping Design

2018 ◽  
Vol 13 (5) ◽  
Author(s):  
Eric Donald Dongmo ◽  
Kayode Stephen Ojo ◽  
Paul Woafo ◽  
Abdulahi Ndzi Njah
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Lyapunov Stability Theory ◽  
The State ◽  
State Variables ◽  
State Variable ◽  
New Type ◽  
The Difference ◽  
Synchronization Scheme ◽  
Hyperchaotic Systems

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

Output Control by Linear Time-Varying Systems using Parametric Identification Methods

2020 ◽  
Vol 21 (7) ◽  
pp. 387-393
Author(s):  
V. Q. Dat ◽  
A. A. Bobtsov
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Control Object ◽  
The State ◽  
Time Varying ◽  
State Variables ◽  
Matrix Differential Equation ◽  
Matrix Differential ◽  
Uniform Observability ◽  
Riccati Matrix

In this paper the problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. This procedure involves the synthesis of linear static feedbacks on state variables in the case of known parameters of the plant. If state variables are not measured, then for the observer design using the matrix Riccati differential equation, using the dual scheme, which provides for the transposition of the state matrix and the replacement of the input matrix by the output matrix. It is well known that an observer of state variables built on the basis of a solution of the Riccati matrix differential equation ensures the exponential stability of a closed loop system in the case of uniform observability. Despite the fact that this type of observer can be classified as universal, its have a number of significant drawbacks. The main problem of such observers is the need for accurate knowledge of the parameters and the requirement for uniform observability, which in practice cannot always be realized. Thus, the problem of the new methods design for constructing observers of state variables of linear non-stationary systems is still relevant. Some time ago, a number of methods for the adaptive observers design of state variables for nonlinear systems were proposed. The main idea of the synthesis of observers was based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. This approach in the English language literature is called PEBO. This paper, based on the PEBO method, proposes a new approach for the observers design of state for non-stationary systems. This approach provides the possibility of obtaining monotonic convergence estimates with transient time tuning.

scholarly journals Identification of highly deformed even–even nuclei in the neutron- and proton-rich regions of the nuclear chart from the B(E2)↑ and E2 predictions in the generalized differential equation model

2015 ◽  
Vol 24 (12) ◽  
pp. 1550091
Author(s):  
R. C. Nayak ◽  
S. Pattnaik
Keyword(s):  
Differential Equation ◽  
Large Deformations ◽  
Transition Probability ◽  
Electric Quadrupole ◽  
Mass Region ◽  
Equation Model ◽  
Generalized Differential Equation ◽  
Predicted Values ◽  
Generalized Differential ◽  
Physical Quantities

We identify here the possible occurrence of large deformations in the neutron- and proton-rich ([Formula: see text]-rich and [Formula: see text]-rich) regions of the nuclear chart from extensive predictions of the values of the reduced quadrupole transition probability [Formula: see text] for the transition from the ground state to the first [Formula: see text] state and the corresponding excitation energy [Formula: see text] of even–even nuclei in the recently developed generalized differential equation (GDE) model exclusively meant for these physical quantities. This is made possible from our analysis of the predicted values of these two physical quantities and the corresponding deformation parameters derived from them such as the quadrupole deformation [Formula: see text], the ratio of [Formula: see text] to the Weisskopf single-particle [Formula: see text] and the intrinsic electric quadrupole moment [Formula: see text], calculated for a large number of both known as well as hitherto unknown even–even isotopes of oxygen to fermium (0 to FM; [Formula: see text]–100). Our critical analysis of the resulting data convincingly support possible existence of large collectivity for the nuclides [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], whose values of [Formula: see text] are found to exceed 0.3 and even 0.4 in some cases. Our findings of large deformations in the exotic [Formula: see text]-rich regions support the existence of another “island of inversion” in the heavy-mass region possibly caused by breaking of the [Formula: see text] subshell closure.

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