Stability for a delayed switched nonlinear system of differential equations in a critical case

2020 ◽  
pp. 1-15
Author(s):  
Daniela Enciu ◽  
Andrei Halanay
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Critical Case ◽  
System Of Differential Equations ◽  
Switched Nonlinear System

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

scholarly journals A Modified SIRD Model to Study the Evolution of the COVID-19 Pandemic in Spain

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 723
Author(s):  
Vicente Martínez
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
System Of Differential Equations ◽  
Short Term ◽  
Exponential Expansion ◽  
Harmful Effects

In this paper, we use an SIRD model to analyze the evolution of the COVID-19 pandemic in Spain, caused by a new virus called SARS-CoV-2 from the coronavirus family. This model is governed by a nonlinear system of differential equations that allows us to detect trends in the pandemic and make reliable predictions of the evolution of the infection in the short term. This work shows this evolution of the infection in various changing stages throughout the period of maximum alert in Spain. It also shows a quick adaptation of the parameters that define the disease in several stages. In addition, the model confirms the effectiveness of quarantine to avoid the exponential expansion of the pandemic and reduce the number of deaths. The analysis shows good short-term predictions using the SIRD model, which are useful to influence the evolution of the epidemic and thus carry out actions that help reduce its harmful effects.

scholarly journals The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
Pavel A. Shamanaev ◽  
Olga S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

PERIODIC ORBITS IN THE NEWTON–LEIPNIK SYSTEM

2002 ◽  
Vol 12 (03) ◽  
pp. 511-523 ◽  
Author(s):  
BENJAMIN A. MARLIN
Keyword(s):  
Periodic Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Closed Orbit ◽  
Numerical Results ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Promising Candidate ◽  
Closed Orbits

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

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