Study of discontinuous differential equations close to the intersection of two discontinuity surfaces

2016 ◽  
Author(s):  
Nicola Guglielmi ◽  
Ernst Hairer
Keyword(s):  
Differential Equations ◽  
Discontinuous Differential Equations ◽  
Discontinuity Surfaces

Lyapunov Instability for Discontinuous Differential Equations

2021 ◽  
Vol 2 (2) ◽  
pp. 49-58
Author(s):  
Iguer Luis Domini dos Santos
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Discontinuous Differential Equations ◽  
Lyapunov Instability

The present work studies the Lyapunov instability for discontinuous differential equations through the use of the notion of Carathéodory solution to differential equations. From Lyapunov's first instability theorem and Chetaev's instability theorem, which deal with instability to ordinary differential equations, two Lyapunov instability results for discontinuous differential equations are obtained.

On the Number of Limit Cycles of Discontinuous Liénard Polynomial Differential Systems

2018 ◽  
Vol 28 (14) ◽  
pp. 1850175
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Yan Wang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Second Order ◽  
Discontinuous Differential Equations ◽  
Differential Systems ◽  
Lower Upper ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Averaging Theorem

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

scholarly journals Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2449
Author(s):  
Flaviano Battelli ◽  
Michal Fečkan
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Singularly Perturbed ◽  
Two Dimensional ◽  
Singularly Perturbed Equation ◽  
Discontinuous Differential Equations ◽  
Scalar Variable ◽  
Perturbed Equation

We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation.

scholarly journals Averaging Methods for Design of Spacecraft Hysteresis Damper

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ricardo Gama ◽  
Anna D. Guerman ◽  
Ana Seabra ◽  
Georgi V. Smirnov
Keyword(s):  
Differential Equations ◽  
Parameter Optimization ◽  
Numerical Optimization ◽  
Gravity Gradient ◽  
Analytic Method ◽  
Discontinuous System ◽  
Discontinuous Differential Equations ◽  
Averaging Methods ◽  
Attitude Stabilization ◽  
Averaging Techniques

This work deals with averaging methods for dynamics of attitude stabilization systems. The operation of passive gravity-gradient attitude stabilization systems involving hysteresis rods is described by discontinuous differential equations. We apply recently developed averaging techniques for discontinuous system in order to simplify its analysis and to perform parameter optimization. The results obtained using this analytic method are compared with those of numerical optimization.

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