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The nonlinear limit-point/limit-circle problem for higher order equations

Miroslav Bartušek, Zuzana Došlá, John R. Graef (1998)

Archivum Mathematicum

We describe the nonlinear limit-point/limit-circle problem for the n -th order differential equation y ( n ) + r ( t ) f ( y , y ' , , y ( n - 1 ) ) = 0 . The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

The nonlinearly damped oscillator

Juan Luis Vázquez (2003)

ESAIM: Control, Optimisation and Calculus of Variations

We study the large-time behaviour of the nonlinear oscillator m x ' ' + f ( x ' ) + k x = 0 , where m , k > 0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case f ( x ' ) = A | x ' | α - 1 x ' with α real, A > 0 . We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

The Nonlinearly Damped Oscillator

Juan Luis Vázquez (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the large-time behaviour of the nonlinear oscillator m x ' ' + f ( x ' ) + k x = 0 , where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case   f ( x ' ) = A | x ' | α - 1 x '  with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.

Two-point oscillatory solutions to system with relay hysteresis and nonperiodic external disturbance

Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva (2024)

Applications of Mathematics

We study an n -dimensional system of ordinary differential equations with a constant matrix, a relay-type nonlinearity, and an external disturbance in the right-hand side. We consider a nonideal relay characteristic. The external disturbance is described by the product of an exponential function and a sine function with an initial phase as a parameter. We assume the matrix of the linear part and the vector at the relay characteristic such that, by a nonsingular transformation, the system is reduced...

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