Periodic solution of a second order, autonomous, nonlinear system

Abstract

There exist a sufficient condition for the existence of at least one periodic solution for a type of second order autonomous ordinary differential equations. The correctness of the condition has been pointed out by Schauder's fixed point theorem. In order to indicate the validity of the assumptions made, two illustrative examples, showing its application in the nonlinear vibration and relaxation oscillation are presented.

Periodic behavior of a nonlinear third order vibrating system

“…Pliss in 1965 and Schmitt in 1971 used the fixed-point theorem and showed its application in finding the condition of periodic solutions of ordinary differential equations [25]. Since then, many investigators have used this method for the existence of periodic solution of differential equations [26][27][28][29][30][31]. It is interesting to know that Bohl was the first scientist that proposed fixed-point theorems in 1904 and then independently by Brouwer in 1911.…”

Oscillation Criteria of Two-Dimensional Time-Scale Systems

“…Proposition 2 Let z 1 be a function from T to R and v be a nondecreasing function from R to R such that v ∘ z 1 is rd-continuous. Suppose also that p ≥ 0 is rd-continuous and α ∈ R: Then Next, note that if x; y ðÞ is a nonoscillatory solution of system (15), then one can easily prove that x is also nonoscillatory. This result was shown by Anderson in [37] when ct ðÞ0: Because the proof when ct ðÞ = 0 is very similar to the proof of the case ct ðÞ0, we leave it to the readers.…”

“…Also Bartolini et al [13,14] consider an uncertain secondorder nonlinear system and propose a new approximate linearization and sliding mode to control such systems. In addition to the nonoscillation for two-dimensional systems of first-order equations, periodic and subharmonic solutions are also investigated in [15][16][17], and significant contributions have been made. Another type of two-dimensional systems of dynamic equations is the Emden-Fowler type equation, named after E. Fowler after he did the mathematical foundation of a second-order differential equation in a series of four papers during 1914-1931 (see [18][19][20][21]).…”

Existence of Periodic Solutions for the Generalized Form of Mathieu Equation

“…The correctness of the periodicity conditions for the existence of a periodic solution for a variety of engineering problems were pointed out by the Schauder's fixed-point theorem [5][6][7][8][9]. Mahmoud and other researchers carried out analytical and numerical investigations into the periodic solution of certain simple systems with two and three coupled nonlinear Hill's equations [10,11].…”