Unpredictable Solutions of Linear Impulsive Systems

Abstract

We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

“…The unpredictable point of the Bebutov dynamics is the unpredictable function. In papers [7][8][9][10][11][12][13][14][15], we provided a dynamical method, how to construct Poisson stable functions. Deterministic and stochastic dynamics have been used.…”

“…In papers [8][9][10]14] and books [7,13], discussing existence of unpredictable solutions, we have developed a new method how to approve Poisson stable solutions, since unpredictable functions are a subset of Poisson stable functions, and to verify the unpredictability one must check, if the Poisson stability is valid. The method is distinctly different than the comparability method by character of recurrence, which was introduced in [20] and later has been realized in several articles [21][22][23][24][25][26][27].…”

“…The method is distinctly different than the comparability method by character of recurrence, which was introduced in [20] and later has been realized in several articles [21][22][23][24][25][26][27]. Unlike papers [7][8][9][10][13][14][15][16][17][18][19], the present research is busy with the new type of Poisson stable functions. Correspondingly, it is the first time in literature, when quasilinear equations with Poisson stable coefficients are under investigation.…”

Unpredictability in Perturbed Quasilinear Systems with Regular Moments of Impulses

“…Unpredictable solutions of linear impulsive systems were investigated in paper [23], where the model admits unpredictable impulse moments. In contrast to paper [23], in this study we consider an impulsive system which comprises nonlinear terms in both the differential and impulse equations. Another difference of our study is the usage of regular impulse moments in the model, i.e., the equation (1.2) is satisfied.…”

“…Another difference of our study is the usage of regular impulse moments in the model, i.e., the equation (1.2) is satisfied. The concept of B-topology was utilized in [23] to define piecewise continuous unpredictable solutions since the unpredictability of the impulse moments causes the discontinuities of the solutions not to coincide under shifts of the time argument. However, in our case, the discontinuities of the solutions coincide under shifts of the time argument by integer multiples of ω because of the regularity of impulse moments.…”