Periodic Liénard-type delay equations with state-dependent impulses

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

“…Remark 5 (Impulsive Melnikov function 'is continuous') Even though M(p, t) is defined for t / ∈ J , the expression (25) indicates that for any t i ∈ J , lim t↑t i M(p, t) = lim t↓t i M(p, t). Thus the {t i } consist of removable singularities; if 'filled in,' M would be continuous in t. The reason for this is that when crossing a jump value t i , both x u ε (p, t) and x s ε (p, t) get reset according to the same jump map, which according to Lemma 1 is continuous.…”

“…ThusF u,s p (s) = 1/(2s), and evaluating both parts of (27) gives R p (t) = e t/2 /2 for t = 0. Therefore, from (25), Some sample calculations for the specific choice of n = 2, t 1 = 0, t 2 = 1, g 1,2 (x 1 , x 2 ) = e t 1 x 1 and g 2,2 (x 1 , x 2 ) = The above can be explicitly integrated, leading to a not particularly illuminating lengthy expression. Its behaviour with p and t is shown in Figure 11, bearing in mind that positive M relates to flow across the heteroclinic from the lower to the upper strip.…”

“…Hence, (1) as it stands forms an ill-defined flow on Ω, a fact which has been highlighted by several authors in the past [21,22,20]. This problem arises because the effect of the impulse is spatially-dependent (sometimes referred to as 'state-dependent impulses' [23,24,25]), as would be reasonable in applications such as underwater explosions. This issue does not arise if the g i are independent of x (as in state-independent kicks [26]), or if the x-dependence is such that there is no ambiguity in the jump (for example where a jump in one spatial variable depends on a different spatial variable which does not encounter a jump [27,28,29], a specification which explicitly uses only x(t − i ) in its state-dependence [30], or under other special conditions [31]).…”

Second order BVPs with state dependent impulses via lower and upper functions

“…[1-3, 9, 10, 12, 13, 16, 17]. There are also some papers dealing with state-dependent impulsive periodic problems for first order differential equations [5,14,20,21,24] or for second order differential equations [6,7]. Other types of boundary value problems with state-dependent impulses have been studied very rarely.…”

Periodic boundary value problems for second-order impulsive integro-differential equations

“…In the simulations of such processes it is frequently convenient to neglect the durations of the rapid changes and to assume that the changes can be represented by state jumps. Appropriate mathematical models for processes of the type described above are so-called systems with impulsive effects, we refer the reader to the monographs [12,37,31] and the articles [1][2][3][4][5][6][7][8][9][13][14][15][16][17][18][19]21,22,24,25,[27][28][29][30]32,34,36,38,39].…”