@article{Sun2025, 
author = {Yanmei Sun},
title = {The coexistence of quasi-periodic and blow-up solutions for a class of impact oscillators without the twist condition},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {5},
pages = {10263-10282},
keywords = {KAM theorem, impact oscillators, boundedness of solutions, quasi-periodic solutions, unbounded solutions},
url = {https://www.sciopen.com/article/10.3934/math.2025467},
doi = {10.3934/math.2025467},
abstract = {The purpose of this paper was the study of the dynamic behavior of impact oscillators:                                                                         {                                                                                                    x                        ″                                            +                      a                      (                      x                      )                                                                    x                                                  2                          n                          +                          1                                                                    +                                              ∑                                                  l                          =                          0                                                                          m                                                                                            p                        l                                            (                      t                      )                                                                    x                                                  l                                                                    =                      0                      ,                                            for                                            x                      (                      t                      )                      >                      0                      ,                                                                                                                        x                      (                      t                      )                      ≥                      0                      ,                                                                                                                          x                        ′                                            (                                              t                                                  0                                                                          +                                                                    )                      =                      −                                              x                        ′                                            (                                              t                                                  0                                                                          −                                                                    )                      ,                                            if                                            x                      (                                              t                                                  0                                                                    )                      =                      0                      ,                                                                                                                              where the positive function    a  (  x  ) is a smooth    T-periodic oscillator violating the monotone twist condition. We have proved that the above equation has an infinite number of bounded solutions as well as a solution that escapes to infinity in a finite amount of time.}
}