Many Biblical scholars believe Jesus’ final journey to Jerusalem began in Luke 9:51. However, in order to maintain this idea, they have to conclude that Luke either uses about nine and one half chapters to show Jesus wandering aimlessly all over the countryside, zigzagging all the way to Jerusalem, and even reversing course to return to Galilee (cf. Luke 17:11), or he places Jesus’ movements in a hodgepodge of seemingly unrelated incidents that took place in various parts of Palestine, having no perceived order in them at all. It is astonishing for me to see how far one will go in order to protect a favorite idea from being disproved, or, perhaps it may be better to say, some scholars have embraced the idea for so long that they believe it must be supported in the Scriptures somewhere.
In Luke 13:22 we are told that Jesus went through certain cities and villages as he journeyed toward Jerusalem, but, as I said above, most scholars believe Jesus is still on the journey he had begun in Luke 9:51. Nevertheless, it seems obvious that Jesus had already been to Jerusalem when Luke tells us he visited Martha and Mary in Bethany (Luke 10:38). Therefore, if Jesus had already come to Jerusalem, which was his intended end in Luke 9:51, this journey Luke speaks of in Luke 13:22 must be a different one from that (Luke 9:51). This would be the easiest way to understand the text.
Luke 9:51 implies that Jesus went to Jerusalem to celebrate the Passover, which Luke 13:1 shows he did do, because it was at a Passover season when Pilate killed those Galilean Jews who were among the crowd that protested his building an aqueduct to bring water to Jerusalem with Temple funds. If this is true, then we have evidence in the Gospel records that Jesus celebrated four Passovers during his Public ministry. The first was in Jerusalem (John 2), and the second was celebrated in Galilee just after John the Baptist was slain by Herod (John 6). The third Passover season that occurred during Jesus’ public ministry was celebrated in Jerusalem one year prior to his crucifixion (Luke 9:51 to Luke 13:9). His final Passover, of course, was celebrated with his crucifixion.
Nevertheless, Luke 13:22 tells us that Jesus went to Jerusalem on another occasion, and if we compare these events with John 10 we will discover similar teaching, which would put Jesus in Jerusalem about three months prior to his death, and there the Gospel of John tells us Jesus celebrated the Feast of Dedication or Hanukkah (cf. John 10:22). The common language involving the door (G2374), in John 10:7, 9, 22 and Luke 13:25 seems to place Jesus in Jerusalem during the Feast of Dedication (Hanukkah) at this point in Luke.
Moreover, considering the season of the year and reason for the feast, namely the Feast of Dedication, then the word strive (G75) seems to take on the idea of not being conformed to this world. Strive to avoid being a part of the crowd that mindlessly follows the dictates of a man or human authority that does not submit to God. This is what occurred during the time of Antiochus Epiphanies when the Jewish nation ran after Grecian syncretism, to the point of abandoning their duties in the Temple. One cannot serve God and man (cf. Luke 16:13).
Again, considering the reason for the feast (John 10:22), Jesus seems to place the many who desire to enter the Kingdom of God without trusting in him (Luke 13:24) in the same category as those who ran after Grecian syncretism and abandoned their duties to God. It was obvious that Jesus was sent by God, but the authorities and the greater part of the nation that followed them rejected Jesus as their Messiah and King (cf. Luke 19:27).
 See Josephus, Antiquities of the Jews 18.3.2.