Smyrna was a beautiful and wealthy city but also a centre of mandated emperor worship. Refusing to comply with this mandate could lead to the loss of legal status, persecution and even martyrdom.
The message to the Church in Smyrna applies prophetically to the church in the post-apostolic era, when Christians were viciously persecuted by the Roman Empire.
The ten days refers to the ten years of the Diocletian persecution from AD303 to AD313, when Constantine the Great issued the Edict of Milan granting Christians religious freedom.
Smyrna was called synagogue of satan because it was the centre of various pagan cults and promoted the cult of emperor worship as compulsory.
Source reference: SDA 1Q19 Quarterly