The Council of Chalcedon in A.D. 451 resulted in the schism of Christendom into two groups. The Catholic (Rome) and Greek (Byzantine) Churches accepted the Council, while the Syrian (Antioch) and Coptic (Alexandria) Churches rejected it.
I would like to add that it was not just a section of the Antiochian and Alexandrian churches rejecting Chalcedon. There was a section in Constantinople and the Church of the East rejecting the council.
Non-Chalcedonians in the East were later organized, but in Constantinople the Church was suppressed.
The whole of Armenian Church also rejected the council. Also, the Churches were not known as Coptic and Syrian Orthodox then. It was Church of Antioch and Church of Alexandria (please refer to the writings of St. Severus for evidence).
St. Anthimus 1, Patriarch of Constantinople ( c. 535-536. ) was Orthodox (non-chalcedonian). Thus the Orthodox or non-chalcedonian movement was not confined to Antioch or Alexandria, it was an universal movement against the Nestorian error accepted in Chalcedon.
There is an official committee setup for dialogue with the RC church. All OO churches and the RC are part of this committee.