West Yorkshire Police would like to speak to David Riley, 36, from Harehills, following a series of incidents in the city on Saturday.
A van hit a cyclist in Sheepscar at 17:30 GMT and then hit several more vehicles before being abandoned.
Another van was then stolen after the occupants were threatened with an axe. It ended up in Headingley where a bicycle was stolen, police said.
It is believed the axe was used to remove the locked bike close to St Chad’s Rise before the suspect fled into nearby woods.
Det Insp Richard Holmes, of Leeds District CID, said: “We have identified David Riley as being the suspect for these incidents and we have been carrying out extensive enquiries to trace and arrest him.”
We would like to hear from anyone who has seen him since these incidents on Saturday or who has any information that could assist.
“While we don’t believe he presents any immediate ongoing danger to the public, we advise anyone who does see him not to approach him and to contact police immediately via 999.”
Riley is described as white, 5ft 8ins and stocky with a black beard. He has a scar across his left eyebrow.