Is the trajectory of a fire rocket inherently chaotic? All of you: a happy and physics-rich new year!
When I looked at the firing rockets launched at 00:10 2020, most of them went straight up. 
Some exploded before time. Some made strange movements. So it's my guess that firing rockets can show chaotic, non-linear behavior. 
The question though is: is this non-linear movement intrinsic (will it show in outer space, assuming it can fire away without oxygen), or are the circumstances responsible? I think the latter is true.
This question looks a bit like this question, which is already a duplicate.
 A: Rocket-propelled fireworks have nozzles made out of pressed cardboard or fired clay, contain fuel and oxidizer that is often not well-mixed or correctly proportioned, contain no guidance control systems, and have bodies and fins that are assembled out of cardboard and glue by poorly-paid workers. It is therefore natural for them to exhibit unpredictably bad behavior, as you noted! 
Regarding inherently chaotic behavior of rockets in general, note that a long, slender rocket with an engine generating thrust at one end is dynamically like a pencil balanced upright on its tip: its natural tendency will be to veer off course in response to any small perturbation. For this reason, all large rockets contain either small rocket engines that can furnish sideways thrust in response to signals from the guidance system's gyroscopes and accelerometers, or gimbal-mounted main engines which can be slightly tilted via hydraulic or electric motors during the rocket's flight, to keep it pointed in the right direction. 
Smaller rockets can rely on fast acceleration to build up enough speed after ignition to the point where fins and rudders can develop enough aerodynamic force to help keep the rocket going straight. A well-made model rocket operating this way can fly a surprisingly straight path with nothing but fins for guidance- and no control system at all. 
A: Rockets are not necessarily unstable. As long as the center of pressure is lower than the center of gravity, aerodynamic forces will act to restore the rocket to its trajectory. Increasing fin size lowers the center of pressure, and adding weight to the nose raises the center of gravity. However, as niels said, most rockets don't rely on aerodynamics to maintain stability, they tend to rely on active control. Source: https://www.grc.nasa.gov/WWW/K-12/rocket/rktstab.html
