The method of Unrestricted Harmonic Balance (UHB) as exposed and applied to purely timedependent systems, stiff systems, and chemical waves in the preceding 3 papers is extended to the case of transcendental functions. So, e.g., exponential functions even in mass action kinetics systems occur, if a wider range of temperature is considered as is the case in continuous stirred tank reactors (CSTR). Here the reaction heat is removed by heat exchangers, but since the volume is in most cases considerable, the system cannot be kept strictly isothermal and instabilities are known to arise. Here the oscillatory case is treated with the method of UHB