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The complex Lyapunov exponent λ plays a vital role in characterizing the dynamics of a physical system. The real part of λ has frequently been related to as just the Lyapunov exponent and has been used for decades to characterize the stability of the system. The imaginary part or the frequency of oscillations can also give valid information about the dynamics of the system, particularly how it behaves near the equilibrium points. In this article we will show that the frequency versus Lyapunov exponent map can give additional information about the very nature of the system and provide background for detailed analysis concerning the applicability of the control technique and its robust nature. As an example of the applicability of the map, an appropriate model to investigate the origin and growth of the auto-oscillations are the circular YIG films. Starting with the low power ferromagnetic resonance spectrum and analyzing the behavior as a function of power the creation and evolution of “shoots” in the map have been demonstrated. The resulting map gives new insights about the relationship between the underlying dynamics of the system and the “growth” of the shoots into auto-oscillation fingers. This approach can explain many features of the auto-oscillation behavior and gives new insights into investigating techniques to control and synchronize chaos as well as to explain desynchronization bursts. © 1998 American Institute of Physics.