Modeling Transitions in Complex Systems by Multiplicative Effect of Temporal Patterns Extracted from Signal Flows

This study presents a mathematical model based on Fourier decomposition of a sequence of internal signals generated in a complex system by a sequence of external pulses (time series) for characterizing suddenly emerging phenomena as nonlinear transitions. Newly created temporal patterns extracted from internal signal flow (mathematically represented as oscillations with long period) interact as new entities in a multiplicative manner with subsequent pulses from the external time series (already existing entities) in order to generate nonlinear transitions within the system. Such effects are enhanced when the period of external pulses creating new patterns is similar to the settling time of the complex system (this being the condition for an efficient external action). For complex systems where both classical and quantum phenomena generated by external time series are involved, this mathematical model can correctly explain the transition from classical to quantum behaviour (corresponding to a more ordered structure) avoiding typical contradictions generated by analysis performed on transient time intervals or by wave superposition.