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Whether there is similarity between two physical processes in the movement ofobjects and the complexity of behavior is an essential problem in science. Howto seek similarity through the adoption of quantitative and qualitativeresearch techniques still remains an urgent challenge we face. To this end, theconcepts of similarity transformation matrix and similarity degree areinnovatively introduced to describe similarity of orbits between twocomplicated discrete dynamical systems that seem to be irrelevant. Furthermore,we present a general optimal principle, giving a strict characterization fromthe perspective of dynamical systems combined with optimization theory. Forwell-known examples of chaotic dynamical systems, such as Lorenz attractor,Chua's circuit, R$\rm\ddot{o}$ssler attractor, Chen attractor, L$\rm\ddot{u}$attractor and hybrid system, with using of the homotopy idea, some numericalsimulation results demonstrate that similarity can be found in richcharacteristics and complex behaviors of chaotic dynamics via the optimalprinciple we presented.

Finding similarity of orbits between two discrete dynamical systems via optimal principle