Complexity analysis on orbits of solutions of nonlinear partial differential equations

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We show nondeterministic recurrent properties of discrete quasi-periodic orbits by estimating the GAP values between upper and lower recurrent dimensions. We also show the relations between the GAP values and the topological entropy, which indicates its chaotic complexity, and then we clarify its unpredictability.We show the unpredictability of the quasi-periodic orbits with multi-frequencies, which are an extremal irrational number and its square, by estimating the positive gaps of their recurrent dimensions.