Computational techniques based on ranks of Hankel matrices (H-ranks) is used to study the convergence to Arnold tongues in the circle map. It appears that the process of convergence to the phase-locked mode of the discrete stationary attractor is far from being trivial. Figures of pseudoranks of Hankel matrices constructed from transient solutions of the circle map carry important physical information about complex nonlinear processes and are also beautiful from the aesthetical point of view.

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