Cyclic animation is widely used in computer graphics and interactive content.It supports seamless playback in games, VR, and interactive simulation,where short clips must repeat smoothly over long durations. Achievingphysically plausible cyclic synthesis from an input sequence is challengingbecause the endpoint states of the observed sequence rarely match exactly,and the governing equations of the underlying system are often unavailable.We therefore propose an equation-free framework that identiffes a Koopmansurrogate from the observed trajectory and computes a cyclic trajectory byapplying a Fourier-parameterized, time-varying control force under a hardtemporal periodicity constraint. The resulting formulation reduces cyclicsynthesis to a linearly constrained quadratic program that can be solvedefffciently through a structured KKT system. Our method is applicable toa diverse range of examples, including N-body systems, cloth, deformableobjects, shallow water, etc.

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