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Biorhythm including neuron firing and protein-mRNA interaction are fundamental activities with diffusive effect. Their well-balanced spatiotemporal dynamics are beneficial for healthy sustainability. Therefore, calibrating both anomalous frequency and amplitude of biorhythm prevents physiological dysfunctions or diseases. However, many works were devoted to modulate frequency exclusively whereas amplitude is usually ignored, although both quantities are equally significant for coordinating biological functions and outputs. Especially, a feasible method coordinating the two quantities concurrently and precisely is still lacking. Here, for the first time, we propose a universal approach to design a frequency-amplitude coordinator rigorously via dynamical systems tools. We consider both spatial and temporal information. With a single well-designed coordinator, they can be calibrated to desired levels simultaneously and precisely. The practical usefulness and efficacy of our method are demonstrated in representative neuronal and gene regulatory models. We further reveal its fundamental mechanism and optimal energy consumption providing inspiration for biorhythm regulation in future.

In this paper, we investigate the dynamics of a fourth-order normal form near a double Takens-Bogdanov bifurcation. The reduced system of this normal form possesses eight pairs of homoclinic orbits for certain parameter values. The nonlinear time transformation method is applied to obtain an analytical approximation of the homoclinic orbit in the perturbed system and to construct the homoclinic bifurcation curve as well. Using numerical continuation, period-doubling and homoclinic-doubling cascades emanating from a codimension-2 bifurcation point are found. A codimension-2 homoclinic-gluing bifurcation point at which several homoclinic orbits concerning the origin glue together to form a new homoclinic orbit is also obtained. It is shown that in the vicinity of these bifurcation points, the system may exhibit chaos and chaotic attractors.