We give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic Riccati equation. In some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones. We also provide an example of the Riccati equation without periodic solutions which appears in the Euler vorticity dynamics.

We give a few sufficient conditions for the existence of two periodic solutions of the Riccati ordinary differential equation in the plane. We give also examples of the equation without periodic solutions.

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