An orientation-preserving diffeomorphism of the circle can be represented in the form \(x\to x+f(x)\), where \(x\in \mathbb R\), \(f(x+2\pi)=f(x)\). Considering the case when \(f(x)\) is a trigonometric polynomial of degree \(n>0\) with real coefficients the author proves that the number of periodic cycles cannot exceed \(2n\). The estimate is sharp for \(x\to x+\epsilon \sin nx\). The problem was raised by V. I. Arnol’d [Usp. Mat. Nauk 38, No. 4(232), 189–203 (1983; Zbl 0541.34035)].