A new technique is given to detect the minimal period of periodic solutions to autonomous systems.
Then, for any integer P > 1, (2) possesses at least a periodic solution with minimal period P. The rest of this paper is divided into two parts.
Then, for any integer P > 1, (2) possesses at least a periodic solution with minimal period P. Theorem 4 Suppose that F satisfies (F5) and (F7).
Then, for any integer P > 1, (2) possesses at least a periodic solution with minimal period P. Now, one weakens the conditions (F5) and (F7).
Then Eq. (2.4) has a T-periodic solution, and there exists a k 1 ≥ 1 such that, for every integer k ≥ k 1, Eq. (1.1) has a periodic solution with a minimal period kT, which makes exactly one revolution around the origin in the period time kT.
So, let be a periodic solution of period (not necessarily minimal).
Then equation (1.1) has at least one 2π-periodic solution, and for each (jin{Bbb {N}}) there is (m_{j}^in{Bbb {N}}), such that for every (kge m_{j}^) with k primes with j, there is at least one periodic solution (x_{k}(cdot)=x_{j,k}(cdot)) with minimal period (2kpi).
We prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. The rest of this paper is organized as follows.
By a subharmonic solution, it means a kT periodic solution with k ⩾ 2 an integer, that is, the minimal period is strictly greater than T. When k = 1, it is a periodic solution or harmonic.
This nontrivial periodic solution is periodic with periodic.
