The Existence of Periodic Solutions of Nonlinear Oscillators

Periodic Solutions for Differential Systems with Symmetries

“…However, these additional requirements are not germane to the discussions in this paper and so we omit them here. One observes that property E with respect to Q implies that the differential equation (2) remains invariant under the substitution ---t, --Q. In fact, using (2) (2), then e(t) Q(-t) is also a solution of (2).…”

“…Periodic solutions of this two dimensional system are easily constructed by employing elementary geometrical and analytical arguments together with the general existence theorem. Since the discussion in [2] is quite extensive, we shall not reconsider this special case. Indeed, our examples here mainly concern higher dimensional systems.…”

“…One observes that property E with respect to Q implies that the differential equation (2) remains invariant under the substitution ---t, --Q. In fact, using (2) (2), then e(t) Q(-t) is also a solution of (2). In particular, if the initial n-vector (0) d0 satisfies Qd0 d0, then (t) and (t) possess the same initial vector and, hence, are one and the same solution, i.e., (4) Q(-t) (t).…”

“…Our main purpose here is to illustrate applications by considering specific examples from a variety of applied problems. An important special case deals with the nonlinear oscillator (1)which has been investigated previously in [2]. Periodic solutions of this two dimensional system are easily constructed by employing elementary geometrical and analytical arguments together with the general existence theorem.…”

“…which has been investigated previously in [2]. Periodic solutions of this two dimensional system are easily constructed by employing elementary geometrical and analytical arguments together with the general existence theorem.…”