Bifurcations from families of periodic solutions in piecewise differential systems

Abstract

Consider a differential system of the formfunctions and T -periodic in the variable t. Assuming that the unperturbed system x ′ = F 0 (t, x) has a d-dimensional submanifold of periodic solutions with d < m, we use the Lyapunov-Schmidt reduction and the averaging theory to study the existence of isolated T -periodic solutions of the above differential system.

Limit cycles of continuous and discontinuous piecewise-linear differential systems in R3

“…Let P be the space formed by the periodic solutions of (9) or (10). If dim(P ) = dim(D) = d then the following result follows directly from Theorem B of [18].…”

“…In [19] the authors extended the averaging theory to discontinuous differential systems. An improvement of this result for a much bigger class of discontinuous differential systems is given in [18].…”

“…If b+g = 0 then the Jacobian determinant (6) will be zero, so we can assume b + g = 0. Then we obtain the solution r 0 = 0, which corresponds to an equilibrium point of the unperturbed system (18), therefore the averaging theory does not provide any periodic solution in this subcase. Subcase (ii.4): Suppose now that h = 0 and m − hi = 0.…”

“…Changing to cylindrical coordinates x = r cos θ, y = r sin θ, z = z, the system (2) writes as (16)ṙ = εg 1 (θ, r, z), θ = 1 + ε r cos θ(e + h|z| + f r cos θ) − sin θ(a + d|z|+ (b − g)r cos θ) − cr sin 2 θ , z = ir cos θ + ε(j + m|z| + kr cos θ + lr sin θ), where g 1 (θ, r, z) = cos θ a + d|z| + br cos θ + sin θ e + h|z|+ (c + f )r cos θ + gr sin 2 θ, and taking θ as the new independent variable system (2) becomes (17) r ′ = εg 1 (θ, r, z), z ′ = ir cos θ + εg 2 (θ, r, z), where the prime denotes derivative with respect to θ, and g 2 (θ, r, z) = j + m|z| − f ir cos 3 θ + lr sin θ − i cos 2 θ e+ h|z| + (g − b)r sin θ + cos θ kr + di|z| sin θ+i sin θ(a + cr sin θ) . The unperturbed system is(18)…”