Abstract
This paper concerns limit cycle bifurcations by perturbing a piecewise linear Hamiltonian system. We first obtain all phase portraits of the unperturbed system having at least one family of periodic orbits. By using the first-order Melnikov function of the piecewise near-Hamiltonian system, we investigate the maximal number of limit cycles that bifurcate from a global center up to first order ofε.
Highlights
Introduction and Main ResultsRecently, piecewise smooth dynamical systems have been well concerned, especially in the scientific problems and engineering applications
By using the first order Melnikov function, they proved that the maximal number of limit cycles on Poincarebifurcations is n up to first-order in ε
This paper focuses on studying the limit cycle bifurcations of system (1) in the case (1) of Figure 2 by using the first order Melnikov function
Summary
Piecewise smooth dynamical systems have been well concerned, especially in the scientific problems and engineering applications. Ẏ = −ax − b, with a2 +b2 ≠ 0 has possibly the following four different phase portraits on the plane (see Figure 1). By using the first order Melnikov function, they proved that the maximal number of limit cycles on Poincarebifurcations is n up to first-order in ε. The maximal number of limit cycles in the case (7) or (8) of Figure 2 is [(n−1)/2] on Poincare, Hopf and Homoclinic bifurcations up to first-order in ε, by using the first order Melnikov function. This paper focuses on studying the limit cycle bifurcations of system (1) in the case (1) of Figure 2 by using the first order Melnikov function.
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