Quantum mechanics

Function (biology)

Pure mathematics

Bifurcation

Biology

Equivalence (formal languages)

Planar

Applied mathematics

Limit cycle

Computer graphics (images)

Hamiltonian system

Computer science

Mathematical optimization

Limit (mathematics)

Hamiltonian (control theory)

Evolutionary biology

Saddle-node bifurcation

Physics

Nonlinear system

Integrable system

Mathematics

Piecewise linear function

Method of averaging

Parameter space

Mathematical analysis

Geometry

Bifurcation diagram

Periodic orbits

Piecewise

Journal of Differential Equations

It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar analytic (or C∞) near-Hamiltonian differential systems. In this paper, we study piecewise smooth near-integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even for systems in high dimensional space. Particularly, we obtain the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems. We finally provide an application example.

Bifurcation methods of periodic orbits for piecewise smooth systems