Limit cycles appearing from perturbations of cubic piecewise smooth center with double invariant real straight lines

Original · EN

This paper investigates the exact number of limit cycles given by the averaging theory of first order for the piecewise smooth integrable non-Hamiltonian system eqnarray* (x,y)=cases (-y(x+a)²+ε f+(x,y),x(x+a)²+ε g+(x,y)),x≥0, (-y(x+b)²+ε f-(x,y),x(x+b)²+ε g-(x,y)), x<0, caseseqnarray* where ab≠ 0, 0<|ε|≪ 1, and f±(x,y) and g±(x,y) are polynomials of degree n. It is proved that the exact number of limit cycles emerging from the period annulus surrounding the origin is linear depending on n and it is at least twice the associated estimation of smooth systems.