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In this paper, we study the limit cycles in the discontinuous piecewise linear planar systems separated by a nonregular line and formed by linear Hamiltonian vector fields without equilibria. Motivated by [Llibre & Teixeira, 2017], where an open problem was posed: Can piecewise linear differential systems without equilibria produce limit cycles? We prove that such systems have at most two limit cycles, and the limit cycles must intersect the nonregular separation line in two or four points. More precisely, the exact upper bound of crossing limit cycles is two, and this upper bound can indeed be reached: either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points. Based on Poincaré map, the stability of various limit cycles is also proved. In addition, we give some concrete examples to illustrate our main results.

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In this paper, we investigate the homoclinic orbits for the three-dimensional continuous piecewise linear generalized Michelson systems via analytical methods and numerical simulation. Based on the Poincaré map and invariant manifold theory, we discuss the existence of homoclinic orbits connecting the saddle-focus equilibrium. Finally, numerical simulations are presented to illustrate our results.

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