Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line

Abstract

This paper provides the classification of the phase portraits in the Poincaré disc of all piecewise linear continuous slow-fast differential systems with two zones separated by a straight line. The sufficient and necessary conditions for existence and uniqueness of limit cycles are also given.

A Bendixon–Dulac theorem for some piecewise systems

“…Hence, we have at most one limit cycle. The existence, for these values of the parameters, is proved in [19]. Finally, as the function B ± 1 is continuous over Σ, with the sign assumptions for α ± and λ ± , Theorem 1.2 proves the hyperbolicity and unstability properties.…”

“…With the weighted blow-up (u, v) → (u, pu 2 ) and rescaling time (dividing by u 2 ), system (19) becomes…”

“…Clearly, Φ + 1 (x + 1 ) = Φ + 2 (x + 1 ). The function Φ + 2 is obtained computing the approximation up to some order k of the unstable separatrix of the saddle located at the origin of system (19), that is, system (6) in the chart U 1 . Although Φ + 2 define a curve without contact, it does not cross the y-axis.…”

Limit Cycles Bifurcating from a Periodic Annulus in Discontinuous Planar Piecewise Linear Hamiltonian differential System with Three Zones

“…Piecewise differential systems with two zones are the most studied, either for their applications in modeling phenomena in general or for their apparent simplicity (see [10,20]). As in the smooth case, the researches are mainly concentrated to the determination of the number and position of the limit cycles of these systems.…”

New iterative methods for dense linear systems

“…Linear system plays an important role of applications in engineering and scientific computing such as boundary element methods, quantum mechanical problems and large least squares problems [1,2,7,12,16,17]. Large sparse linear systems can be solved efficiently by iterative methods, especially those based on a Krylov subspace.…”