Periodic orbits of continuous-discontinuous piecewise differential systems with four pieces separated by the curve \(xy=0\) and formed by linear Hamiltonian systems (Q6586847)

Let \( Q_1, Q_2, Q_3, Q_4 \) be the quadrants in the phase plane notated anticlockwise beginning with the positive quadrant \(Q_1\). Consider the planar autonomous system defined by\N\[\N\begin{array}{l} \frac{{dx}}{{dt}} = -a_2-a_4x-2a_5y, \quad \frac{{dy}}{{dt}} = a_1+2a_3x+a_4 y, \quad (x,y) \in Q_1, \\\N\frac{{dx}}{{dt}} = -b_2-b_4x-2b_5y, \quad \,\, \frac{{dy}}{{dt}} = b_1+2b_3x+b_4 y, \quad (x,y) \in Q_2, \\\N\frac{{dx}}{{dt}} = -b_2-b_4x-2c_5y, \quad \,\, \frac{{dy}}{{dt}} = b_1+2b_3x+b_4 y, \quad (x,y) \in Q_3, \\\N\frac{{dx}}{{dt}} = -a_2-a_4x-2d_5y, \quad \frac{{dy}}{{dt}} = a_1+2a_3x+a_4 y, \quad (x,y) \in Q_4, \\\N\end{array}\tag{1}\N\]\Nwhere \( a_i, b_i. c_5, d_5\) are real numbers. System (1) is continuous along the \(x\)-axis and discontinuous along the \(y\)-axis. The authors prove that system (1) has at most one limit cycle.