Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

Figure 1.  Intersection of the curves $ P(x,y) = 0 $ (in blue) and $ Q(x,y) = 0 $ (in magenta). The PM boxes $ I_1\times I_5 $, $ I_2\times I_4 $ (left and right respectively, in red)

Figure 2.  Intersection the curves $ g_1(x,y;0.9) = 0 $ (in blue) and $ g_2(x,y;0.9) = 0 $ (in magenta). It can be seen that there are seven intersections corresponding to one fixed point and two different $ 3 $-periodic orbits. In red, a PM box of one of the solutions of system (4)

Figure 3.  The PM box of a solution of system (6) used in the proof of Theorem 4.1 (in red). It corresponds to the intersection of the curves defined by the curves $ T_{5,1}(x,y) = 0 $ (in blue) and $ T_{5,2}(x,y) = 0 $ (in magenta)

Figure 4.  Two $ 5 $-periodic orbits of the map (5) in $ \mathrm{Int}(\triangle) $ (Orbit 1 in red and Orbit 2 in green). They correspond to the intersection of the curves defined by the curves $ T_{5,1}(x,y) = 0 $ (in blue) and $ T_{5,2}(x,y) = 0 $ (in magenta). The fixed point $ (1,2) $ (in brown). The PM box containing the point $ P_{9,10} $ used in the proof of Theorem 4.1 (in red)

Figure 5.  Three $ 6 $-periodic orbits of the map (5) in $ \mathrm{Int}(\triangle) $ (Orbit 1, 2 and 3 in brown, red and green, respectively). They correspond to the intersection of the curves defined by the curves $ T_{6,1}(x,y) = 0 $ (in blue) and $ T_{6,2}(x,y) = 0 $ (in magenta). The fixed point $ (1,2) $ (in black)

Figure 6.  Left part: Intersection points between $ g_{1}(u,v) = 0 $ (in blue) and $ g_{2}(u,v) = 0 $ (in magenta) and some PM boxes containing them. Right part: the 3 limit cycles of system (10)

Figure 7.  Left figure: graph of $ f(\theta) $ in $ (0,\pi] $. Right figure: intersection of the curves $ g_1(u,v) = 0 $ (in blue) and $ g_2(u,v) = 0 $ (in magenta). In red, a PM box