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

  • * Corresponding author: Armengol Gasull

    * Corresponding author: Armengol Gasull 
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  • We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piecewise linear planar vector field; a new counterexample of Kouchnirenko conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the $ (1+4) $-body problem.

    Mathematics Subject Classification: Primary: 37C25, 39A23; Secondary: 13P15, 34D23, 70F15, 70K05.


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  • 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

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