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The secant map applied to a real polynomial with multiple roots

This work has been partially supported by MINECO-AEI grants MTM-2017-86795-C3-2-P and MTM-2017-86795-C3-3-P, the Maria de Maeztu Excellence Grant MDM-2014-0445 of the BGSMath and the AGAUR grant 2017 SGR 1374

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  • We investigate the plane dynamical system given by the secant map applied to a polynomial $ p $ having at least one multiple root of multiplicity $ d>1 $. We prove that the local dynamics around the fixed points related to the roots of $ p $ depend on the parity of $ d $.

    Mathematics Subject Classification: Primary: 37G35, 37N30; Secondary: 37C70.

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  • Figure 1.  Dynamical plane of the secant map applied to $ p(x) = (x+2)x(x-1)^d $ for several values of $ d $. We show in red (dark grey) the basin of attraction of the multiple root of $ p $ corresponding to the fixed point of the secant map located at $ (1,1) $, in green (light grey) the basin of attraction of $ (-2,-2) $ and in blue (black) the basin of attraction of $ (0,0) $. The white regions that appear in each of the pictures are in the basin of a critical point of $ p $. The range of the pictures (a), (c), (e) and (f) is [-3, 3]x[-3, 3]

    Figure 2.  Dynamics of T near a simple focal point Q

  • [1] E. Bedford and P. Frigge, The secant method for root finding, viewed as a dynamical system, Dolomites Res. Notes Approx., 11 (2018), 122-129. 
    [2] G.-I. BischiL. Gardini and C. Mira, Plane maps with denominator. I. Some generic properties, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 119-153.  doi: 10.1142/S0218127499000079.
    [3] G.-I. BischiL. Gardini and C. Mira, Plane maps with denominator. Ⅱ. Noninvertible maps with simple focal points, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2253-2277.  doi: 10.1142/S021812740300793X.
    [4] G.-I. BischiL. Gardini and C. Mira, Plane maps with denominator. Ⅲ. Nonsimple focal points and related bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 451-496.  doi: 10.1142/S0218127405012314.
    [5] A. Garijo and X. Jarque, Global dynamics of the real secant method, Nonlinearity, 32 (2019), 4557-4578.  doi: 10.1088/1361-6544/ab2f55.
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