# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020133

## The secant map applied to a real polynomial with multiple roots

 1 Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Catalonia 2 Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Catalonia

* Corresponding author: antonio.garijo@urv.cat

Received  July 2019 Revised  September 2019 Published  February 2020

Fund Project: 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

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 $d$.

Citation: Antonio Garijo, Xavier Jarque. The secant map applied to a real polynomial with multiple roots. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020133
##### References:
 [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.   Google Scholar [2] G.-I. Bischi, L. 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.  Google Scholar [3] G.-I. Bischi, L. 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.  Google Scholar [4] G.-I. Bischi, L. 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.  Google Scholar [5] A. Garijo and X. Jarque, Global dynamics of the real secant method, Nonlinearity, 32 (2019), 4557-4578.  doi: 10.1088/1361-6544/ab2f55.  Google Scholar

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##### References:
 [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.   Google Scholar [2] G.-I. Bischi, L. 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.  Google Scholar [3] G.-I. Bischi, L. 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.  Google Scholar [4] G.-I. Bischi, L. 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.  Google Scholar [5] A. Garijo and X. Jarque, Global dynamics of the real secant method, Nonlinearity, 32 (2019), 4557-4578.  doi: 10.1088/1361-6544/ab2f55.  Google Scholar
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]
Dynamics of T near a simple focal point Q
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