Dynamics of A Re-Parametrization of A 2-Dimensional Mapping Derived from Double Discrete Sine-Gordon Mapping
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Lampung, Indonesia.
Johan Matheus Tuwankotta
Analysis and Geometry Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia.
Received on June 24, 2019
Accepted on November 14, 2019
We study the dynamics of a two dimensional map which is derived from another two dimensional map by re-parametrizing the parameter in the system. It is shown that some of the properties of the original map can be preserved by the choice of the re-parametrization. By means of performing stability analysis to the critical points, and also studying the level set of the integrals, we study the dynamics of the re-parametrized map. Furthermore, we present preliminary results on the existence of a set where iteration starts at a point in that set, in which it will go off to infinity after finite step.
Keywords- Re-parametrizing, 2-dimensional mapping, Generalized double discrete sine-Gordon, Integral.
Zakaria, L., & Tuwankotta, J. M. (2020). Dynamics of A Re-Parametrization of A 2-Dimensional Mapping Derived from Double Discrete Sine-Gordon Mapping. International Journal of Mathematical, Engineering and Management Sciences, 5(2), 363-377. https://doi.org/10.33889/IJMEMS.2020.5.2.030.
Conflict of Interest
The authors confirm that there is no conflict of interest for this publication.
LZ acknowledges supports from a The Research and Community Services of Lampung University through Lampung
Celledoni, E., Evripidou, C., McLaren, D.I., Owren, B., Quispel, G.R.W., & Tapley, B.K. (2019). Discrete Darboux polynomials and the search for preserved measures and integrals of rational maps. Arxiv Preprint Arxiv: 1902.04685v1.
Duistermaat, J.J. (2010). Discrete integrable systems: QRT maps and elliptic surfaces. Springer Monographs in Mathematics, Springer-Verlag, New York.
Joshi, N., & Kassotakis, P. (2019). Re-factorising a QRT map. Arxiv Preprint Arxiv:1906.00501v1[nlin.SI]
Kulenovic, M.R.S., & Merino, O. (2002). Discrete dynamical systems and difference equations with mathematica. Chapman and Hall/CRC, Boca Raton, Florida. USA.
Quispel, G.R.W., Capel, H.W, & Roberts, J.A.G. (2005). Duality for discrete integrable systems.Journal of Physics A: Mathematical and General, 38(18), 3965.
Quispel, G.R.W., Capel, H.W, Papageorgiou V.G, & Nijhoff, F.W (1991). Integrable mappings derived from soliton equations. Physica A: Statistical Mechanics and its Applications, 173(1-2), 243–266.
Quispel, G.R.W., Roberts, J.A.G., & Thompson, C.J. (1988). Integrable mappings and soliton equations. Physics Letters A, 126(7), 419-421.
Quispel, G.R.W., Roberts, J.A.G., & Thompson, C.J. (1989). Integrable mappings and soliton equations II. Physica D, 34(1-2), 183-192.
Roberts, J.A.G., Iatrou A., & Quispel, G.R.W. (2002). Interchanging parameters and integrals in dynamical systems: the mapping case, Journal of Physics A: Mathematical and General, 35(9), 2309-2325.
Tuwankotta, J.M., Van der Kamp, P., Quispel, G.R.W., & Saputra, K.V.I. (2019). Generating a chain of maps which preserve the same integral as a given map. arXiv Preprint arXiv:1902.05206.
Tuwankotta, J.M., Quispel G.R.W., & Tamizhmani, K.M. (2004). Dynamics and bifurcations of a three-dimensional piecewise-linear integrable map. Journal of Physics A: Mathematical and General, 37(50), 12041.
Van der Kamp, P.H., & Quispel, G.R.W. (2010). The staircase method: integrals for periodic reductions of integrable lattice equations. Journal of Physics A: Mathematical and Theoretical, 43(46) , 465207.
Van der Kamp, P.H., Rojas, O., & Quispel, G.R.W. (2007). Closed-form expressions for integrals of mKdV and sine-Gordon maps. Journal of Physics A: Mathematical and Theoretical, 40(42), 12789.
Zakaria, L., & Tuwankotta, J.M. (2016). Dynamics and bifurcations in a two-dimensional maps derived from a generalized ΔΔsine-Gordon equation. Far East Journal of Dynamical Systems, 28(3), 165-194.