DYNAMICS AND BIFURCATIONS IN A TWO-DIMENSIONAL MAP DERIVED FROM A GENERALIZED \Delta\Delta-SINE-GORDON EQUATION

Zakaria, La; Tuwankotta, J.M.

doi:10.17654/ds028030165

Cited by 5 publications

(10 citation statements)

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“…where z 1 and z 2 are relatively prime integers (see [26,27] for other applications). Substituting Equation (7) into Equation (4), the following discrete mapping can be obtained:…”

Section: Traveling Wave Solution For P∆e Mkdv (O∆e Mkdv)mentioning

confidence: 99%

A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography

2021

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Cryptography is the science and study of protecting data in computer and communication systems from unauthorized disclosure and modification. An ordinary difference equation (a map) can be used in encryption–decryption algorithms. In particular, the Arnold’s cat and the sine-Gordon linear maps can be used in cryptographic algorithms for encoding digital images. In this article, a two-dimensional linear mKdV map derived from an ordinary difference mKdV equation will be used in a cryptographic encoding algorithm. The proposed encoding algorithm will be compared with those generated using sine-Gordon and Arnold’s cat maps via the correlations between adjacent pixels in the encrypted image and the uniformity of the pixel distribution. Note that the mKdV map is derived from the partial discrete mKdV equation with Consistency Around the Cube (CAC) properties, whereas the sine-Gordon map is derived from the partial discrete sine-Gordon equation, which does not have CAC properties.

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“…where z 1 and z 2 are relatively prime integers (see [26,27] for other applications). Substituting Equation (7) into Equation (4), the following discrete mapping can be obtained:…”

Section: Traveling Wave Solution For P∆e Mkdv (O∆e Mkdv)mentioning

confidence: 99%

A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography

2021

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“…This equation is derived from the compatibility condition of the generalized Lax pair of the classical discrete sine-Gordon equation (see Zakaria and Tuwankotta, 2016). The travelling wave solutions of (1) can be obtained by considering the following form:…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The number of independent integrals are preserved by the transformation which implies that the integrability is preserved. In our article (Zakaria and Tuwankotta, 2016) a straight-forward generalization by adding parameters in the Lax pair of the ordinary discrete sine-Gordon partial difference equation has done. By using the standard staircase method, the resulted equation is then reduced to system of ordinary difference equations (see Van der Kamp et al, 2007 for the method).…”

Section: Introductionmentioning

confidence: 99%

Dynamics of A Re-Parametrization of A 2-Dimensional Mapping Derived from Double Discrete Sine-Gordon Mapping

Zakaria¹,

Tuwankotta²

2020

Int J Math, Eng, Manag Sci

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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.

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“…In 2016, Zakaria and Tuwankotta described a procedure to derive a twodimensional traveling wave solution mapping of ΔΔ-sine Gordon equation [8]. Using the same procedure, a two-dimensional traveling wave solution mapping of ΔΔ-mKdV equation 7…”

Section: A the Traveling Wave Solution Of The Generalized δδ-Mkdv Eqmentioning

confidence: 99%

“…Tuwankotta and Quispel [7] have made the derived discrete dynamic system through the effort of multiplying the parameters on the Lax matrix system. It is intended that the system's integrity properties are always maintained in examining more dynamics occurring from a discrete dynamic system (see [7] and [8]). By the same procedure, the following shows the attempt to elaborate (1).…”

Section: Introductionmentioning

confidence: 99%

THE INTEGRAL NORMAL FORM OF A THREE-DIMENSIONAL TRAVELING WAVE SOLUTION MAPPING DERIVED FROM GENERALIZED ΔΔ-mKdV EQUATION

Zakaria¹,

Notiragayu²,

Nuryaman³

et al. 2018

ADECP

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The integrability of traveling wave solution mappings can be obtained as reductions of the discrete generalized ΔΔ modified Korteweg-de Vries (ΔΔ-mKdV) equation. The properties of the integrable discrete dynamical system can be examined through the level set of integral function. In this paper, we show that the integral of a threedimensional traveling wave solution mapping derived from generalized ΔΔ-mKdV equation can be made in the normal form.

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DYNAMICS AND BIFURCATIONS IN A TWO-DIMENSIONAL MAP DERIVED FROM A GENERALIZED \Delta\Delta-SINE-GORDON EQUATION

Cited by 5 publications

References 0 publications

A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography

A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography

Dynamics of A Re-Parametrization of A 2-Dimensional Mapping Derived from Double Discrete Sine-Gordon Mapping

THE INTEGRAL NORMAL FORM OF A THREE-DIMENSIONAL TRAVELING WAVE SOLUTION MAPPING DERIVED FROM GENERALIZED ΔΔ-mKdV EQUATION

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