The Analysis of Singularities and Bifurcation of Heteroclinic Loops of Two-Dimensional Cubic Polynomial Systems

2019 ◽  
Vol 09 (05) ◽  
pp. 578-584
Author(s):  
文雅 姜
Keyword(s):  
Polynomial Systems ◽  
Two Dimensional ◽  
Cubic Polynomial

scholarly journals A Novel Method for Polar Form of Any Degree of Multivariate Polynomials with Applications in IoT

Sensors ◽  
2019 ◽  
Vol 19 (4) ◽  
pp. 903 ◽  
Author(s):  
Sedat Akleylek ◽  
Meryem Soysaldı ◽  
Djallel Boubiche ◽  
Homero Toral-Cruz
Keyword(s):  
Polynomial Systems ◽  
Multivariate Polynomials ◽  
Polar Form ◽  
Identification Schemes ◽  
Property Identification ◽  
Cubic Polynomial ◽  
Novel Method ◽  
Iot Devices ◽  
Polar Forms ◽  
Rfid Applications

Identification schemes based on multivariate polynomials have been receiving attraction in different areas due to the quantum secure property. Identification is one of the most important elements for the IoT to achieve communication between objects, gather and share information with each other. Thus, identification schemes which are post-quantum secure are significant for Internet-of-Things (IoT) devices. Various polar forms of multivariate quadratic and cubic polynomial systems have been proposed for these identification schemes. There is a need to define polar form for multivariate dth degree polynomials, where d ≥ 4 . In this paper, we propose a solution to this need by defining constructions for multivariate polynomials of degree d ≥ 4 . We give a generic framework to construct the identification scheme for IoT and RFID applications. In addition, we compare identification schemes and curve-based cryptoGPS which is currently used in RFID applications.

scholarly journals Existence of generic cubic homoclinic tangencies for Hénon maps

2012 ◽  
Vol 33 (4) ◽  
pp. 1029-1051 ◽  
Author(s):  
SHIN KIRIKI ◽  
TERUHIKO SOMA
Keyword(s):  
Arbitrary Order ◽  
Homoclinic Orbits ◽  
Steklov Inst ◽  
Strange Attractors ◽  
Heteroclinic Cycle ◽  
Two Dimensional ◽  
Henon Map ◽  
Cubic Polynomial ◽  
New Phenomena ◽  
Homoclinic Tangencies

AbstractIn this paper, we show that the Hénon map $\varphi _{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko, Shilnikov and Turaev [On models with non-rough Poincaré homoclinic curves. Physica D 62(1–4) (1993), 1–14; Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6(1) (1996), 15–31; On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle. Proc. Steklov Inst. Math.216 (1997), 70–118; Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. 67 (1999), 69–128, translation in J. Math. Sci. 105 (2001), 1738–1778; Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20 (2007), 241–275]. Combining this fact with theorems in Kiriki and Soma [Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies. Nonlinearity 21(5) (2008), 1105–1140], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

Asymptotic behavior of eigen energies of non-Hermitian cubic polynomial systems

2007 ◽  
Vol 85 (12) ◽  
pp. 1473-1480 ◽  
Author(s):  
A Nanayakkara
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial System ◽  
Expansion Method ◽  
Polynomial Systems ◽  
Level Spacing ◽  
Spacing Distribution ◽  
Cubic Polynomial ◽  
Asymptotic Energy ◽  
04.20 Jb ◽  
Energy Expansion

The asymptotic behavior of the eigenvalues of a non-Hermitian cubic polynomial system H = (P2/2) + µx3 + ax2 + bx, where µ, a, and b are constant parameters that can be either real or complex, is studied by extending the asymptotic energy expansion method, which has been developed for even degree polynomial systems. Both the complex and the real eigenvalues of the above system are obtained using the asymptotic energy expansion. Quantum eigen energies obtained by the above method are found to be in excellent agreement with the exact eigenvalues. Using the asymptotic energy expansion, analytic expressions for both level spacing distribution and the density of states are derived for the above cubic system. When µ = i, a is real, and b is pure imaginary, it was found that asymptotic energy level spacing increases with the coupling strength a for positive a while it decreases for negative a. PACS Nos.: 03.65.Ge, 04.20.Jb, 03.65.Sq, 02.30.Mv, 05.45

scholarly journals Faces of Relativistic Toda Chain

1997 ◽  
Vol 12 (15) ◽  
pp. 2675-2724 ◽  
Author(s):  
S. Kharchev ◽  
A. Mironov ◽  
A. Zhedanov
Keyword(s):  
Orthogonal Polynomial ◽  
Toda Lattice ◽  
Toda Chain ◽  
Polynomial Systems ◽  
Function Approach ◽  
Unitary Matrix ◽  
Lax Representation ◽  
Two Dimensional ◽  
Two Component ◽  
Toda Lattice Hierarchy

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of the Lax representation, orthogonal polynomial systems, and τ-function approach. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. Some applications to biorthogonal polynomial systems are discussed.

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