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Global vector-field reconstruction of nonlinear dynamical systems from a time series with SVD method and validation with Lyapunov exponents

第十三届全国非线性振动暨第十届全国非线性动力学和运动稳...

作者：王喜英许勇张慧清顾仁财 Department of Applied Mathematics Northwestern Polytechnical University

This paper is to investigate the Lyapunov exponents for nonlinear systems with Lévy process.A method of equivalent linearization is proposed to reduce and simplify the original systems and the mean square responses are carried out to verify the effectiveness of the proposed approach.Then Lyapunov exponents will be defined and computed to explore the stochastic stability,and two examples are presented to demonstrate the procedure of equivalent linearization and stochastic stability is considered for these two special examples.The results show that the technique of equivalent linearization can be used to study nonlinear systems excited by Lévy noise.

关键词： Lyapunov exponents The Equivalent Linearization Stochastic Differential Equations（SDEs） Lévy process

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Chinese Physics B 2003年第12期12卷 1366-1373页

作者：刘卫东 K.F.Ren S.Meunier-Guttin-Cluzel G.Gouesbet DepartmentofAerospaceTechnology NatlonalUniversityofDefenseTechnologyChangsha410073China L.E.S.P. U.M.R.6614INSAdeRouen76801SaintEtienneduRouvrayFrance

A method for the global vector-field reconstruction of nonlinear dynamical systems from a time series is studied in this paper. It employs a complete set of polynomials and singular value decomposition (SVD) to estimate a standard function which is central to the algorithm. Lyapunov exponents and dimension, calculated from the differential equations of a standard system, are used for the validation of the reconstruction. The algorithm is proven to be practical by applying it to a Rossler system.

关键词： global vector-field reconstruction Lyapunov exponents model validation

Surface Billiards and Nonvanishing Lyapunov exponents

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Acta Mathematica Sinica,English Series 1997年第1期13卷 21-34页

作者： Cheng Jian (Department of Mathematics,Nanjing University,Nanjing 210008,China) 南京大学数学系江苏南京

We study in this paper the billiards on surfaces with mix-valued Gaussian curvature and the condition which gives nonvanishing Lyapunov exponents of the system.We introduce a criterion upon which a small perturbation of the surface will also produce a system with positive Lyapunov exponents.Some examples of such surfaces are given in this article.

关键词： Billiards Lyapunov exponents Curvature

Reduced differential transform and Sumudu transform methods for solving fractional financial models of awareness

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Applied Mathematics(A Journal of Chinese Universities) 2023年第3期38卷 338-356页

作者： A.M.S.Mahdy K.A.Gepreel Kh.Lotfy A.El-Bary Department of Mathematics and Statistics College of ScienceTaif UniversityP.O.Box 11099Taif21944Saudi Arabia Department of Mathematics Faculty of ScienceZagazig UniversityP.O.Box 44519ZagazigEgypt Arab Academy for Science Technology and Maritime Transport1029AlexandriaEgypt

In that paper,we new study has been carried out on previous studies of one of the most important mathematical models that describe the global economic movement,and that is described as a non-linear fractional financial model of awareness,where the studies are represented at the steps following:One:The schematic of the model is suggested.Two:The disease-free equilibrium point(DFE)and the stability of the equilibrium point are discussed.Three:The stability of the model is fulfilled by drawing the Lyapunov exponents and Poincare map.Fourth:The existence of uniformly stable solutions have discussed.Five:The Caputo is described as the fractional derivative.Six:Fractional optimal control for NFFMA is discussed by clarifying the fractional optimal control through drawing before and after control.Seven:Reduced differential transform method(RDTM)and Sumudu Decomposition Method(SDM)are used to take the resolution of an NFFMA.Finally,we display that SDM and RDTM are highly identical.

关键词： financial of awareness stability Lyapunov exponents Poincare map fractional optimal control Hamiltonian

A new fractional order 6D chaotic model:Study of model dynamics,system structure graph,electronic circuit realization and fractional control

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Journal of Ocean Engineering and Science 2024年第2期9卷 112-125页

作者： M.Higazy Norah Almalki Shabbir Muhammad A.Al-Ghamdi Department of Mathematics and Statistics College of ScienceTaif UniversityP.O.Box 11099Taif 21944Saudi Arabia Department of Physics and Engineering Mathematics Faculty of Electronic EngineeringMenoufia UniversityMenouf 32952Egypt Research Center for Advanced Materials Science(RCAMS) King Khalid UniversityP.O.Box 9004Abha 61413Saudi Arabia Department of Chemistry College of ScienceKing Khalid UniversityP.O.Box 9004Abha 61413Saudi Arabia Department of Physics Faculty of ScienceKing Abdulaziz UniversityP.O.Box 80203Jeddah 21589Saudi Arabia

A new fractional 6D chaotic model is constructed in this paper.The new fractional 6D chaotic model has six positive parameters plus the fractional order with eight nonlinear terms.The complicated chaotic dy-namics of the new fractional 6D model is presented and analyzed.The basic properties of this model are studied and its chaotic attractors,dissipative feature,symmetry,equilibrium points,Lyapunov exponents are investigated.The new dynamics of the 6D fractional model is numerically simulated using Matlab software.In addition,utilizing the graph theory tools certain structural characteristics are calculated.An electrical circuit is built to implement the new 5.4 fractional order 6D model.Finally,an active fractional order controller is proposed to control the new model at different fractional orders.The chaos of the new model is very useful and can be used to produce random keys for data encryption.

关键词： Chaotic systems Fractional calculus Lyapunov exponents Chaotic attractors Adjacency matrix Hermitian matrix Electronic circuit realization

Livšic Theorem for Matrix Cocycles Over An Axiom A Flow

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Communications in Mathematics and Statistics 2022年第4期10卷 681-704页

作者： Zeng Lian Jianhua Zhang College of Mathematical Sciences Sichuan UniversityChengduSichuan 610016People’s Republic of China

In this paper,we prove a version of Livšic theorem for a class of matrix cocycles over a C^(2)Axiom A flow.As a by-product,an approximative theorem on Lyapunov exponents is also obtained which assets that lyapunov exp... 详细信息

关键词： Matrix cocycles Axiom A flow Lyapunov exponents

Optimal and Memristor-Based Control of A Nonlinear Fractional Tumor-Immune Model

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Computers, Materials & Continua 2021年第6期67卷 3463-3486页

作者： Amr M.S.Mahdy Mahmoud Higazy Mohamed S.Mohamed Department of Mathematics College of ScienceTaif UniversityTaif21944Saudi Arabia Department of Mathematics Faculty of ScienceZagazig UniversityZagazig44519Egypt Department of Physics and Engineering Mathematics Faculty of Electronic engineeringMenoufia UniversityMenouf32952Egypt Department of Mathematics Faculty of ScienceAl-Azher UniversityNasr City11884Egypt

In this article,the reduced differential transform method is introduced to solve the nonlinear fractional model of Tumor-Immune.The fractional derivatives are described in the Caputo sense.The solutions derived using this method are easy and very accurate.The model is given by its signal flow diagram.Moreover,a simulation of the system by the Simulink of MATLAB is given.The disease-free equilibrium and stability of the equilibrium point are calculated.Formulation of a fractional optimal control for the cancer model is calculated.In addition,to control the system,we propose a novel modification of its model.This modification is based on converting the model to a memristive one,which is a first time in the literature that such idea is used to control this type of diseases.Also,we study the system’s stability via the Lyapunov exponents and Poincare maps before and after control.Fractional order differential equations(FDEs)are commonly utilized to model systems that have memory,and exist in several physical phenomena,models in thermoelasticity field,and biological paradigms.FDEs have been utilized to model the realistic biphasic decline manner of elastic systems and infection of diseases with a slower rate of change.FDEs are more useful than integer-order in modeling sophisticated models that contain physical phenomena.

关键词： RDTM tumor-immune optimal control caputo derivative signal flow simulink disease-free equilibrium stability memristive Lyapunov exponents poincare map

A New Four-Wing Chaotic System and Its Unified Generalized Projective Synchronization

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Wuhan University Journal of Natural Sciences 2020年第3期25卷 256-266页

作者： ZHANG Li PENG Jiankui The Basic Courses Department Lanzhou Institute of TechnologyLanzhou 730050GansuChina The Institute of Education Lanzhou University of Arts and ScieneeLanzhou 730000GansuChina

In this paper,some basic properties of a new four-dimensional(4 D)continuous autonomous chaotic system,in which each equation contains a cubic cross-product term,are further analyzed.The new system has 9 equilibria displaying graceful symmetry with respect to the origin and coordinate planes,and the stability of them are discussed.Then detailed bifurcation analysis is given to demonstrate the evolution processes of the system.Numerical simulations show that the system evolves chaotic motions through period-doubling bifurcation or intermittence chaos while the system parameters vary.We design a new scheme of generalized projective synchronization,so-called unified generalized projective synchronization,whose response signal synchronizes with the linear combination of drive signal.The design has the advantages of containing complete synchronization,anti-synchronization and disorder synchronization over the usual generalized projective synchronization,such that it can provide greater security in secure communication.Based on lyapunov stability theorem,some sufficient conditions for the new synchronization are inferred.Numerical simulations demonstrate the effectiveness and feasibility of the method by employing the four-wing chaotic system.

关键词： four-wing chaotic attractor four-dimensional(4D)chaotic system Lyapunov exponents bifurcation generalized projective synchronization

Trivial and Simple Spectrum for SL(d,R) Cocycles with Free Base and Fiber Dynamics

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Acta Mathematica Sinica,English Series 2015年第7期31卷 1113-1122页

作者： Mrio BESSA Paulo VARANDAS Departamento de Matemtica Universidade da Beira InteriorRua Marquês d'vila e Bolama6201-001 CovilhPortugal Departamento de Matemtica Universidade Federal da BahiaAv.Ademar de Barros s/n40170-110 SalvadorBrazil

Let ACD（M, SL（d,R）） denote the pairs （f, A） so that f∈ A C Diff^1（M） is a C^1-Anosov transitive diffeomorphisms and A is an SL（d,R） cocycle dominated with respect to f. We prove that open and densely in ACD（M, SL（d,R））, in appropriate topologies, the pair （f,A） has simple spectrum with respect to the unique maximal entropy measure μf. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AUtLeb（M） × LP（M, SL（d, R））.

关键词： Linear cocycles Lyapunov exponents Anosov diffeomorphisms topological conjugacy,maximal entropy measures