Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay
|
|
- Mervin Houston
- 5 years ago
- Views:
Transcription
1 Applied Mathematical Sciences, Vol 11, 2017, no 22, HIKARI Ltd, wwwm-hikaricom Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay Luca Guerrini Department of Management Polytechnic University of Marche, Italy Copyright c 2017 Luca Guerrini This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract This paper presents the stability and bifurcation analysis of a dynamical model of the heartbeat with time delay The existence of Hopf bifurcations at the positive equilibrium is established by analyzing the distribution of the characteristic values Moreover, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determinated by applying the center manifold theorem and the normal form theory Mathematics Subject Classification: 34K18, 91B62 Keywords: dynamical heart model; time delay; Hopf bifurcation 1 Introduction Differential equations with time delay play an important role in economy, engineering, biology and social sciences, because great deal of problems may be described with their help The effects of time delays are of great interest as well, since their presence may include complex behavior Recently, Páez-Hernández et al [2] have investigated a model of the heartbeat using nonlinear dynamics and considering the time delays inherent in the system The resulting model
2 1090 Luca Guerrini is given by ẋ = 1 ε (x3 T x + y ξ ), ẏ = x ξ + x d, (1) where x denotes the length of the muscle fiber, y represents a variable related to electrochemical activity, ε is a small positive constant associated to the local stability of an endoreversible system with the fast eigenvalue of the system, x d is a scalar quantity representing a typical length of muscle fiber in the diastolic state, T is tension in the muscle fiber, and ξ is a time delay However, system (1) was only analyzed in the particular case ξ = π/2 The aim of this paper is to give the explicitly mathematical analysis for stability and periodicity of system (1) for an arbitrary time delay ξ 2 Stability and existence of Hopf bifurcation The corresponding characteristic equation of the linear system of (1) at its unique equilibrium (x, y ), where x = x d and y = x 3 d T x d, is 1 (λ, ξ) = λ 2 + Aλ + Be 2λξ = 0, (2) where A = 1 ε (3x2 d T ), B = 1 ε When ξ = 0, Eq (2) reduces to λ 2 + Aλ + B = 0 An analysis of the sign of the roots of this equation implies that the equilibrium (x, y ) is unstable if T 3x 2 d and stable if T < 3x2 d Next we shall explore the distribution of characteristic roots of (2) when ξ > 0 For computational purpose, we rewrite (2) as 1 (λ, ξ) = e λξ 1 (λ, ξ) = ( λ 2 + Aλ ) e λξ + Be λξ = 0 (3) Let λ = iω (ω > 0) be a root of (2) Then, substituting it into (3) yields ( ω 2 + B ) cos ωξ = Aω sin ωξ, ( ω 2 + B ) sin ωξ = Aω cos ωξ (4) Squaring and adding Eqs (4) we obtain ω 4 ( A 2 + 2B ) ω 2 + B 2 = 0 (5) Lemma 21 1) If A = 0 holds, then Eq (5) has a unique positive root ω 0, where ω 0 = B
3 Hopf bifurcation analysis of a dynamical heart model with time delay ) If A 0 holds, then Eq (5) has two positive roots ω < ω +, where A 2 + 2B ± (A 2 + 2B) 2 4B 2 ω ± = 2 Proof The statement follows noting that the discriminant of Eq (5) is given by(a 2 + 2B) 2 4B 2, and recalling that B > 0 Notice that solving Eqs (4) directly, one can derive the values ξj 0 and ξ ± j (j = 0, 1, 2, ) of ξ at which Eq (3) has a pair of purely imaginary roots ±iω 0 and ±iω ±, respectively Proposition 22 Let ξ ξj 0, ξ j, } ξ+ j and ω ω 0, ω, ω + } If λ (ξ) is the root of (3) near ξ = ξ such that Re(λ(ξ )) = 0 and Im(λ(ξ )) = ω, then [ ] dre(λ) > 0 ξ=ξ,ω=ω Furthermore, λ = iω is a simple root of the characteristic equation (3) Proof By substituting λ (ξ) into equation (3) and differentiating both sides of this equation with respect to ξ, we obtain (2λ + A) e λξ + ( λ 2 + Aλ ) ξe λξ Bξe λξ} dλ = λ [ ( λ 2 + Aλ ) e λξ + Be λξ] (6) If λ = iω is not a simple root of the characteristic equation ( (3), then, from (6) and using (3), ie (λ 2 + Aλ) e λξ = Be λξ, yields iω ) 2Be iω ξ = 0, leading to an absurd According to (6), we can get Recalling that sign ( dλ d (Reλ) ) 1 = ξ=ξ,ω=ω 2λ + A 2Bλ (λ 2 + Aλ) ξ λ } = sign Re ( ) } 1 dλ, ξ=ξ,ω=ω we obtain sign d (Reλ) ξ=ξ,ω=ω } } 2ω 2 = sign + A 2 2Bω 2 (ω 2 + A 2 )
4 1092 Luca Guerrini From the previous discussions and the Hopf bifurcation theorem, we can obtain the following results Theorem 23 1) Let A = 0 i) If the equilibrium (x, y ) of system (1) is locally asymptotically stable in absence of delay, then it is locally asymptotically stable for 0 ξ < ξ 0 0 and unstable for ξ > ξ 0 0 System (1) undergoes a Hopf bifurcation at (x, y ) when ξ = ξ 0 0 ii) If the equilibrium (x, y ) of system (1) is unstable in absence of delay, then it remains unstable for ξ 0 2) Let A 0 The equilibrium (x, y ) of system (1) has many stability switches and a Hopf bifurcation occurs at each stability switch 3 Stability and direction of bifurcating periodic solutions In this section, formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solution of system (1) shall be presented by employing the normal form method and center manifold theorem introduced by Hassard et al [1] Let ξ = ξ c + µ, µ R, where ξ c is a value for ξ where a Hopf bifurcation occurs Let us rewrite (1) in the form u = L µ (u t ) + f µ (µ, u t ), (7) where u = (u 1, u 2 ) T R 2, u t (θ) = u(t+θ) C = C([ ξ, 0], R 2 ), and L µ : C R 2, f : R C R 2 are defined, respectively, by L µ (ϕ) = P 0 ϕ(0) + P 1 ϕ( ξ) with and P 0 = 1 ε (3x2 d T ) f µ (ϕ) = and P 1 = 0 1 ε 0 0 6x d ε ϕ(0)2 1 ε ϕ(0)3 0,
5 Hopf bifurcation analysis of a dynamical heart model with time delay 1093 By the Riesz representation theorem, there is a bounded variation function η(θ, µ) for θ [ ξ, 0] such that L µ (ϕ) = 0 ξ dη(θ, µ)ϕ(θ), for ϕ C We can choose η(θ, µ) = P 0 δ(θ) + P 1 δ(θ + ξ), where δ( ) is the Dirac function For ϕ C 1 = C 1 ([ ξ, 0], R 2 ), we define two operators dϕ(θ), θ [ ξ, 0), dθ A(µ)(ϕ) = 0 dη(s, µ)ϕ(s), θ = 0, and R(µ)(ϕ) = ξ In this way, we can transform (7) into 0, θ [ ξ, 0), f(µ, ϕ), θ = 0 u = A(µ)u t + R(µ)u t (8) For ψ C 1 ([0, ξ], R 2 ), we define the operator A as dψ(s), s (0, ξ], A ds ψ(s) = 0 dη T (r, µ)ψ( r), s = 0, ξ and consider the following bilinear form < ψ, ϕ >= ψ T (0)ϕ(0) 0 θ θ= ξ r=0 ψ T (r θ)dη(θ)ϕ(r)dr, where η(θ) = η(θ, 0) and the overline stands for complex conjugate Since A and A are adjoint operators, if ±iω c ξ c are eigenvalues of A, then they are also eigenvalues of A One can compute the eigenvectors q(θ) and q (θ ) of A and A corresponding to iω c ξ c and iω c ξ c, respectively, which satisfy the normalized conditions < q, q >= 1, and < q, q >= 0 Hence, we set q(θ) = q(0)e iωcθ, q (θ ) = q (0)e iωcθ, for θ [ ξ, 0), θ (0, ξ], and q(0) = (q 1, q 2 ) T, q (0) = (1/ρ)(q1, q2) T Next, using q and q, we can compute the coordinates describing the center manifold at µ = 0 Let u t be the solution of (8) with µ = 0 Define z =< q, u t >, W (t, θ) = u t (θ) 2Re [zq(θ)]
6 1094 Luca Guerrini On the center manifold, one has W (t, θ) = W (z, z, θ), where W (z, z, θ) = W 20 (θ) z2 2 + W 11(θ)z z + W 02 (θ) z2 2 +, z and z are the local coordinates for the center manifold in the direction of q and q, respectively One has with g(z, z) = q T (0)f 0 (z, z) = g 20 z 2 z = iω c z + g(z, z), 2 + g 11z z + g 02 z 2 Notice that, from the definition of f(µ, u t ), we have Hence, we get 2 + g 21 z 2 f 0 (z, z) = f z z 2 f z f z 2 z z zz z + f z2 z 2 + z 2 z 2 + g 20 = q (0)f z 2, g 02 = q (0)f z 2, g 11 = q (0)f z z, g 21 = q (0)f z2 z We remark that g 21 will depend on the W 20 (θ) and W 11 (θ) Note that W 20 (θ) = ig 20 ω c q(0)e iωcθ ḡ02 3iω c q(0)e iωcθ + E 1 e 2iωcθ, W 11 (θ) = g 11 iω c q(0)e iωcθ ḡ11 iω c q(0)e iωcθ + E 2, where E 1 = (E (1) 1, E (2) 1 ) R 2 is a constant vector that can be determined with a further analysis Based on the above analysis, we have all formulas to find the values of g 20, g 02, g 11 and g 21 Thus, we can explicitly compute the following quantities: c 1 (0) = i 2ω c [ ] g 11 g 20 2 g 11 2 g g 21 2, µ 2 = Re [c 1(0)] Re [λ (ξ c )], β 2 = 2Re [c 1 (0)], τ 2 = Im [c 1(0)] + µ 2 Im [λ (ξ c )] ω c, where λ is the root of characteristic equation (3) Theorem 31 Let ξ c be a value for ξ where system (1) has a Hopf bifurcation
7 Hopf bifurcation analysis of a dynamical heart model with time delay ) If µ 2 > 0 (µ 2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ξ > ξ c (ξ < ξ c ) 2) The Floquet exponent β 2 determines the stability of bifurcating periodic solutions The bifurcating periodic solutions are stable (unstable) if β 2 < 0 (β 2 > 0) 3) The quantity τ 2 determines the period of the bifurcating periodic solutions The period increases (decreases) if τ 2 > 0 (τ 2 < 0) References [1] B Hassard, N Kazarinoff and Y Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981 [2] R Páez-Hernández et al, Effects of time delays in a dynamical heart model, Journal of Chemistry and Chemical Engineering, 8 (2014), Received: March 4, 2017; Published: April 15, 2017
Direction and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists
International Journal of Mathematical Analysis Vol 9, 2015, no 38, 1869-1875 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma201554135 Direction and Stability of Hopf Bifurcation in a Delayed Model
More informationStability and nonlinear dynamics in a Solow model with pollution
Nonlinear Analysis: Modelling and Control, 2014, Vol. 19, No. 4, 565 577 565 http://dx.doi.org/10.15388/na.2014.4.3 Stability and nonlinear dynamics in a Solow model with pollution Massimiliano Ferrara
More informationHOPF BIFURCATION CONTROL WITH PD CONTROLLER
HOPF BIFURCATION CONTROL WITH PD CONTROLLER M. DARVISHI AND H.M. MOHAMMADINEJAD DEPARTMENT OF MATHEMATICS, FACULTY OF MATHEMATICS AND STATISTICS, UNIVERSITY OF BIRJAND, BIRJAND, IRAN E-MAILS: M.DARVISHI@BIRJAND.AC.IR,
More informationHOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY
Vol. 37 17 No. J. of Math. PRC HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY LI Zhen-wei, LI Bi-wen, LIU Wei, WANG Gan School of Mathematics and Statistics,
More informationThe Cai Model with Time Delay: Existence of Periodic Solutions and Asymptotic Analysis
Appl Math Inf Sci 7, No 1, 21-27 (2013) 21 Applied Mathematics & Information Sciences An International Journal Natural Sciences Publishing Cor The Cai Model with Time Delay: Existence of Periodic Solutions
More informationHopf bifurcation analysis for a model of plant virus propagation with two delays
Li et al. Advances in Difference Equations 08 08:59 https://doi.org/0.86/s366-08-74-8 R E S E A R C H Open Access Hopf bifurcation analysis for a model of plant virus propagation with two delays Qinglian
More informationBIFURCATION ANALYSIS ON A DELAYED SIS EPIDEMIC MODEL WITH STAGE STRUCTURE
Electronic Journal of Differential Equations, Vol. 27(27), No. 77, pp. 1 17. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) BIFURCATION
More informationDynamical analysis of a delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response
Liu Advances in Difference Equations 014, 014:314 R E S E A R C H Open Access Dynamical analysis of a delayed predator-prey system with modified Leslie-Gower and Beddington-DeAngelis functional response
More informationResearch Article Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function
Abstract and Applied Analysis Volume 2013 Article ID 738460 6 pages http://dx.doi.org/10.1155/2013/738460 Research Article Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped
More informationResearch Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System
Abstract and Applied Analysis Volume, Article ID 3487, 6 pages doi:.55//3487 Research Article Hopf Bifurcation Analysis and Anticontrol of Hopf Circles of the Rössler-Like System Ranchao Wu and Xiang Li
More informationStability and bifurcation analysis of Westwood+ TCP congestion control model in mobile cloud computing networks
Nonlinear Analysis: Modelling and Control, Vol. 1, No. 4, 477 497 ISSN 139-5113 http://dx.doi.org/1.15388/na.16.4.4 Stability and bifurcation analysis of Westwood+ TCP congestion control model in mobile
More informationStability and Hopf Bifurcation Analysis of the Delay Logistic Equation
1 Stability and Hopf Bifurcation Analysis of the Delay Logistic Equation Milind M Rao # and Preetish K L * # Department of Electrical Engineering, IIT Madras, Chennai-600036, India * Department of Mechanical
More informationOn a Certain Representation in the Pairs of Normed Spaces
Applied Mathematical Sciences, Vol. 12, 2018, no. 3, 115-119 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712362 On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida
More information2 The model with Kaldor-Pasinetti saving
Applied Mathematical Sciences, Vol 6, 2012, no 72, 3569-3573 The Solow-Swan Model with Kaldor-Pasinetti Saving and Time Delay Luca Guerrini Department of Mathematics for Economic and Social Sciences University
More informationHopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel
Math. Model. Nat. Phenom. Vol. 2, No. 1, 27, pp. 44-61 Hopf Bifurcation Analysis of Pathogen-Immune Interaction Dynamics With Delay Kernel M. Neamţu a1, L. Buliga b, F.R. Horhat c, D. Opriş b a Department
More informationTHE ANALYSIS OF AN ECONOMIC GROWTH MODEL WITH TAX EVASION AND DELAY
Key words: delayed differential equations, economic growth, tax evasion In this paper we formulate an economic model with tax evasion, corruption and taxes In the first part the static model is considered,
More informationSTABILITY AND HOPF BIFURCATION OF A MODIFIED DELAY PREDATOR-PREY MODEL WITH STAGE STRUCTURE
Journal of Applied Analysis and Computation Volume 8, Number, April 018, 573 597 Website:http://jaac-online.com/ DOI:10.11948/018.573 STABILITY AND HOPF BIFURCATION OF A MODIFIED DELAY PREDATOR-PREY MODEL
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationAN AK TIME TO BUILD GROWTH MODEL
International Journal of Pure and Applied Mathematics Volume 78 No. 7 2012, 1005-1009 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu PA ijpam.eu AN AK TIME TO BUILD GROWTH MODEL Luca Guerrini
More informationBifurcation Analysis of a Population Dynamics in a Critical State 1
Bifurcation Analysis of a Population Dynamics in a Critical State 1 Yong-Hui Xia 1, Valery G. Romanovski,3 1. Department of Mathematics, Zhejiang Normal University, Jinhua, China. Center for Applied Mathematics
More informationResearch Article Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus
Abstract and Applied Analysis Volume, Article ID 84987, 6 pages doi:.55//84987 Research Article Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus Tao Dong,, Xiaofeng Liao,
More information(8.51) ẋ = A(λ)x + F(x, λ), where λ lr, the matrix A(λ) and function F(x, λ) are C k -functions with k 1,
2.8.7. Poincaré-Andronov-Hopf Bifurcation. In the previous section, we have given a rather detailed method for determining the periodic orbits of a two dimensional system which is the perturbation of a
More informationChaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback
International Journal of Automation and Computing 12(2, April 215, 182-191 DOI: 1.17/s11633-14-852-z Chaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback Chang-Jin
More informationAndronov Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types IV and II
Electronic Journal of Qualitative Theory of Differential Equations 018 No 78 1 7; https://doiorg/10143/ejqtde018178 wwwmathu-szegedhu/ejqtde/ Andronov Hopf and Bautin bifurcation in a tritrophic food chain
More informationStability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations
Stability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations Sreelakshmi Manjunath Department of Electrical Engineering Indian Institute of Technology Madras (IITM), India JTG Summer
More informationResearch Article Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network
e Scientific World Journal Volume 4, Article ID 944, 4 pages http://dx.doi.org/.55/4/944 Research Article Dynamics of a Delayed Model for the Transmission of Malicious Objects in Computer Network Zizhen
More informationExistence, Uniqueness Solution of a Modified. Predator-Prey Model
Nonlinear Analysis and Differential Equations, Vol. 4, 6, no. 4, 669-677 HIKARI Ltd, www.m-hikari.com https://doi.org/.988/nade.6.6974 Existence, Uniqueness Solution of a Modified Predator-Prey Model M.
More informationLinearization of Two Dimensional Complex-Linearizable Systems of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 9, 2015, no. 58, 2889-2900 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121002 Linearization of Two Dimensional Complex-Linearizable Systems of
More informationDiophantine Equations. Elementary Methods
International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,
More informationChaos control and Hopf bifurcation analysis of the Genesio system with distributed delays feedback
Guan et al. Advances in Difference Equations 212, 212:166 R E S E A R C H Open Access Chaos control and Hopf bifurcation analysis of the Genesio system with distributed delays feedback Junbiao Guan *,
More informationMethod of Generation of Chaos Map in the Centre Manifold
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 16, 795-800 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.51097 Method of Generation of Chaos Map in the Centre Manifold Evgeny
More informationHOPF BIFURCATION CALCULATIONS IN DELAYED SYSTEMS
PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 48, NO. 2, PP. 89 200 2004 HOPF BIFURCATION CALCULATIONS IN DELAYED SYSTEMS Gábor OROSZ Bristol Centre for Applied Nonlinear Mathematics Department of Engineering
More informationStability and Hopf Bifurcation for a Discrete Disease Spreading Model in Complex Networks
International Journal of Difference Equations ISSN 0973-5321, Volume 4, Number 1, pp. 155 163 (2009) http://campus.mst.edu/ijde Stability and Hopf Bifurcation for a Discrete Disease Spreading Model in
More information5.2.2 Planar Andronov-Hopf bifurcation
138 CHAPTER 5. LOCAL BIFURCATION THEORY 5.. Planar Andronov-Hopf bifurcation What happens if a planar system has an equilibrium x = x 0 at some parameter value α = α 0 with eigenvalues λ 1, = ±iω 0, ω
More informationQualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions
Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 1, 45-59 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.3819 Qualitative Theory of Differential Equations and Dynamics of
More informationA Note on Open Loop Nash Equilibrium in Linear-State Differential Games
Applied Mathematical Sciences, vol. 8, 2014, no. 145, 7239-7248 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49746 A Note on Open Loop Nash Equilibrium in Linear-State Differential
More informationDynamical System of a Multi-Capital Growth Model
Applied Mathematical Sciences, Vol. 9, 2015, no. 83, 4103-4108 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.53274 Dynamical System of a Multi-Capital Growth Model Eva Brestovanská Department
More informationPhase Synchronization
Phase Synchronization Lecture by: Zhibin Guo Notes by: Xiang Fan May 10, 2016 1 Introduction For any mode or fluctuation, we always have where S(x, t) is phase. If a mode amplitude satisfies ϕ k = ϕ k
More informationResearch Article Stability Switches and Hopf Bifurcations in a Second-Order Complex Delay Equation
Hindawi Mathematical Problems in Engineering Volume 017, Article ID 679879, 4 pages https://doi.org/10.1155/017/679879 Research Article Stability Switches and Hopf Bifurcations in a Second-Order Complex
More informationStability of Limit Cycle in a Delayed IS-LM Business Cycle model
Applied Mathematical Sciences, Vol. 2, 28, no. 5, 2459-247 Stability of Limit Cycle in a Delayed IS-LM Business Cycle model A. Abta, A. Kaddar H. Talibi Alaoui Université Chouaib Doukkali Faculté des Sciences
More informationOn a Principal Ideal Domain that is not a Euclidean Domain
International Mathematical Forum, Vol. 8, 013, no. 9, 1405-141 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.013.37144 On a Principal Ideal Domain that is not a Euclidean Domain Conan Wong
More informationA Note on Cohomology of a Riemannian Manifold
Int. J. Contemp. ath. Sciences, Vol. 9, 2014, no. 2, 51-56 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.311131 A Note on Cohomology of a Riemannian anifold Tahsin Ghazal King Saud
More informationHopf bifurcation analysis of Chen circuit with direct time delay feedback
Chin. Phys. B Vol. 19, No. 3 21) 3511 Hopf bifurcation analysis of Chen circuit with direct time delay feedback Ren Hai-Peng 任海鹏 ), Li Wen-Chao 李文超 ), and Liu Ding 刘丁 ) School of Automation and Information
More informationStability, convergence and Hopf bifurcation analyses of the classical car-following model
1 Stability, convergence and Hopf bifurcation analyses of the classical car-following model Gopal Krishna Kamath, Krishna Jagannathan and Gaurav Raina Department of Electrical Engineering, Indian Institute
More informationSums of Tribonacci and Tribonacci-Lucas Numbers
International Journal of Mathematical Analysis Vol. 1, 018, no. 1, 19-4 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ijma.018.71153 Sums of Tribonacci Tribonacci-Lucas Numbers Robert Frontczak
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More informationOn a Boundary-Value Problem for Third Order Operator-Differential Equations on a Finite Interval
Applied Mathematical Sciences, Vol. 1, 216, no. 11, 543-548 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.512743 On a Boundary-Value Problem for Third Order Operator-Differential Equations
More informationThird and Fourth Order Piece-wise Defined Recursive Sequences
International Mathematical Forum, Vol. 11, 016, no., 61-69 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.016.5973 Third and Fourth Order Piece-wise Defined Recursive Sequences Saleem Al-Ashhab
More informationDynamicsofTwoCoupledVanderPolOscillatorswithDelayCouplingRevisited
Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 7 Issue 5 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals
More informationNon Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical Systems
Applied Mathematical Sciences, Vol. 12, 2018, no. 22, 1053-1058 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.87100 Non Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical
More informationQualitative analysis for a delayed epidemic model with latent and breaking-out over the Internet
Zhang and Wang Advances in Difference Equations (17) 17:31 DOI 1.1186/s1366-17-174-9 R E S E A R C H Open Access Qualitative analysis for a delayed epidemic model with latent and breaking-out over the
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationAn Improved Hybrid Algorithm to Bisection Method and Newton-Raphson Method
Applied Mathematical Sciences, Vol. 11, 2017, no. 56, 2789-2797 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.710302 An Improved Hybrid Algorithm to Bisection Method and Newton-Raphson
More informationLocal stability and Hopf bifurcation analysis for Compound TCP
This article has been accepted for publication in a future issue of this journal, but has not been fully edited Content may change prior to final publication Citation information: DOI 1119/TCNS17747839,
More informationPoincaré`s Map in a Van der Pol Equation
International Journal of Mathematical Analysis Vol. 8, 014, no. 59, 939-943 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.411338 Poincaré`s Map in a Van der Pol Equation Eduardo-Luis
More information1.7. Stability and attractors. Consider the autonomous differential equation. (7.1) ẋ = f(x),
1.7. Stability and attractors. Consider the autonomous differential equation (7.1) ẋ = f(x), where f C r (lr d, lr d ), r 1. For notation, for any x lr d, c lr, we let B(x, c) = { ξ lr d : ξ x < c }. Suppose
More informationAlternate Locations of Equilibrium Points and Poles in Complex Rational Differential Equations
International Mathematical Forum, Vol. 9, 2014, no. 35, 1725-1739 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.410170 Alternate Locations of Equilibrium Points and Poles in Complex
More informationOn Two New Classes of Fibonacci and Lucas Reciprocal Sums with Subscripts in Arithmetic Progression
Applied Mathematical Sciences Vol. 207 no. 25 2-29 HIKARI Ltd www.m-hikari.com https://doi.org/0.2988/ams.207.7392 On Two New Classes of Fibonacci Lucas Reciprocal Sums with Subscripts in Arithmetic Progression
More informationPólya-Szegö s Principle for Nonlocal Functionals
International Journal of Mathematical Analysis Vol. 12, 218, no. 5, 245-25 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ijma.218.8327 Pólya-Szegö s Principle for Nonlocal Functionals Tiziano Granucci
More informationExplicit Expressions for Free Components of. Sums of the Same Powers
Applied Mathematical Sciences, Vol., 27, no. 53, 2639-2645 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ams.27.79276 Explicit Expressions for Free Components of Sums of the Same Powers Alexander
More informationNumerical Solution of Heat Equation by Spectral Method
Applied Mathematical Sciences, Vol 8, 2014, no 8, 397-404 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams201439502 Numerical Solution of Heat Equation by Spectral Method Narayan Thapa Department
More informationQuadratic Optimization over a Polyhedral Set
International Mathematical Forum, Vol. 9, 2014, no. 13, 621-629 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.4234 Quadratic Optimization over a Polyhedral Set T. Bayartugs, Ch. Battuvshin
More informationThe Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive
More informationDouble Total Domination on Generalized Petersen Graphs 1
Applied Mathematical Sciences, Vol. 11, 2017, no. 19, 905-912 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7114 Double Total Domination on Generalized Petersen Graphs 1 Chengye Zhao 2
More informationChaos Control for the Lorenz System
Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 181-188 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8413 Chaos Control for the Lorenz System Pedro Pablo Cárdenas Alzate
More informationImprovements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method
International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919-1928 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54124 Improvements in Newton-Rapshon Method for Nonlinear
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationNew Generalized Sub Class of Cyclic-Goppa Code
International Journal of Contemporary Mathematical Sciences Vol., 206, no. 7, 333-34 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ijcms.206.6632 New Generalized Sub Class of Cyclic-Goppa Code
More informationSome Reviews on Ranks of Upper Triangular Block Matrices over a Skew Field
International Mathematical Forum, Vol 13, 2018, no 7, 323-335 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/imf20188528 Some Reviews on Ranks of Upper Triangular lock Matrices over a Skew Field Netsai
More informationThe Expansion of the Confluent Hypergeometric Function on the Positive Real Axis
Applied Mathematical Sciences, Vol. 12, 2018, no. 1, 19-26 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712351 The Expansion of the Confluent Hypergeometric Function on the Positive Real
More informationHamad Talibi Alaoui and Radouane Yafia. 1. Generalities and the Reduced System
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 22, No. 1 (27), pp. 21 32 SUPERCRITICAL HOPF BIFURCATION IN DELAY DIFFERENTIAL EQUATIONS AN ELEMENTARY PROOF OF EXCHANGE OF STABILITY Hamad Talibi Alaoui
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationBinary Relations in the Set of Feasible Alternatives
Applied Mathematical Sciences, Vol. 8, 2014, no. 109, 5399-5405 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.47514 Binary Relations in the Set of Feasible Alternatives Vyacheslav V.
More informationApproximation to the Dissipative Klein-Gordon Equation
International Journal of Mathematical Analysis Vol. 9, 215, no. 22, 159-163 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.5236 Approximation to the Dissipative Klein-Gordon Equation Edilber
More informationConvex Sets Strict Separation in Hilbert Spaces
Applied Mathematical Sciences, Vol. 8, 2014, no. 64, 3155-3160 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.44257 Convex Sets Strict Separation in Hilbert Spaces M. A. M. Ferreira 1
More informationDouble Total Domination in Circulant Graphs 1
Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1623-1633 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811172 Double Total Domination in Circulant Graphs 1 Qin Zhang and Chengye
More informationOn Strong Alt-Induced Codes
Applied Mathematical Sciences, Vol. 12, 2018, no. 7, 327-336 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8113 On Strong Alt-Induced Codes Ngo Thi Hien Hanoi University of Science and
More information2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ
1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.
More informationStability Analysis of Plankton Ecosystem Model. Affected by Oxygen Deficit
Applied Mathematical Sciences Vol 9 2015 no 81 4043-4052 HIKARI Ltd wwwm-hikaricom http://dxdoiorg/1012988/ams201553255 Stability Analysis of Plankton Ecosystem Model Affected by Oxygen Deficit Yuriska
More information8.385 MIT (Rosales) Hopf Bifurcations. 2 Contents Hopf bifurcation for second order scalar equations. 3. Reduction of general phase plane case to seco
8.385 MIT Hopf Bifurcations. Rodolfo R. Rosales Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts MA 239 September 25, 999 Abstract In two dimensions a Hopf bifurcation
More informationResearch Article Hopf Bifurcation in a Cobweb Model with Discrete Time Delays
Discrete Dynamics in Nature and Society, Article ID 37090, 8 pages http://dx.doi.org/0.55/04/37090 Research Article Hopf Bifurcation in a Cobweb Model with Discrete Time Delays Luca Gori, Luca Guerrini,
More informationDynamical Study of a Cylindrical Bar
Applied Mathematical Sciences, Vol. 11, 017, no. 3, 1133-1141 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.017.73104 Dynamical Study of a Cylindrical Bar Fatima Zahra Nqi LMN Laboratory National
More informationNonexistence of Limit Cycles in Rayleigh System
International Journal of Mathematical Analysis Vol. 8, 014, no. 49, 47-431 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.4883 Nonexistence of Limit Cycles in Rayleigh System Sandro-Jose
More informationNovel Approach to Calculation of Box Dimension of Fractal Functions
Applied Mathematical Sciences, vol. 8, 2014, no. 144, 7175-7181 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49718 Novel Approach to Calculation of Box Dimension of Fractal Functions
More informationOrder-theoretical Characterizations of Countably Approximating Posets 1
Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 9, 447-454 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4658 Order-theoretical Characterizations of Countably Approximating Posets
More informationA Numerical Solution of Classical Van der Pol-Duffing Oscillator by He s Parameter-Expansion Method
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 15, 709-71 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.355 A Numerical Solution of Classical Van der Pol-Duffing Oscillator by
More informationCaristi-type Fixed Point Theorem of Set-Valued Maps in Metric Spaces
International Journal of Mathematical Analysis Vol. 11, 2017, no. 6, 267-275 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.717 Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric
More informationSolution for a non-homogeneous Klein-Gordon Equation with 5th Degree Polynomial Forcing Function
Advanced Studies in Theoretical Physics Vol., 207, no. 2, 679-685 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/astp.207.7052 Solution for a non-homogeneous Klein-Gordon Equation with 5th Degree
More informationLocal stability and Hopf bifurcation analysis. for Compound TCP
Local stability and Hopf bifurcation analysis for Compound TCP Debayani Ghosh, Krishna Jagannathan and Gaurav Raina arxiv:604.05549v [math.ds] 9 Apr 06 Abstract We conduct a local stability and Hopf bifurcation
More informationCatastrophe Theory and Postmodern General Equilibrium: Some Critical Observations
Applied Mathematical Sciences, Vol. 11, 2017, no. 48, 2383-2391 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.77242 Catastrophe Theory and Postmodern General Equilibrium: Some Critical
More informationDetermination of Young's Modulus by Using. Initial Data for Different Boundary Conditions
Applied Mathematical Sciences, Vol. 11, 017, no. 19, 913-93 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.017.7388 Determination of Young's Modulus by Using Initial Data for Different Boundary
More informationarxiv: v4 [nlin.cd] 23 Apr 2018
Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear Ben Niu and Jiaming Zhang, Junjie Wei Department of Mathematics, arxiv:1608.03394v4 [nlin.cd] 3 Apr 018
More informationStability Analysis and Numerical Solution for. the Fractional Order Biochemical Reaction Model
Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 11, 51-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/nade.16.6531 Stability Analysis and Numerical Solution for the Fractional
More informationRestrained Independent 2-Domination in the Join and Corona of Graphs
Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3171-3176 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.711343 Restrained Independent 2-Domination in the Join and Corona of Graphs
More informationGlobal Stability Analysis on a Predator-Prey Model with Omnivores
Applied Mathematical Sciences, Vol. 9, 215, no. 36, 1771-1782 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.512 Global Stability Analysis on a Predator-Prey Model with Omnivores Puji Andayani
More informationTHE most studied topics in theoretical ecology is the
Engineering Letters :3 EL 3_5 Stabilit Nonlinear Oscillations Bifurcation in a Dela-Induced Predator-Pre Sstem with Harvesting Debaldev Jana Swapan Charabort Nadulal Bairagi Abstract A harvested predator-pre
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More informationDynamical Analysis of a Harvested Predator-prey. Model with Ratio-dependent Response Function. and Prey Refuge
Applied Mathematical Sciences, Vol. 8, 214, no. 11, 527-537 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/12988/ams.214.4275 Dynamical Analysis of a Harvested Predator-prey Model with Ratio-dependent
More informationOn the Three Dimensional Laplace Problem with Dirichlet Condition
Applied Mathematical Sciences, Vol. 8, 2014, no. 83, 4097-4101 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.45356 On the Three Dimensional Laplace Problem with Dirichlet Condition P.
More informationKKM-Type Theorems for Best Proximal Points in Normed Linear Space
International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, 603-609 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.81069 KKM-Type Theorems for Best Proximal Points in Normed
More information