Applied Mathematical Sciences, Vol 11, 2017, no 22, 1089-1095 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/ams20177271 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
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
Hopf bifurcation analysis of a dynamical heart model with time delay 1091 2) 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 )
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 ) 0 0 0 f µ (ϕ) = and P 1 = 0 1 ε 0 0 6x d ε ϕ(0)2 1 ε ϕ(0)3 0,
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)]
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 2 2 + z 2 f z 2 2 + 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 02 2 3 + 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
Hopf bifurcation analysis of a dynamical heart model with time delay 1095 1) 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), 320-323 https://doiorg/1017265/1934-7375/201403015 Received: March 4, 2017; Published: April 15, 2017