Dynamics of a continuous Hénon model

Research Article Mathematical Methods in the Applied Sciences Received XXXX (www.interscience.wiley.com) DOI: 10.1002/sim.0000 MOS subject classification: 34C28; 34K23 Dynamics of a continuous H´enon model Tom´as Caraballo∗, Renato Colucci and Luca Guerrini We study a continuous H´enon system obtained by considering the discrete original model in continuous time. While the dynamics of the continuous model is trivial, we are able to recover the complexity of the discrete model by the introduction of time delays. In particular high period limit cycles and chaotic attractors are observed. We illustrate the results with some numerical simulations. Copyright c 2009 John Wiley & Sons, Ltd. Keywords: Chaotic Systems; Delay differential equations; H´enon Map 1. Introduction In his famous work [9], M. H´enon introduced a discrete system which exhibits chaotic behaviour. The target of the author was to find the simplest model which presents the same essential properties as those encountered for the Lorenz system ([10]). The system is constructed through the composition of three mappings of the plane into itself, with the purpose to have that the volume is stretched in one direction, and at the same time folded over itself, in the course of one revolution. The obtained system is the following: (xn+1 = 1 −ax2 n+yn, yn+1 =bxn.(1.1) where aand bare real parameters. The system possesses two fixed points if a > a0=(1−b)2 4, one of them is always linearly stable, the other is unstable for a > a1=3(1−b)2 4. Through numerical simulations they found that for a < a0,or a > a3, where a3depends on b, the solutions always diverge, while for a0< a < a3, depending on the initial values either the trajectories diverge or converge to an attractor. For a0< a < a1, the attractor is the stable fixed point, while for a > a1the attractor consists in periodic set of ppoints. The value of pincreases as ais increased till reaching a critical value a2. For a2< a < a3, the attractor appears to be chaotic and the dynamics is described for the case a= 1.4, b = 0.3. The above numerical results have been rigorously proved in [11]. In particular the author proved the existence of a transversal homoclinic orbit for some parameter values, thus proving the existence of a chaotic attractor. In the present paper we consider a continuous time version of the H´enon map, and we will show that the dynamics of the continuous system is trivial compared to that of the discrete system. In fact the continuous system presents two fixed point (as in the discrete case) whose stability does not depend on the parameters, that is, one fixed point is always locally stable and the Tom´as Caraballo: Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain (Email: ca[email protected]) Renato Colucci: Department of Engineering, Niccol´o Cusano University, via Don Carlo Gnocchi 3, 00166 Roma, Italy (Email: [email protected]) Luca Guerrini: Department of Management, Universit´a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy (Email: [email protected]) ∗Correspondence to: Email: ca[email protected]. Postal Address: Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain. Tel: +34 954 55 7998 Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. Prepared using mmaauth.cls [Version: 2009/09/02 v2.00] Mathematical Methods in the Applied Sciences Dynamics of a continuous H´enon model other is unstable. Then the solutions either converge to the stable fixed point or diverge to −∞ (see next section). Of course, this behavior is not surprising, since the discrete model describes the intersections of the solutions of a three dimensional differential system with a two dimensional surface and then a more complex behavior is expected. Instead of considering the continuous model in higher dimension, we consider the addition of time delay. This strategy is motivated by the results of Berezowski (see [1], [2] and the applications [5]-[7], [12]) who studied the effects of time delay in continuous physical system. These studies are based on the consideration that most physical systems are characterized by the presence of delays of various types. It can be shown that, even small values of the delays, may lead to qualitative changes in the dynamical behaviour of the system. In particular in [1] they proved the presence of quasi-periodicity and chaos for a system that describes a tubular chemical reactor with superimposed recirculation of mass and/or heat. We will see that, for the continuous H´enon model, the introduction of time delay may recover the complexity of the discrete model. In fact we will see (through Hopf Bifurcation and stability switches analysis) the appearance of stable limit cycles with increasing period and even the presence of a strange attractor which resembles the famous H´enon Attractor for the values a= 1.3 and b= 0.3. In the next section we introduce the continuous model and describes its dynamics, while in Section 3 we study the system with one delay parameter. Sections 4 and 5 are devoted to the case in which we have two delay parameters while in Section 6 we present several numerical simulations in order to illustrate the results of the paper. In the last section we present some consideration about the attractor proving the existence of a trapping region (positively invariant set) and of an absorbing set. 2. The continuous model In this section we introduce the continuous model with two time delay parameters which will be the objective of our paper. First of all, note that if we substract xnand ynrespectively in the equations of (1.1), we obtain: (xn+1 −xn=−xn+1−ax2 n+yn, yn+1 −yn=−yn+ (bxn), which can be considered as the time-discretization of the following continuous (H´enon) model (σ1˙x=−x+1−ax2+y, σ2˙y=−y+ (bx), where σ1>0, σ2>0 are some time-scale parameters. In the present work we will consider a generalization of the previous continuous model including two time delay parameters. Namely, we will analyze the problem (σ1˙x=−x+1−ax2 τ1+yτ1, σ2˙y=−y+ (bxτ2), where a, b ∈R,σ1>0 and σ2>0,are the previous time-scale parameters, τ1>0 and τ2>0 are time delays, and we use the notation xτ=x(t−τ) for short. In order to compare our results with those obtained by H´enon for a= 1.4 and b= 0.3, we restrict the constants a, b ∈Rto be positive. In addition, for simplicity, we will work with σ1=σ2=σ. For results ensuring the global existence and uniqueness of initial value problems associated to the previous system the reader can see, e.g., Hale [8]. Let us now rewrite our system as follows for mathematical convenience          ˙x=−1 σx+1 σ−a σx2 τ1+1 σyτ1, ˙y=−1 σy+b σxτ2. (2.1) Like its discrete counterpart, system (2.1) has two steady states P+= (x+ ∗, y+ ∗) and P−= (x− ∗, y− ∗) , where x± ∗are the solutions of the equation ax2+ (1 −b)x−1 = 0, and y± ∗=bx± ∗. In details we have: x± ∗=b−1±p(1 −b)2+ 4a 2a.(2.2) Obviously, they are real since (1 −b)2+ 4a > 0.Having assumed a > 0, a direct calculation shows that x+ ∗>0 and x− ∗<0. Let (x∗, y∗) be a steady state of (2.1). In order to determine the stability of (x∗, y∗), we linearize (2.1) at (x∗, y∗), and find that the associated characteristic equation takes the form λ2+2 σλ+1 σ2+2ax∗ σ2+2ax∗ σλe−λτ1−b σ2e−λ(τ1+τ2)= 0.(2.3) 2Copyright c 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2009, 00 1–19 Prepared using mmaauth.cls Dynamics of a continuous H´enon model Mathematical Methods in the Applied Sciences Lemma 1 In absence of delay, the equilibrium point (x+ ∗, y+ ∗)for model (2.1)is locally asymptotically stable, while the equilibrium point (x− ∗, y− ∗)is unstable. Proof. When time delays are not considered, i.e. τ1=τ2= 0,the characteristic polynomial for model (1.1) reduces to λ2+2(1 + ax∗) σλ+1−b+ 2ax∗ σ2= 0.(2.4) It is seen from (2.4) that the determinant of the functional Jacobian matrix is (1 −b+ 2ax∗)/σ2and its trace is −2(1 + ax∗)/σ. As a result, we derive that both eigenvalues of the characteristic polynomial (2.4) are negative when x∗=x+ ∗and the eigenvalues have different signs when x∗=x− ∗.The statement follows. We recall that P−is always on the left of P+, its stable curve S(t) splits the plane into two regions (see figure 1): on the right of S(t) the solutions converge to P+while on the left of S(t) the solutions diverges. The latter behaviour is suggested by the preeminence of the term −a σx2in the first equations that let x→ −∞. Then, in absence of delay the dynamics is trivial. In figure 1we represent the vector field of the system for the classical case a= 1.4, b= 0.3. -4-2 0 2 4 -4 -2 0 2 4 x y Figure 1. The vector field of the system for σ= 1, a= 1.4, b= 0.3. We will now explore how the time delays affect the dynamics of (2.1) and will see how to recover the complexity of dynamics observed for the discrete case. We first observe that it is well known that a root λof (2.3) depends on time delays continuously. If it will ever leave the left half plane and enter the right half plane on the complex plane as time delays increase, it must cross the purely imaginary axis and a stability switch may occur (through Hopf bifurcation). We need to analyse the boundary of the stability region determined by the equations λ= 0 and λ=±iω. The case λ= 0 cannot occur since from (2.3) we obtain the absurd 0 = 2ax∗+ 1 −b > 0. Therefore, only the case λ=iω (ω > 0) has to be analysed. We will analyze in details this problems in the following sections. 3. Case τ1= 0 and τ2>0 In this section we consider the case in which τ1= 0 and τ2>0. The characteristic polynomial (2.3) becomes λ2+Aλ +B+Cλe−λτ2= 0,(3.1) where the coefficients are given by the following expression. A=2(1 + ax∗) σ, B =1 + 2ax∗ σ2and C=−b σ2<0.(3.2) For the sake of simplicity we prove the following lemma which analyzes in details the sign of the coefficient of the characteristic polynomial depending on the control parameters. Lemma 2 1) Let x∗=x+ ∗>0.Then A > 0, B > 0and C < 0. Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. 3 Prepared using mmaauth.cls Mathematical Methods in the Applied Sciences Dynamics of a continuous H´enon model 2) Let x∗=x− ∗<0. i) If b > 1+4a 2, then A > 0, B > 0and C < 0. ii) If b=1+4a 2, then A > 0, B = 0 and C < 0. iii) If a < b < 1+4a 2, then A > 0, B < 0and C < 0. iv) If b=a, then A= 0, B < 0and C < 0. v) If b < a, then A < 0, B < 0and C < 0. Proof. 1.is immediate; 2.follows from sign(A) = sign(1 + ax− ∗) and sign(B) = sign(1 + 2ax− ∗),(3.3) noting that from (2.2) one has 1 + 2ax− ∗=b−p(1 −b)2+ 4a, 1 + ax− ∗=b+ 1 −p(1 −b)2+ 4a 2.(3.4) Assuming that Eq. (3.1) has a purely imaginary solution of the form λ=iω, (ω > 0), substituting it into (3.1) and separating the real and imaginary parts lead to −ω2+B=−Cω sin ωτ2, Aω =−Cω cos ωτ2.(3.5) Squaring and adding both equations of (3.5), we obtain the following equation of ω2 ω4+A2−C2−2Bω2+B2= 0,(3.6) In the next propositions we study the existence of positive roots for equation 3.6 depending on the coefficients (3.2). Proposition 1 1) If A2−C2−2B≥0,or if A2−C2−2B < 0, B < 0and A2−C2>4B, or if A2−C2−2B < 0, B > 0 and A2−C2≥0,then Eq. (3.6)has no positive roots. 2) If A2−C2−2B < 0and B= 0,then Eq. (3.6)has a unique positive root ω0=p−(A2−C2). 3) If A2−C2−2B < 0, B < 0and A2−C2<4B, or if A2−C2−2B < 0, B > 0and A2−C2<0,Eq. (3.6)has two positive roots ω1,2=v u u t−(A2−C2−2B)±q(A2−C2−2B)2−4B2 2, where ω1< ω2. Proof. The statement is straightforward if A2−C2−2B≥0. Let A2−C2−2B < 0.If B= 0,then Eq. (3.6) reduces to ω4+A2−C2ω2= 0,which has a unique positive root. Let B < 0.Notice that A2−C2−2B < 0 implies A2−C2<0. Hence, A2−C2−2B2−4B26= 0.We find that A2−C2−2B2−4B2>0 when A2−C2<4B, so that there exist two positive solutions for Eq. (3.6). On the other hand, A2−C2−2B2−4B < 0 when A2−C2>4B, and so there is no positive solution for Eq. (3.6). Let B > 0.If A2−C2≥0,then Eq. (3.6) does not have positive roots. If A2−C2<0,then Eq. (3.6) has two positive roots. This completes the proof. From the previous proposition we are able to study the existence/quantity of positive roots of equation (3.6) depending on the parameters of the system. Proposition 2 1) Let x∗=x+ ∗. i) If σ≥b [2(1+ax+ ∗)], then Eq. (3.6)has no positive roots. ii) If σ < b [2(1+ax+ ∗)], then Eq. (3.6)has two different positive roots. 2) Let x∗=x− ∗. i) If σ≥b q(1+2ax− ∗)2+1 or if σ < b q(1+2ax− ∗)2+1 , with b > 1+4a 2and σ≥b [2(1+ax− ∗)], or a < b < 1+4a 2and σ≥b [2(1+ax− ∗)], or b=a, or b < a and σ > b (−2ax− ∗)or a < b < 1+4a 2and σ > b (−2ax− ∗), then Eq. (3.6)has no positive roots. ii) If σ < b q(1+2ax− ∗)2+1 with b > 1+4a 2and σ < b [2(1+ax− ∗)], or a < b < 1+4a 2and σ < b (−2ax− ∗), or b < a and σ < b −2ax− ∗, then Eq. (3.6)has two different positive roots ω1,2. iii) If σ < b q(1+2ax− ∗)2+1 , with b=1+4a 2,σ < b [2(1+ax− ∗)] i.e. with σ < 1+4a 4, then Eq. (3.6)has a unique positive root ω0. Proof. A direct computation shows A2−C2−2B=(1 + 2ax∗)2+ 1 σ2−b2 σ4. 4Copyright c 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2009, 00 1–19 Prepared using mmaauth.cls Dynamics of a continuous H´enon model Mathematical Methods in the Applied Sciences Consequently, A2−C2−2B≥0 if σ≥b q(1 + 2ax∗)2+ 1 , and A2−C2−2B < 0 if σ < b q(1 + 2ax∗)2+ 1 . The statement follows after a long and tedious analysis using the previous Lemma 2, Proposition 1, (3.3) and (3.4). The critical values τr 2,j (r= 0,1,2; j= 0,1,2, ...) of the delay τ2corresponding to ωrare obtained solving equations in (3.5). This yields sin(ωrτr 2,j ) = ω2 r−B Cωr ,cos(ωrτr 2,j ) = −A C. Thus, τr 2,j =           1 ωrcos−1−A C+ 2jπ,if ω2 r−B≥0, 1 ωr2π−cos−1−A C+ 2jπ,if ω2 r−B < 0. (3.7) The next proposition analyzes the direction of the bifurcation. Proposition 3 λ=iωr(r= 0,1,2) is a simple root of the characteristic equation (3.1)satisfying dRe(λ) dτ2τ2=τ0 2,j ,ω=ω0 >0,dRe(λ) dτ2τ2=τ1 2,j ,ω=ω1 <0,dRe(λ) dτ2τ2=τ2 2,j ,ω=ω2 >0. Proof. Let λ(τ2) be the root of (3.1) near τ2=τr 2,j such that Re(λ(τr 2,j )) = 0 and Im(λ(τr 2,j )) = ωr. By differentiating (3.1) with respect to τ2, we obtain h2λ+A+ (C−Cτ2λ)e−λτ2idλ dτ2 =Cλ2e−λτ2.(3.8) Therefore, dλ dτ2−1 =2λ+A+ (C−Cτ2λ)e−λτ2 Cλ2e−λτ2=(2λ+A)eλτ2 Cλ2+1 λ2−τ2 λ. Recalling (3.1), eλτ2=−(Cλ)/(λ2+Aλ +B),and so we have dλ dτ2−1 =−2λ+A λ(λ2+Aλ +B)+1 λ2−τ2 λ. A calculation yields Re dλ dτ2−1 τ2=τr 2,j ,ω=ωr =2ω2 r+A2−2B (B−ω2 r)2+A2ω2 r−1 ω2 r . Notice that (3.6) implies C2ω2 r= (B−ω2 r)2+A2ω2 r. As a result, Re dλ dτ2−1 τ2=τr 2,j ,ω=ωr =2ω2 r+A2−C2−2B C2ω2 r , which, using (3.7), gives sign (d(Reλ) dτ2τ2=τr 2,j ,ω=ωr)=sign (Re dλ dτ −1 τ2=τr 2,j ,ω=ωr) =sign 2ω2 r+A2−C2−2B. If ωr=ω0, then ω2 0+A2−C2−2B= 0. Thus, sign 2ω2 r+A2−C2−2B=sign ω2 0. Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. 5 Prepared using mmaauth.cls Mathematical Methods in the Applied Sciences Dynamics of a continuous H´enon model If ωr=ω1,2, then 2ω2 r+A2−C2−2B=±q(A2−C2−2B)2−4B2. Hence, it is positive when ωr=ω1and negative when ωr=ω2. It remains to prove that λ=iωris a simple root of (3.1). If we suppose by contradiction that λ=iωris a multiple root of (3.1), then from (3.8) we obtain Cω2 re−iωrτr 2,j = 0, leading to an absurd. This completes the proof. Consider the fixed point P+for model (2.1) that is locally asymptotically stable for τ2= 0. From the previous discussion we have that when Eq. (3.6) has no positive real root, all eigenvalues of the characteristic equation (3.1) have negative real parts for arbitrary τ2>0. The equilibrium point remains stable for all τ2>0.When Eq. (3.6) has only the positive root ω0,then λ=iω0satisfies [dRe(λ)/dτ2]ω=ω0>0,and therefore there exists a critical delayed value τ0 2,0such that all the eigenvalues of the characteristic equation (3.1) have negative real parts for τ2∈(0, τ0 2,0) and at least one root with a positive real part for τ2> τ0 2,0. The equilibrium point remains stable in (0, τ0 2,0) and loses its stability via a Hopf bifurcation at τ2=τ0 2,0. When Eq. (3.6) has two positive roots ω1< ω2,then [dRe(λ)/dτ2]ω=ω1<0 and [dRe(λ)/dτ2]ω=ω2>0,and there exists a finite number of τ2intervals in which all eigenvalues of the characteristic equation (3.1) have negative real parts. If time delay is fixed into these intervals, the equilibrium point is locally asymptotically stable, while it is unstable for values outside these delayed ranges. Therefore, the system dynamics switches from stable to unstable, and then back to stable when delay increases and crosses the critical delayed values. Consider now the fixed point P−of the model (2.1) that is unstable for τ2= 0. If Eq. (3.6) has no positive solutions or the unique solution ω0,then no stability switches exist. If Eq. (3.6) has two positive solutions ω1and ω2, then many stability switches may exist. Based on the above analysis, we have the following results on the stability of the equilibria (x+ ∗, y+ ∗) and (x− ∗, y− ∗). Theorem 1 Let τ0 2,0be defined as in (3.7). 1) The equilibrium (x+ ∗, y+ ∗)of system (2.1)is locally asymptotically stable for τ2≥0if σ≥b [2(1+ax+ ∗)], and it has stability switches for τ2>0if σ < b [2(1+ax+ ∗)]. A Hopf bifurcation occurs at each stability switch. 2) The equilibrium (x− ∗, y− ∗)of system (2.1)is unstable for τ2≥0if one of the following conditions is fulfilled: ⋄σ≥b q(1+2ax− ∗)2+1 , ⋄σ < b q(1+2ax− ∗)2+1 ,b > (1+4a) 2and σ≥b [2(1+ax− ∗)], ⋄σ < b q(1+2ax− ∗)2+1 ,a < b < (1+4a) 2and σ≥b [2(1+ax− ∗)], ⋄σ > b (−2ax− ∗), ⋄σ < b q(1+2ax− ∗)2+1 ,b < a and σ > b (−2ax− ∗), ⋄σ < b q(1+2ax− ∗)2+1 , with b=(1+4a) 2and σ < b [2(1+ax− ∗)] i.e. with σ < (1+4a) 4. 3) The equilibrium (x− ∗, y− ∗)has stability switches for τ2>0if one of the following conditions is fulfilled ⋄σ < b q(1+2ax− ∗)2+1 ,b > 1+4a 2and σ < b [2(1+ax− ∗)], ⋄σ < b q(1+2ax− ∗)2+1 ,b < (1+4a) 2and b6=a. A Hopf bifurcation occurs at each stability switch. 4. Case τ1>0, τ2in its stable intervals In this case we consider both delay parameters different from zero. In order to simplify the analysis we consider the characteristic equation (2.3) with τ2in its stable intervals and regard τ1as a parameter. For convenience, we rewrite (2.3) as λ2+Mλ +N+ (P+Qλ)e−λτ1+Re−λ(τ1+τ2)= 0,(4.1) where M=2 σ, N =1 σ2, P =2ax∗ σ2, Q =2ax∗ σ, R =−b σ2. Let λ=iω (ω > 0) be a root of (4.1). Then, we obtain −ω2+N= (Rsin ωτ2−Qω) sin ωτ1−(Rcos ωτ2+P) cos ωτ1(4.2) and Mω = (Rsin ωτ2−Qω) cos ωτ1+ (Rcos ωτ2+P) sin ωτ1.(4.3) 6Copyright c 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2009, 00 1–19 Prepared using mmaauth.cls Dynamics of a continuous H´enon model Mathematical Methods in the Applied Sciences Squaring and adding Eqs. (4.3) and (4.4) lead to g(ω) = ω4+M2−Q2−2Nω2+ (2QR sin ωτ2)ω +N2−P2−R2−2PR cos ωτ2= 0.(4.4) It is easy to check that g(+∞) = +∞.In case g(0) = N2−(P+R)2<0, there is at least a positive ωsatisfying Eq. (4.4). Notice that g(0) <0 for |N|<|P+R|, i.e. when −1±p(1 −b)2+ 4a>1. We find that g(0) <0 if the equilibrium point (x∗, y∗) of system (2.1) is (x− ∗, y− ∗), while g(0) is not necessarily negative if (x∗, y∗) = (x+ ∗, y+ ∗).Henceforth, we assume that Eq. (4.4) has finitely many positive roots ω1, ω2, ..., ωN.For every fixed ωl, l= 1,2, ..., N, there exists a sequence τj 1,l >0 (j= 1,2, ...) such that (4.4) holds. Let τc 1= min nτj 1,l , l = 1,2, ..., N, j = 1,2, ...o.(4.5) When τ1=τc 1,then the characteristic equation (4.1) has a pair of purely imaginary roots ±iωc. Let λ(τ1) be the root of (4.2) near τ1=τc 1satisfying Re(λ(τc 1)) = 0 and Im(λ(τc 1)) = ωc. Substituting λ(τ1) into the left side of (4.1) and taking the derivative with respect to τ1gives h2λ+M+Qe−λτ1−(P+Qλ)τ1e−λτ1−R(τ1+τ2)e−λ(τ1+τ2)idλ dτ1 =h(P+Qλ)λ+Rλe−λτ2ie−λτ1.(4.6) Proceeding similarly to the case τ1= 0 and τ2>0,we can verify the transversality condition of Hopf bifurcation to be given by sign dRe(λ) dτ1τ1=τc 1 =sign "Re dλ dτ1−1#τ1=τc 1 =sign (a1b1−a2b2), where a1=M−Qωcτc 1sin ωcτc 1+ (Q−Pτc 1) cos ωcτc 1+R(τc 1+τ2) sin ωcτc 1sin ωcτ2−R(τc 1+τ2) sin ωcτc 1cos ωcτ2,(4.7) a2=−ωc(2 −Qτc 1cos ωcτc 1) + (Q−Pτc 1) sin ωcτc 1−R(τc 1+τ2) sin ωcτc 1cos ωcτ2−R(τc 1+τ2) cos ωcτc 1sin ωcτ2,(4.8) and b1=ωc[(Rsin ωcτ2−Qωc) cos ωcτc 1+ (P+Rcos ωcτ2) sin ωcτc 1],(4.9) b2=−ωc[(Rsin ωcτ2−Qωc) sin ωcτc 1−(P+Rcos ωcτ2) cos ωcτc 1].(4.10) Then, if sign (a1b1−a2b2)>0 (resp. sign (a1b1−a2b2)<0), each crossing of the real part of characteristic roots at τc 1is from left (resp. right) to right (resp. left). Finally, notice that λ=iωcis a simple root of (4.2). Otherwise, from (4.6) we have (P+Qiωc)iωc+Riωce−iωcτ2, which means Rcos ωτ2+P= 0 and Rsin ωτ2−Qω = 0. From (4.2) and (4.3), one would get the absurd −ω2+N= 0 and Mω = 0. According to the general Hopf bifurcation theorem for Functional Differential Equations in [8], we derive the following results. Theorem 2 Let g(ω), τc 1and a1, b1, a2, b2be defined as in (4.4),(4.5),(4.7)-(4.10), respectively, and τ2in its stable intervals. Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. 7 Prepared using mmaauth.cls Mathematical Methods in the Applied Sciences Dynamics of a continuous H´enon model 1) If g(ω)has no positive root, then the equilibrium (x∗, y∗)of system (2.1)is locally asymptotically stable or unstable for any given time delay τ1, depending on whether or not the system free of time delay is stable. 2) Suppose that the fixed point (x∗, y∗)for the system without delay is locally asymptotically stable (resp. unstable). Moreover, suppose that the system with delay satisfies the following conditions: ⋄g(ω)has only one positive root ⋄sign (a1b1−a2b2)>0(resp. sign (a1b1−a2b2)<0), then there exists one critical time delay τc 1>0such that the equilibrium (x∗, y∗)of system (2.1)is locally asymptotically stable (resp. unstable) for τ1∈[0, τc 1)and becomes unstable (resp. locally asymptotically stable) when τ1> τc 1. System (2.1) undergoes a Hopf bifurcation at the equilibrium (x∗, y∗)for τ1=τc 1. Suppose that the fixed point (x∗, y∗)for the system without delay is locally asymptotically stable (resp. unstable) and sign (a1b1−a2b2)<0(resp. sign (a1b1−a2b2)>0), then the equilibrium (x∗, y∗)of system (2.1) remains locally asymptotically stable (resp.unstable) as τ1increases. 3) If g(ω)has at least two positive roots, then a finite number of stability switches may occur as the time delay τ1increases from zero to the positive infinity, and the system becomes unstable at last. A Hopf bifurcation occurs at each stability switch. 5. Case τ1=τ2=τ In this section we consider the case in which τ1=τ2. The characteristic equation (2.3) becomes λ2+Aλ +B+ (Cλ +D)e−λτ +Ee−2λτ = 0,(5.1) where its coefficients are given by A=2 σ, B =1 σ2, C =2ax∗ σ, D =2ax∗ σ2, E =−b σ2. In order to analyze the distribution of characteristic roots of (5.1) we will use the method proposed in [4]. If λ=iω, (ω > 0) is a root of (5.1), then −ω2+Aiω +B+ (Ciω +D)e−iωτ +Ee−2iωτ = 0.(5.2) Let us consider the following two cases: (ωτ)/26= (π/2) + jπ, j ∈N0=N∪{0}, and (ωτ)/2 = (π/2) + jπ, j ∈N0. Suppose the first condition is verified. Letting θ= tan [(ωτ)/2], we have e−iωτ = (1 −iθ)/(1 + iθ). Separating the real and imaginary parts in (5.2), we see that θsatisfies (ω2−B+D−Eθ2−2Aωθ =ω2−B−D−E, (C−A)ωθ2+−2ω2+ 2B−2Eθ=−(A+C)ω. (5.3) Denote D(ω) =  ω2−B+D−E−2Aω (C−A)ω−2ω2+ 2B−2E ,(5.4) E(ω) =  ω2−B−D−H−2Aω −(A+C)ω−2ω2+ 2B−2E ,(5.5) F(ω) =  ω2−B+D−E ω2−B−D−E (C−A)ω−(A+C)ω .(5.6) 8Copyright c 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2009, 00 1–19 Prepared using mmaauth.cls Dynamics of a continuous H´enon model Mathematical Methods in the Applied Sciences If D(ω)6= 0, one can solve from Eqs. (5.3) that θ2=E(ω) D(ω), θ =F(ω) D(ω). Furthermore, from (5.4)-(5.6), we obtain that ωsatisfies D(ω)E(ω) = [F(ω)]2,(5.7) which is a polynomial equation for z=ω2with degree 4, z4+s1z3+s2z2+s3z+s4= 0,(5.8) with s1= 2A2−4B−C2, s2= 6B−2E2−4A2B−D2+A4−A2C2+ 2BC2+ 2C2E, s3= 2BD2−A2D2−4B3+ 2A2B2−B2C2−2BC2E+ 4ACDE −2D2E+ 4BE2−2A2E2−C2E2, s4= (B−E)2[−D2+ (B+E)2]. Notice that if D(ω) = 0, in order to make sure the solvability of Eqs. (5.3) for θ, we have E(ω) = F(ω) = 0. Then, ωis still a solution of Eq. (5.7). Now suppose that (ωτ)/2 = (π/2) + jπ, j ∈N0. In this case, A=Cand ω2=B+E−D, and thus D(ω) = F(ω) = 0. Hence, ω2satisfies Eq. (5.8) in this case as well. The next lemma (see [4]) provides the relation between solutions of the characteristic polynomial (5.1) and the roots of (5.8). Lemma 3 If ±iω,(ω > 0) is a pair of purely imaginary roots of the characteristic equation (5.1), then ω2is a positive root of the quartic polynomial equation (5.8). The next lemma (see [4]) provides the algorithm of solving the critical delay values for purely imaginary roots of (5.1). Lemma 4 If Eq. (5.8) has a positive root ω2 N(ωN>0) and D(ωN)6= 0, then system (5.3)has a unique real root θN=F(ωN) D(ωN), when ω=ωN. Hence, the characteristic equation (5.1)has a pair of purely imaginary roots ±iωNwhen τ=τj N=2 tan−1(θN) + 2jπ ωN . Following again strategy in [4], the transversality condition for the roots moving across the imaginary axis can be formulated as follows. Proposition 4 Let G(ω, θ) = D(1 + θ2) + 2E(1 −θ2)[2ω(1 −θ2) + 2Aθ]−[Cω(1 + θ2)−4Eθ][A(1 −θ2)−4ωθ +C(1 + θ2)]. If G(ωN, θN)6= 0, then iωNis a simple root of the characteristic equation (5.1)for τ=τj Nand there exists λ(τ) = α(τ) + iω(τ) which is the unique root for τ∈(τj N−ε, τj N+ε)for some small ε > 0satisfying α(τj N) = 0, ω(τj N) = ωN,and α′(τj N)>0. Moreover, [dRe(λ)/dτ]τ=τj N >0,G(ωN, θN)>0, and [dRe(λ)/dτ]τ=τj N <0,G(ωN, θN)<0. Hence, from the above discussion, we can establish the following result. Theorem 3 If the equilibrium (x∗, y∗)of system (2.1) is locally asymptotically stable (resp. unstable) when τ= 0 and one has that G(ωN, θN)>0 (resp. G(ωN, θN)<0), then there exists τ∗>0such that the equilibrium (x∗, y∗)is locally asymptotically stable (resp. unstable) when τ∈[0, τ∗)and it is unstable (resp. locally asymptotically stable) when τ∈(τ∗, τ∗+ε)for ε > 0and small. Furthermore, system (2.1) undergoes a Hopf bifurcation at (x∗, y∗)when τ=τ∗. As τincreases from τ=τ∗, stability switches may occur. Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. 9 Prepared using mmaauth.cls Mathematical Methods in the Applied Sciences Dynamics of a continuous H´enon model 7.2. The case with delay For simplicity, we only consider the case in which τ1=τ2. In this situation we use a different approach, we will find a positive invariant set in which a local attractor lies and which attracts all the bounded sets of the positive invariant set. First of all we observe that Proposition 5is still valid. Next we prove the following preliminary result: Proposition 6 Suppose that x(t)and y(t)are bounded and suppose that the initial data satisfy x(θ)≤1 1−b, y(θ)≤b 1−b,for all θ∈[−τ, 0]. Then xM= max t≥−τ{x(t)} ≤ 1 1−b, yM= max t≥−τ{y(t)} ≤ b 1−b. Proof. From the second equation we have that ˙y=b σx(t−τ)−1 σy≤b σxM−1 σy, from which it follows immediately that yM≤b·xM. Now suppose Tis such that x(T) = xM, then ˙x(T) = 0, and from the first equation of the system we obtain: −1 σxM+1 σ−1 σax2(t−τ) + 1 σy(t−τ) = 0. Then xM= 1 −ax2 τ+yτ≤1 + yM≤1 + bxM, from which we obtain xM≤1 1−b. The previous proposition provides bounds for solutions whenever they are bounded. In the next proposition we will prove that any solution is bounded. Proposition 7 The solutions of system (2.1), with τ1=τ2, are bounded. Proof. In order to simplify the computations we perform the following transformation ˜x=x−x− ∗,˜y=y−y− ∗, which transforms P−into (0,0) and P+into √∆ a,b√∆ a, where ∆ = (1 −b)2+ 4a. Moreover, the positive invariant set for the associated problem is R2 +. We rewrite the system in the new coordinates without replacing the notation:        ˙x=−1 σx+1 σyτ−a σx2 τ+2a σ|x∗ −|xτ, ˙y=−1 σy+b σxτ. We sum both equations, term by term, and obtain: ˙x+ ˙y+1 σ(x+y) = 1 σ(xτ+yτ) + a σxτ√∆ a−xτ. Observe that the function f(q) := a σq√∆ a−q 16 Copyright c 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2009, 00 1–19 Prepared using mmaauth.cls Dynamics of a continuous H´enon model Mathematical Methods in the Applied Sciences is positive for q∈(0,√∆ a) and achieves its maximum for q=√∆ 2awith max f=∆ 4aσ =1 σ(1 −b)2 4a+ 1:= 1 σF. We consider two different cases: Case 1: Suppose x(t) + y(t)≥x(t−τ) + y(t−τ),(7.2) then ˙x(t) + ˙y(t)≤     0,if x(t−τ)≥√∆ a, 1 σF, if x(t−τ)<√∆ a. If there exists a time Tsuch that x(T−τ) = √∆ athen x(t) + y(t) is decreasing in a right neighbourhood of T, and remains decreasing till x(t−τ)≥√∆ aand (7.2) holds. Then x+yis bounded in this time interval. Suppose on the contrary that x(t−τ)<√∆ a,for all t≥T, then x(t)<√∆ a,for all t≥T+τ. From the second equation of the system we have that ˙y≤ −1 σy+1 σb√∆ a,for all t≥T, then we conclude that y(t) is bounded for all t≥T. We observe that if x(t−τ)≤√∆ ain the interval [t1, t2], then y(t2)≤y(t1)e−1 σ(t2−t1)+1 σb√∆ a (t2−t1) e1 σt2 . From the second equation of the system, it is not possible to have a sequence of such intervals in which y(t)→ ∞ being x(t−τ)<√∆ a. Case 2: Suppose x(t) + y(t)< x(t−τ) + y(t−τ).(7.3) If the previous expression is valid in (T, +∞) we obtain that x+yis bounded by x(−τ) + y(−τ). If, on the contrary, there exists a infinite sequence of intervals [tn 1, tn 2] such that (7.3) holds, then x+yis decreasing in each interval. We have to exclude the case in which the sequence of local maxima of x+yblow up in infinite time. Suppose that Tnis the sequence of time for which the function x+yattains the local maxima with Tn→+∞. For any local maximum we have x(Tn) + y(Tn) = x(Tn−τ) + y(Tn−τ) + ax(Tn−τ)√∆ a−x(Tn−τ). If x(tn−τ)>√∆ a, then x(Tn) + y(Tn)< x(Tn−τ) + y(Tn−τ), and, as a consequence, at Tn−τthere is a local maximum or there exists ε > 0 such that at Tn−τ−εthere is a local maximum. Then we repeat the same reasoning with Tnreplaced by Tn−τor Tn−τ−εand we recursively bound each local maximum by a previous one. If x(Tn−τ)<√∆ a, then x(Tn) + y(Tn)> x(Tn−τ) + y(Tn−τ), Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. 17 Prepared using mmaauth.cls Mathematical Methods in the Applied Sciences Dynamics of a continuous H´enon model and the previous expression remains true in a small right neighbourhood of Tn, then we are in case 1. Then if Tnkis the subsequence of local maximum such that the previous two inequalities are verified, we have that from the second equation of the system we cannot have that y(Tnk)→ ∞. From the previous propositions we obtain that the set Σ := x∗ −,1 1−b×y∗ −,b 1−b is positively invariant, that is, if solutions start in Σ, they remain inside Σ for all time. As a consequence a local attractor is also contained in it (see figure (8)). Theorem 6 The system (2.1)with τ1=τ2, possesses a local attractor inside the set Σ. Proof. Since Σ is positively invariant then it contains the omega limit sets of the solutions starting on it and as a consequence Σ contains a local attractor. P2 P1 P 3 P- -1.0 -0.5 0.5 1.0 1.5 x -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 y Figure 8. The attractor, for a= 1.4, b= 0.3, σ= 0.1, τ1τ2= 40, is inside the set Σ. Here P−= (x− ∗, y− ∗), P1= ( 1 1−b, y− ∗), P2= ( 1 1−b,b 1−b), P3= (x− ∗,b 1−b) 8. Conclusions In the present paper we have considered a continuous system derived from the H´enon map, in particular we have shown that the complex dynamics of the discrete model is not displayed by the continuous one. In order to recover the complexity of the solutions set it is necessary to consider time delay. Several interesting questions are left open and may be considered a natural continuation of the present work. Among them it is of interest to consider the problem of multi-stability, that is the coexistence of several local attractors (see [13] and references cited therein for a general discussion). The complicate conditions founded for stability of fixed points and for Hopf Bifurcation suggest that multi-stability is possible for the system, however it is far from the scope of the present article and we will be addressed in future research. Acknowledgements. This work has been partially supported by grant MTM2015-63723-P (MINECO/FEDER, EU) and Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492. References 1. Berezowski, M., Effect of delay time on the generation of chaos in continuous systems. One-dimensional model. Two-dimensional model - tubular chemical reactor with recycle. Chaos, Solitons and Fractals 12, 83-89; 2001. 2. 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Physics Reports, Volume 540, Issue 4, 30 July 2014, Pages 167-218. Math. Meth. Appl. Sci. 2009, 00 1–19 Copyright c 2009 John Wiley & Sons, Ltd. 19 Prepared using mmaauth.cls