/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 4, 2014, 442-461 ISSN 1307-5543 – www.ejpam.com Analyzing Periodic Solutions of an ODE Suspension Bridge Model using Difference Equations and Polynomial Methods Sukanya Basu1 1 Independent, Grand Rapids, Michigan, USA Abstract. In [13], McKenna and Moore studied oscillations in a suspension bridge by investigating pe- riodic solutions to a differential equations model for the bridge and its linearized version numerically. In this paper, the author seeks to build a rigorous mathematical foundation for the numerical experi- ments of McKenna and Moore in [13] by studying an associated discrete difference equations model using an interplay of ideas from engineering, discrete dynamical systems, algebraic geometry and the theory of polynomials. 2010 Mathematics Subject Classifications: 39A05, 39A11 Key Words and Phrases: suspension bridge, torsional angle, discrete model, periodic solution, equi- librium, bifurcation, elliptic curve, basin of attraction, global attractivity, eigenvalues 1. Introduction Suspension bridge dynamics such as factors leading to the 1940 Tacoma Narrows Bridge Collapse in Washington, USA, have been widely studied by civil engineers, architects and ap- plied mathematicians all across the world (see [1, 6–8, 13, 14]). Some researchers prefer to use suspension bridge models involving ordinary differential equations (see [1, 13, 14]) while others prefer to use partial differential equations models (see [6–8]) to study the dynamics of suspension bridges. Often both groups of researchers turn to numerical algorithms to gain further insight into their model dynamics. More specifically, they run their numerical algo- rithms on discretized versions of their original model for a finite number of initial conditions to get approximate solutions. Some exceptions to this rule are paper [14] by A. Pascoletti and F. Zanolin and paper [8] by Z. Ding in which they present rigorous mathematical proofs to back their results. A danger in relying too heavily on numerical algorithms to study the dynamics of actual suspension bridges is that one can very easily miss those few useful initial points which lead to extraordinary or unusual bridge dynamics from the point of view of applications (see [13]) since computers do not possess the mathematical intuition needed to look for special initial Email address: sukanyabasu49@yahoo.in http://www.ejpam.com 442 c© 2014 EJPAM All rights reserved. S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 443 conditions. It is also extremely difficult to predict exact bifurcation values for parameters by relying solely on computer-aided numerical simulations (see [13]). In this paper, the author studies the dynamics of a discrete difference equations model for a suspension bridge using a rigorous mathematical approach involving the theory of difference equations (see [2–4, 9– 12, 15]) to get useful global attractivity results and parameter bifurcation values backed by mathematical theorems and proofs. In [13], P. J. McKenna and K. S. Moore studied oscillations in a suspension bridge by ana- lyzing periodic solutions to the system of nonlinear ordinary differential equations θ ′′ =− 6K m cosθ sinθ −δ1θ ′ +λ sinµt y ′′ =− 2K m y −δ2 y ′ + g (1) where δ1, δ2 are damping constants, K , m, λ and µ are positive parameters, g is the force due to gravity and λ sinµt is an external force at time t. They used a numerical continuation algorithm to demonstrate the existence of three periodic solutions. The paper relied heavily on numerical experiments and did not focus as much on developing the mathematical theory to back their numerical observations. It was also missing three key aspects, namely, (a) bounds on the number of real equilibria and real periodic solutions, (b) existence conditions for the real equilibria and periodic solutions, and (c) global attractivity results for these solutions including basins of attraction and precise bifurcation values for the parameters λ and µ. In this paper, the author will set up a rigorous mathematical foundation for the McKenna- Moore suspension bridge model (1) by first discretizing it and then employing analytical and geometrical methods from the theory of difference equations (see [2–4, 9–12, 15]) to analyze equilibria and periodic solutions of the resulting difference equation. For the rest of this paper, the author will use the term ’periodic solutions’ to mean periodic solutions of minimal period two. Using this approach, the author will successfully come up with missing mathematical explanations for numerical phenomena observed by McKenna and Moore in [13]. She will also successfully resolve the three key aspects missing from [13], namely, bounds, existence conditions and global attractivity of the real equilibria and periodic solutions of equation (1) in this paper. This paper is organized as follows. In Section 2, we introduce a discretization of the the McKenna-Moore suspension bridge model (1) involving a second-order nonlinear difference equation. In Section 3, we look at the linearization of the discrete nonlinear model introduced in Section 2. In Section 4, we introduce a modification to our discrete model to make it more realistic. In Section 5, we establish the number of real equilibria and real periodic solutions one can expect to see in our modified discrete model. In Section 6, we establish local and global attractivity results for our model, including basins of attraction and bifurcation values for the parameters λ and µ. In Section 7, we give a physical interpretation of our mathematical results for our modified discrete suspension bridge model from Section 4. S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 444 2. A Discrete Nonlinear Suspension Bridge Model We start with the McKenna-Moore suspension bridge model (1) from the previous section. Replacing the parameters in (1) by actual numerical values from engineers’ reports of the 1940 Tacoma Narrows Bridge Collapse which were used by McKenna and Moore in their paper [13] and discretizing the first equation in (1) by setting θ ′ := θn+1 − θn, we get the nonlinear nonautonomous difference equation θn+1 = 1.99θn − 0.99θn−1 − 2.4 cosθn−1 sinθn−1 +λ sinµ(n− 1) (2) where λ, µ are positive parameters and −π/2< θn < π/2 for all n ∈ N. Note that the periodic external forcing term λ sinµ(n− 1) depends on the step number n − 1. More precisely, it depends on the position of the suspension bridge and hence on its torsional angle θn−1 at step number n−1. For example, if the periodic external forcing term is due to blowing wind, then the effect of the wind on the bridge would depend on the position of the bridge with respect to the direction of the wind gusts. Incorporating this in (2), we get the updated autonomous difference equation θn+1 = 1.99θn − 0.99θn−1 − 2.4 cosθn−1 sinθn−1 +λ sinµθn−1 =: f (θn−1,θn) (3) Applying the transformation θn+1 := un and un+1 := f (θn,un) to (3) changes it to the nonlinear system of two difference equations ¨ θn+1 = un un+1 = 1.99un − 0.99θn − 2.4 cosθn sinθn +λ sinµθn (4) whose associated map T (θ ,u) is defined as T � θ u � = � u 1.99u− 0.99θ − 2.4 cosθ sinθ +λ sinµθ � (5) Note that the periodic solutions of (3) are precisely the intersection points of the equilibrium curves of the map T2(θ ,u) := (F(θ ,u), G(θ ,u)) which are defined by the equations ¨ F(θ ,u) = θ G(θ ,u) = u (6) In particular, the map T2(θ ,u) for equation (3) has the form T2 � θ u � =   −0.99(−1.0101λ sinµθ + θ + 2.42424 sinθ cosθ − 2.0101u) 2.9701(0.670011λ sinµθ + 0.336689λ sinµu− 0.663311θ− 1.60803 sinθ cosθ + u− 0.808054 sin u cos u)   (7) S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 445 with its associated equilibrium curves given by    E1 : −0.99(−1.0101λ sinµθ + θ + 2.42424 sinθ cosθ − 2.0101u)− θ = 0 E2 : 2.9701(0.670011λ sinµθ + 0.336689λ sinµu− 0.663311θ −1.60803 sinθ cosθ − 0.808054 sin u cos u) = 0 (8) Replacing sinθ , cosθ , sin u and cos u by their Taylor series expansions in (8), one can think of the curves E1 and E2 as polynomials of infinite degrees, that is, of degrees n1 and n2 where n1→∞ and n2→∞. It follows from Bézout’s Theorem [16] that the number of intersections of E1 and E2 is bounded by the product n1n2 →∞. In other words, one can expect to see infinitely many periodic solutions for equation (3). However this is not a realistic scenario from the applications point of view. In Section 4, we will consider a modification of equation (3) with a finite number of periodic solutions. In the next section, we consider a linearization of (3) which will be used to give mathematical explanations for certain numerical observations of McKenna and Moore in [13]. 3. A Linearization of the Discrete Nonlinear Model Recall that for small θ values, cosθ ≈ 1 and sinθ ≈ θ . Putting these changes in equation (3) gives the following linearization of (3) θn+1 = 1.99θn − 0.99θn−1 − 2.4θn−1 +λµθn−1 (9) where λ, µ are positive parameters and −π/2 < θn < π/2 for all n ∈ N. Its associated linearized system has the form ¨ θn+1 = un un+1 = 1.99un − 0.99θn − 2.4θn +λµθn (10) It is straightforward to see that the equilibrium curves (6) of the map T2(θ ,u) of (10) belong to straight lines passing through the origin with explicit formulas ¨ E1 : u= 0.502513(4.39−λµ)t E2 : u= 1.99(3.39−λµ) λµ−0.4299 t Moreover it is easy to see that E1 and E2 intersect at infinitely many points exactly when they have the same slope, that is exactly when λµ = 2.4,6.38. For all other λµ-values, E1 and E2 intersect only at the origin which also happens to be the equilibrium solution of the map T2(θ ,u). It follows that the linearization (9) of equation (3) has infinitely many periodic solutions for λµ= 2.4 and 6.38, and no nontrivial periodic solutions for all other λµ-values. This suggests that linearization is not a very practical way to study oscillations in a sus- pension bridge since the dynamics in this case are too extreme to occur in real life. But it does help to explain mathematically the numerical observations of McKenna and Moore in [13], an S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 446 explanation that was missing from their paper. More specifically, McKenna and Moore noted in their paper that for their linearized differential equation θ ′′ = −2.4θ − 0.01θ ′ +λ sinµt (11) periodic solutions occur for µ ≈ 1.55 and for small λ whose exact range was unspecified in the paper. Moreover for µ > 1.55, they observed no bifurcation from one to many periodic solutions. They also noted that for µ < 1.55 but close to this value, their numerical algorithm did not converge to a periodic solution. They did not address the case where µ < 1.55 and far away from this value at all in their paper [13]. In short, the case µ < 1.55 was pretty much left unanalyzed in their paper. For their numerical experiments, McKenna and Moore used the four sets of λµ-values given below to study the occurrence of periodic solutions in (11). {λ,µ} = {0.0126,1}, {0.0117,1.2}, {0.0088,1.4}, {0.0197,1.5} In the first three cases, they observed bifurcations from single to multiple periodic solutions while in the fourth case, they simply noted the existence of multiple periodic solutions. A quick check shows that in the first three cases, the product λµ is roughly constant with λµ ≈ 0.013 or 0.014 with 3-digit rounding and in the fourth case, λµ≈ 0.03. Drawing analogy to the bifurcation analysis of the discrete linearized difference equations model (9) presented earlier in this section, one can conclude that the McKenna-Moore lin- earized equation (11) has infinitely many periodic solutions for λµ ≈ 0.013 (or 0.014) and 0.03, and no nontrivial periodic solutions for all other λµ-values. This would explain why their numerical algorithm did not converge to a periodic solution in some cases (no nontrivial periodic solutions in theses cases!) while in other cases they observed no bifurcation from one to many periodic solutions (infinitely many periodic solutions in these cases!). To summarize, the authors in [13] missed the fact that the observed bifurcations from one to multiple peri- odic solutions in equation (11) were being caused not separately by the parameters λ or µ but rather by their joint product λµ. Analyzing its associated discrete linearized difference equa- tion (9) as shown above helped to figure this out. In the next section, we look at a modification of equation (3). 4. A Modified Discrete Nonlinear Suspension Bridge Model In this section, we will consider a modified nonlinear discrete difference equations model for a suspension bridge with a finite number of periodic solutions. The goal is to come up with a more realistic model from an applications point of view than the original discrete nonlinear model (3) with the infinite number of periodic solutions which was introduced in Section 2. The strategy will be to replace cosθ and sinθ by their second-order Taylor polynomial approximations in (3) and analyze the resulting model mathematically. The proofs for higher- order Taylor polynomial approximations are essentially the same and will be omitted. One way to decide which order Taylor Polynomial to use is to see which order best ap- proximates an actual data set of torsional angle values for a given suspension bridge taken S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 447 at different time intervals. Replacing cosθ and sinθ by their respective second-order Taylor polynomial approximations 1− θ2 and θ in (3), we get the new equation θn+1 = 1.99θn − 0.99θn−1 − 2.4(1− θ2 n−1)θn−1 +λµθn−1 (12) where λ, µ are positive parameters and −π/2 < θn < π/2 for all n ∈ N. Its associated 2D- system is ¨ θn+1 = un un+1 = 1.99un − 0.99θn − 2.4(1− θ2 n )θn +λµθn (13) with associated map T (θ ,u) defined as T � θ u � = � u 1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ � (14) Setting T (θ ,u) = (θ ,u) and solving for θ in (14) gives the formulas for the three equilibria of (12) as E1 := (0,0) E2 := (0.645497 Æ 2.4−λµ, 0.645497 Æ 2.4−λµ) E3 := (−0.645497 Æ 2.4−λµ,−0.645497 Æ 2.4−λµ) To find the number of real periodic solutions of (12), note that these are precisely the inter- section points of the equilibrium curves of the map T2(θ ,u) whose explicit formulas can be found by solving the equation T2(θ ,u) = (θ ,u) where T2 � θ u � = � 1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ 1.99(1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ ) +λµu− 2.4 � 1− u2 � u− 0.99u � (15) A simple calculation shows that the equilibrium curves of T2(θ ,u) are elliptic curves with formulas    E1 : −0.99(−1.0101λµθ + θ + 2.42424θ (1− θ2)− 2.0101u)− θ = 0 E2 : 2.9701(0.670011λµθ + 0.336689λµu− 0.663311θ − 1.60803θ (1− θ2) −0.808054u(1− u2) = 0 (16) The equilibrium curves for the maps T (θ ,u) and T2(θ ,u) are shown in Figure 1. According to Bézout’s Theorem, the maximum number of intersections of the third-degree elliptic curves E1 and E2 counting complex intersections and multiplicities is 3×3= 9. Thus Bézout’s Theorem gives an upper bound for the number of real periodic solutions to (12), which is what we are interested in for studying real oscillations in a suspension bridge. In the next section, we will apply Descartes’ Rule of Signs for real roots of polynomials (see [16]) to predict the number of real periodic solutions of (12) for different λµ-parameter regions. S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 448 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 (a) -2 -1 0 1 2 -3 -2 -1 0 1 2 3 (b) Figure 1: Equilibrium curves for the maps T (θ ,u) (on the left) and T2(θ ,u) (on the right). The black dots represent equilibria (left) and periodic solutions (right) of equation (12). 5. Number of Real Equilibria and Periodic Solutions In [13], McKenna and Moore used a numerical continuation algorithm to demonstrate the existence of three periodic solutions to their continuous suspension bridge model (1) through a finite number of numerical experiments (five in their paper). However, they did not give a rigorous mathematical proof for the existence of (or lack of) three periodic solutions in all cases not covered by their finite number of simulations. More specifically, they did not address the question of whether to expect more or less than three periodic solutions for some λµ- parameter regions. They also did not distinguish between trivial periodic solutions, namely, equilibrium solutions, and nontrivial periodic solutions for their model (1). In this section, we will rigorously prove that our suspension bridge model (12) can have at most three real equilibria and at most four real nontrivial periodic solutions for 0< λµ≤ 6.38. Moreover, we will show that it is possible to have λµ-parameter regions where at most one or even no real nontrivial periodic solutions exist. The main theorem of this section is as follows. Theorem 1. The number of real equilibria and real nontrivial periodic solutions of equation (12) for various λµ-parameter regions is as shown in Table 1. Table 1: Table showing the number of real equilibria and real nontrivial periodic solutions of equation (12). λµ - region No. of real equilibria No. of real periodic solns. 0< λµ < 2.4 three ≤ four pairs λµ= 2.4 (0,0) unique equilibrium ≤ three pairs 2.4< λµ < 4.39 (0,0) unique equilibrium ≤ three pairs λµ= 4.39 (0,0) unique equilibrium ≤ one pair 4.39< λµ < 6.38 (0,0) unique equilibrium ≤ one pair λµ= 6.38 (0,0) unique equilibrium none λµ > 6.38 (0,0) unique equilibrium none S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 449 Proof. The number of real equilibria in Table 1 follows directly from their formulas given in the previous section. Recall that the periodic solutions of equation (12) are precisely the intersection points of the elliptic curves E1 and E2 which were defined in (16). Solving the equations for E1 and E2 simultaneously for the variable θ gives rise to the ninth-degree poly- nomial equation g(θ ) =− 4.21004θ[θ8 + (−5.4875+ 1.25λµ)θ6 + (10.0376− 4.57292λµ+ 0.520833λ2µ2)θ4 + (−7.3777+ 4.46878λµ− 0.952691λ2µ2 + 0.072338λ3µ3)θ2 + (1.82765− 1.04799λµ+ 0.119361λ2µ2)] =:− 4.21004θh(θ ) (17) where h(θ ) =θ8 + (−5.4875+ 1.25λµ)θ6 + (10.0376− 4.57292λµ+ 0.520833λ2µ2)θ4 + (−7.3777+ 4.46878λµ− 0.952691λ2µ2 + 0.072338λ3µ3)θ2 + (1.82765− 1.04799λµ+ 0.119361λ2µ2) (18) Observe that the coefficients of all terms except the leading θ8-term in h(θ ) are polynomials in the ’new’ variable λµ. Renaming λµ→ s, one can rewrite h(θ ) in (18) as follows. h(θ ) = θ8 + P1(s)θ 6 + P2(s)θ 4 + P3(s)θ 2 + P4(s) (19) where P1(s) := −5.4875+ 1.25s P2(s) := 10.0376− 4.57292s+ 0.520833s2 P3(s) := −7.3777+ 4.46878s− 0.952691s2 + 0.072338s3 P4(s) := 1.82765− 1.04799s+ 0.119361s2 The graphs of P1(s), P2(s), P3(s) and P4(s) are shown in Figure 2. Solving the equations P1(s) = 0, P2(s) = 0 and P3(s) = 0 simultaneously, we see that the common s-intercept of the three polynomial curves P1(s), P2(s) and P3(s) is at s = 4.39, that is, at λµ = 4.39. Moreover, the parabola P2(s) has a real, repeated root at s = 4.39 and is positive everywhere else. The line P1(s) and the cubic P3(s) are clearly positive for s > 4.39 and negative elsewhere. Moreover, the two s-intercepts of the parabola P4(s) are given by s = 2.4 and s = 6.38. In particular, P4(s) is negative between these two s-values and positive everywhere else. These observations are summarized in Table 2 in terms of the coefficients of the polynomial h(θ ) given in (19). S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 450 -5 5 10 -2 2 4 6 8 10 Figure 2: Diagram showing the polynomials P1(s) in black, P2(s) in red, P3(s) in blue and P4(s) in green. Table 2: Table showing the signs of the coefficients of h(θ ) for various λµ-parameter regions. λµ - region θ8 Coeff. θ6 Coeff.: P1(s) θ4 Coeff.: P2(s) θ2 Coeff.: P3(s) Constant: P4(s) 0< λµ < 2.4 + - + - + λµ= 2.4 + - + - 0 2.4< λµ < 4.39 + - + - - λµ= 4.39 + 0 0 0 - 4.39< λµ < 6.38 + + + + - λµ= 6.38 + + + + 0 λµ > 6.38 + + + + + It follows from Table 2 that for 0 < λµ < 2.4, there are four sign changes between the coefficients of h(θ ). Hence by Descartes’ Rule of Signs [16], there exist at most four positive periodic solutions of equation (12). We also saw in Section 4 that (12) has a positive equilib- rium θ = 0.645497 p 2.4−λµ for λµ < 2.4. Since equilibria are trivially periodic solutions, equation (12) has at most 4− 1 = 3 nontrivial positive periodic solutions for λµ < 2.4. Also note that h(−θ ) has the same number of sign changes between its coefficients as h(θ ). Hence one has from Descartes’ Rule of Signs that (12) has at most 3 nontrivial negative periodic so- lutions for λµ < 2.4. Since periodic solutions of discrete 2D-systems occur in pairs, one can conclude that there exist at most three pairs of real nontrivial periodic solutions of equation (12) for 0< λµ < 2.4. Similar arguments can be given for the remaining λµ-parameter cases in Table 2. To summarize, in this section we looked at the exact number of equilibria for equation (12) and computed strict upper bounds for the number of real nontrivial periodic solutions of (12) in the parameter region 0 < λµ < 2.4. In the next section, we will address global attractivity and basins of attraction of these real equilibria and periodic solutions. S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 451 6. Global Attractivity and Basins of Attraction of the Real Equilibria and Periodic Solutions An important question that McKenna and Moore failed to address in [13] was global at- tractivity properties of the real equilibria and periodic solutions of their continuous suspension bridge model (1), including basins of attraction of these solutions. One can expect global at- tractivity properties of real periodic solutions to play a very important role in predicting the stability of a swaying bridge, with an attracting periodic solution implying stable bridge oscil- lations in the long run and a repelling or saddle point periodic solution signifying the onset of unstable or even chaotic bridge oscillations in the long run possibly leading to disastrous consequences such as the 1940 Tacoma Narrows Bridge Collapse in Washington, USA. In this section, we will compute for our discrete suspension bridge model (12) precise λµ-parameter regions where there exist a unique equilibrium, multiple equilibria, a unique real periodic solution and multiple real periodic solutions. We will also compute basins of attraction for these. We start by stating a local stability result for the equilibria E1, E2, and E3 of (12) whose formulas were first introduced in Section 4 and are given below for easy reference. E1 := (0,0) E2 := (0.645497 Æ 2.4−λµ, 0.645497 Æ 2.4−λµ) E3 := (−0.645497 Æ 2.4−λµ,−0.645497 Æ 2.4−λµ) (20) Lemma 1. The equilibria E1, E2, and E3 are unstable for all admissible λµ-parameter regions. Proof. It is easy to check that the eigenvalues λ1 and λ2 (with λ1 ≤ λ2) of the jacobian of the map T (θ ,u) evaluated at the three equilibria are as follows: E1 : λ1 = 1− Æ λµ− 2.4,λ2 = 1+ Æ λµ− 2.4 E2 : λ1 = 1− 1.414 Æ 2.4−λµ,λ2 = 1+ 1.414 Æ 2.4−λµ E3 : λ1 = 1− 1.414 Æ 2.4−λµ,λ2 = 1+ 1.414 Æ 2.4−λµ Note that E2 and E3 have the same set of eigenvalues. Moreover, an easy calculation shows that the eigenvalues of E1 satisfy the following inequalities for the given λµ-parameter regions:    0< λµ < 2.4 : |λ1|> 1, |λ2|> 1 (a) λµ= 2.4 : |λ1|= |λ2|= 1 (b) 2.4< λµ < 6.38 : |λ1|< 1, |λ2|> 1 (c) λµ= 6.38 : |λ1|= 1, |λ2|> 1 (d) λµ > 6.38 : |λ1|> 1, |λ2|> 1 (e) (21) Hence E1 is a repeller in cases (a) and (e), a saddle point equilibrium in case (c) and a nonhy- perbolic equilibrium in cases (b) and (d). Similarly, one can show that the eigenvalues of E2 S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 452 and E3 satisfy    0< λµ < 0.41 : |λ1|> 1, |λ2|> 1 λµ= 0.41 : |λ1|= 1, |λ2|> 1 0.41< λµ < 2.4 : |λ1|< 1, |λ2|> 1 (22) Hence E2 and E3 are both repellers, saddle point equilibria or nonhyperbolic equilibria. The next theorem follows directly from (20), (21) and (22). Theorem 2. The following are true for the equilibria E1, E2 and E3 of equation (12): 1. If 0 < λµ < 2.4, then all three equilibria E1, E2 and E3 exist. The zero equilibrium E1 is a repelling equilibrium. The nonzero equilibria E2 and E3 satisfy: i. If 0< λµ < 0.41, then they are both repelling equilibria. ii. If 0.41< λµ < 2.4, then they are both saddle point equilibria. iii. If λµ= 0.41, then they are both nonhyperbolic equilibria. 2. If 2.4≤ λµ < 6.38, then E1 is a unique saddle point equilibrium. 3. If λµ≥ 6.38, then E1 is a repelling equilibrium. Note that the south-east partial ordering “�se” and the north-east partial ordering “�ne” from the theory of cooperative and competitive maps (see see [5, 11, 12, 15]) are defined as follows: (x1, y1)�se (x2, y2) if and only if x1 ≤ x2 and y1 ≥ y2 (x1, y1)�ne (x2, y2) if and only if x1 ≤ x2 and y1 ≤ y2 (23) It is a well-known fact that cooperative maps preserve the “�ne” ordering (see [5, 11, 12, 15]). Our next lemma gives precise parameter and initial value conditions for T2(θ ,u) to be cooperative. It will play a key role in the proof of our main theorem for this section. Lemma 2. The map T2(θ ,u) is cooperative exactly when one of the following holds: i. 3.39≤ λµ≤ 6.38 ii. 0≤ λµ < 3.39 and θ 6∈ (θ1,θ2), where θ1 and θ2 are given by the formulas θ1 := −0.372678 p 3.39−λµ and θ2 := 0.372678 p 3.39−λµ Proof. Note that the Jacobian of the map T2(θ ,u) is given by JacT2 = � 7.2θ2 +λµ− 3.39 1.99 1.99(7.2θ2 +λµ− 3.39) λµ+ 7.2u2 + 0.5701 � (24) If 3.39 ≤ λµ ≤ 6.38, then all entries of the jacobian are positive which implies that the asso- ciated map T2(θ ,u) is cooperative (see [11, 12, 15]). If 0< λµ < 3.39, then the 1,1-entry of the jacobian in (24), f (θ ) := 7.2θ2 +λµ− 3.39 S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 453 is a parabola with positive leading coefficient whose θ -axis intercepts are given by the formulas θ1,θ2 = ±0.372678 Æ 3.39−λµ, θ1 < θ2 (25) Clearly f (θ ) ≥ 0 for θ ∈ (−∞,θ1] ⋃ [θ2,∞). Hence the map T2(θ ,u) is cooperative in this θ -range. The main theorem of this paper is as follows: Theorem 3. Under the hypotheses of Lemma 2, equation (12) has exactly two real nontrivial periodic solutions {{p1,q1}, {p2,q2}} and {{s1, t1}, {s2, t2}} where {p1,q1} �se (0,0)�se {p2,q2} and {s1, t1} �ne (0,0)�ne {s2, t2} {{p1,q1}, {p2,q2}} is a repelling periodic solution and {{s1, t1}, {s2, t2}} is locally asymptotically stable. The global dynamics of equation (12) is as follows: 1. If 0< λµ < 2.4, then both pairs of periodic solutions are present. Moreover, i. If 0< λµ < 0.41, then every solution converges to the stable periodic solution {{s1, t1}, {s2, t2}}. ii. If 0.41≤ λµ < 2.4, then every solution converges to the stable periodic solution {{s1, t1}, {s2, t2}} or to the saddle point equilibria E2 or E3. 2. If 2.4 ≤ λµ < 6.38, then the repelling periodic solution {{p1,q1}, {p2,q2}} is the only one present. Every solution converges to the zero saddle point equilibrium E1. 3. If λµ ≥ 6.38, then equation (12) does not possess any periodic solutions. The zero equilibrium E1 is a repelling equilibrium. Every solution escapes to infinity. The proof of Theorem 3 follows from the statements of Theorem 2, Lemmas 3-5, Propo- sitions 1-2, Corollaries 1-2 and the fact that all orbits of a bounded cooperative map must converge to a locally asymptotically stable equilibrium or to a saddle point equilibrium (see for example [5, 11, 12, 15]). The next lemma gives bounds for the coordinates of the map T2(θ ,u). It will be crucial for the rest of the proofs in this section. Lemma 3. Under the assumptions −π/2 ≤ θ ≤ π/2 and 0 < λµ ≤ 6.38, there exist two real numbers m1 and m2 for which the map T2(θ ,u) = (F(θ ,u), G(θ ,u)) satisfies −m1 ≤ F(θ ,u) ≤ m1 and −m2 ≤ G(θ ,u)≤ m2 Proof. Note that under the given assumptions, the first coordinate of the map T2 � θ u � = � 1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ 1.99(1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ ) +λµu− 2.4 � 1− u2 � u− 0.99u � S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 454 which was defined in (15) has bounds given by −17.7528≤ 1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ ≤ 17.7528 (26) and its second coordinate has bounds given by −9.09491≤1.99(1.99u− 0.99θ − 2.4(1− θ2)θ +λµθ ) +λµu− 2.4 � 1− u2 � u −0.99u≤9.09491 (27) Choose m1 = 17.7528 and m2 = 9.09491 in the statement of the lemma. Define the four regions R1, R2, R3 and R4 as follows. R1 := [0, m1]× [0, m2] R2 := [−m1, 0]× [0, m2] R3 := [−m1, 0]× [−m2, 0] R4 := [0, m1]× [−m2, 0] (28) The following lemma establishes an invariant attracting set for the map T2(θ ,u) when the latter is cooperative. It will play a key role in establishing global attractivity results for equation (12) later on in this section. Lemma 4. Under the hypotheses of Lemma 2, the regions R1, R2, R3 and R4 satisfy: T2(R1) ⊆R1, T (R2) ⊆R1, T2(R3) ⊆R3, T (R4) ⊆R3 In particular, R1 ∪R3 is an invariant attracting set for the map T2(θ ,u). Proof. To see that T2(R1) ⊆R1, observe that all (θ ,u) ∈ R1 satisfy (0,0)�ne (θ ,u)�ne (m1, m2) Under the hypotheses of Lemma 2, the map T2(θ ,u) is cooperative. Since it also has (0,0) as a fixed point, it follows from the “�ne” order-preserving property of cooperative maps (see [12, 15]) that (0,0) = T2(0,0)�ne T2(θ ,u)�ne T2(m1, m2)�ne (m1, m2) which is what we wanted to prove. The proof of T2(R3) ⊆R3 is similar and we skip it. To see that T (R2) ⊆R1, note that the equilibrium curves θ = f (θ ,u) and u= g(θ ,u) of the map T (θ ,u) = ( f (θ ,u), g(θ ,u)) defined in (14) are both increasing curves passing through the origin. In particular, the region R2 lies entirely above them. Moreover, all (θ ,u) ∈ R2 satisfy (θ ,u) �ne T (θ ,u). For example, one can check that (0,u) �ne T (0,u) for u > 0. It is also easy to check that T (θ ,u) is a cooperative map by looking at its jacobian. Hence for all (θ ,u) ∈ R2, one has (θ ,u)�ne T (θ ,u) �ne T2(θ ,u)�ne . . . S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 455 The monotone increasing sequence given above must enter the region R1 since otherwise it would have to converge to a fixed point of T (θ ,u) in R2, contradicting the fact that the equilibrium curves of T (θ ,u) do not lie in this region by our previous discussion. The proof of T (R4) ⊆R3 is similar and we skip it. Our next lemma establishes the fact that any region bounded by two fixed points of a cooperative map in the “�ne” ordering must contain a locally asymptotically stable fixed point of this map. Lemma 5. Suppose eT : R→ R is a cooperative map. In addition, suppose there exist two points (x∗1, y∗1) and (x∗2, y∗2) with (x∗1, y∗1)�ne (x ∗ 2, y∗2) in R×R such that eT (x∗1, y∗1) = (x ∗ 1, y∗1) and eT (x∗2, y∗2) = (x ∗ 2, y∗2) Then there exists a point (p∗,q∗) ∈ [x∗1, x∗2]× [y ∗ 1 , y∗2] such that eT (p∗,q∗) = (p∗,q∗). Moreover, (p∗,q∗) must be a locally asymptotically stable fixed point of the map eT. Proof. It is easy to see that [x∗1, x∗2]× [y ∗ 1 , y∗2] is an invariant region for eT since (x∗1, y∗1) = eT (x∗1, y∗1)�ne (x , y)�ne eT (x , y)�ne eT (x∗2, y∗2) = (x ∗ 2, y∗2) for all (x , y) ∈ [x∗1, x∗2]× [y ∗ 1 , [y∗2]. The proof follows from this and the theory of cooperative maps. The next two propositions give existence and uniqueness conditions for the real periodic solutions of equation (12). They complete the proof of Theorem 3 and are as follows. Proposition 1. Suppose the hypotheses of Lemma 2 hold. If 0 < λµ < 6.38, then there always exists a unique periodic solution {{p1,q1}, {p2,q2}} of equation (12) in the bounded regionR2∪ R4 defined by (28). This periodic solution is repelling in nature and disappears for λµ≥ 6.38. Proof. Observe that the θ -intercepts of the equilibrium curves E1 and E2 of the map T2(θ ,u) defined in (16) are respectively given by E1 : θ = 0, ±0.645497 Æ 4.39−λµ E2 : θ = 0, ±0.645497 Æ 3.39−λµ (29) It follows that if 0 < λµ < 3.39, then both E1 and E2 have a positive θ -axis intercept and a negative θ -axis intercept in addition to the (0,0)-intercept. If 3.39≤ λµ < 4.39, then only E1 has the two nonzero θ -axis intercepts. And if 4.39≤ λµ < 6.38, then both E1 and E2 have only the (0,0)-intercept. Moreover, it is straightforward to check that both E1 and E2 have negative slopes inR2∪R4 which satisfy Slope(E1) < Slope (E2) at all three θ -intercepts. Hence E1 and E2 must intersect transversally inR2∪R4 whenever 0< λµ < 6.38. If λµ≥ 6.38, then this is no longer true since the slopes of E1 and E2 now satisfy Slope(E1) = Slope (E2) at the origin. The various scenarios are shown in Figure 3. The repelling nature of {{p1,q1}, {p2,q2}} is a direct consequence of the fact that T (R2) ⊆R1 and T (R4) ⊆R3 by Lemma 4. S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 456 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 (a) 0< λµ < 3.39 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 (b) 3.39≤ λµ < 4.39 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 (c) 4.39≤ λµ < 6.38 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 (d) λµ≥ 6.38 Figure 3: Diagram showing the nontrivial periodic solution in R2 ∪R4 along with the (0,0) equilibrium. Proposition 2. Suppose the hypotheses of Lemma 2 hold. If 0 < λµ < 2.4, then there always exists a unique periodic solution {{s1, t1}, {s2, t2}} of equation (12) in the bounded regionR1∪R3 defined by (28). This periodic solution is locally asymptotically stable and disappears for λµ≥ 2.4. Proof. We prove uniqueness of the locally asymptotically stable periodic solution {{s1, t1}, {s2, t2}} by contradiction. Suppose there exists a second periodic solution, say {{v1, w1}, {v2, w2}}, of equation (2) in R1 ∪R3. Then it must satisfy {v1, w1} �ne {s1, t1} �ne (0,0)�ne {s2, t2} �ne {v2, w2} (30) Otherwise if {v1, w1} �se {s1, t1}, then the fact that {v1, w1} �ne (0,0), an unstable fixed point of T2(θ ,u) by Lemma 1, along with Lemma 5 would imply that {v1, w1} is locally asymptotically stable. This would make {v1, w1} and {s1, t1} locally asymptotically stable neighbors in the “�se” ordering, contradicting a theorem by Dancer and Hess in [5] which says that stable and unstable fixed points of an order-preserving map must alternate. Hence {{v1, w1}, {v2, w2}}must be an unstable periodic solution of (28) satisfying the “�ne” ordering S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 457 in (30). However in this case, the invariant region defined by [−m1, v1]× [−m2, w1] cannot contain a locally asymptotically stable fixed point of the cooperative map T2, contradicting Lemma 5. There also cannot exist an additional stable periodic solution to get around this problem because if it did, then we would have two unstable periodic solutions including the repelling equilibrium (0,0) alternating with two stable periodic solutions including {{s1, t1}, {s2, t2}} in the �ne ordering of periodic solutions. However, this would leave no room to fit the two nonzero unstable equilibria E2 and E3, which also exist in this case by Theorem 2 part 1, in this ordering without having two unstable equilibria as neighbors, thus violating Dancer and Hess’ result in [5] once again. The fact that the unique periodic solution {{s1, t1}, {s2, t2}} must be locally asymptotically stable follows directly from the theory of cooperative maps which guarantees the existence of a stable fixed point of the cooperative map T2(θ ,u) in the invariant attracting regionR1∪R3. To see that the periodic solution disappears for λµ ≥ 2.4, one just needs to observe that in this case, the critical points and θ -intercepts of the equilibrium curve E2 are smaller in magnitude than those of the equilibrium curve E1. Since the critical points of both curves lie in R1 ∪R3, it follows that they cannot intersect here. The next corollary is a direct consequence of Theorem 3. Corollary 1. If the hypotheses of Lemma 2 are not satisfied, i.e., if the map T2(θ ,u) is not cooperative or is unbounded, then the solutions of equation (12) may exhibit unpredictable or even chaotic behavior. The next corollary addresses bifurcation values of the parameters λ and µ. Corollary 2. At λµ = 2.4 and 6.38, the point (0,0) is a unique nonhyperbolic equilibrium and undergoes a Neimark-Sacker bifurcation. Moreover if λµ > 6.38, then every nonzero solution escapes to infinity. Proof. The proof of the first statement is a direct consequence of Table 1 and (21)-(22) in the proof of Lemma 1. In particular, note that at λµ = 6.38, the (0,0) equilibrium changes from a repeller to a saddle point and a new pair of periodic solutions is created. Similarly at λµ = 2.4, (0,0) changes from a saddle point to a repeller and a new pair of of periodic solutions is created. Thus the nonhyperbolic equilibrium (0,0) undergoes a Neimark-Sacker bifurcation at both these λµ-values. It is easy to see that the map T2(θ ,u) in (15) is unbounded for λµ > 6.38. This and the fact that T2(θ ,u) is cooperative guarantee unbounded growth for the orbits of T2(θ ,u). As a result, every nonzero solution escapes to infinity. In the next section, we give a summary of our mathematical results in this paper and their physical interpretation for our discrete suspension bridge model (12). 7. Conclusion and Model Interpretation In this paper, we mathematically analyzed a discrete difference equations version of McKenna and Moore’s continuous nonlinear differential equations model for a suspension bridge in [13]. S. Basu / Eur. J. Pure Appl. Math, 7 (2014), 442-461 458 More specifically, we computed the exact number of real equilibria and real nontrivial peri- odic solutions for our discrete suspension bridge model under some very general hypotheses. We also gave precise parameter regions for our model to have (a) unique and multiple equi- libria, and (b) unique and multiple real periodic solutions. In addition, we established the different types of local and global attractivity behaviors that these equilibria and real periodic solutions exhibit along with the precise parameter regions where they exhibit these behaviors. Finally, we used a linearization of our discrete model to gain valuable insights that helped us offer missing mathematical explanations for phenomena that McKenna and Moore observed for their linearized continuous model in [13] via numerical simulations. Our results are sum- marized in Table 3. The physical interpretation of our mathematical results for the discrete suspension bridge model (12) is that the suspension bridge will eventually stop oscillating and come to rest at its original position for all initial conditions θ ∈ � −π2 , π2 � if 2.4< λµ≤ 6.38. It will either come to a complete rest at one of two nonzero equilibrium positions or its oscillations will eventually settle down to one of two stable periodic motions for all initial conditions θ ∈ � −π2 , π2 � if 0.41 < λµ ≤ 2.4. Its oscillations will eventually settle down to one of two stable periodic motions for all initial conditions θ ∈ � −π2 , π2 � if 0 ≤ λµ ≤ 0.41. The model breaks down for λµ > 6.38. ACKNOWLEDGEMENTS The author is deeply indebted to Dr. Edward F. Aboufadel from the Math Department of Grand Valley State University in Allendale, MI, USA, for his constant en- couragement, support and mentoring over the years. She would also like to thank Dr. Orlando Merino from the Math Department of the University of Rhode Island in Kingston, RI, USA, for making her the mathematician that she is today. S . B a su / E u r. J. P u re A p p l. M a th , 7 (2 0 1 4 ), 4 4 2 -4 6 1 4 5 9 Table 3: Table summarizing our results in this paper. λµ - parameter region No. of equilibria Local attractivity No. of periodic solns Local attractivity Global dynamics 0< λµ < 0.41 3 3 repelling equilibria 2 At least one is locally asymptotically stable Every Solution ↓ λµ= 0.414 3 (0,0) a repeller, 2 nonhyperbolic equilibria 2 At least one is locally asymptotically stable converges to ↓ 0.41< λµ < 2.4 3 (0,0) a repeller, 2 saddle point equilibria 2 At least one is locally asymptotically stable an equilibrium or a periodic soln. ↓ λµ= 2.4 1 (0,0) nonhyperbolic, Neimark-Sacker bifurcation occurs 1 It is a repelling periodic solution for θ -values not in ↓ 2.4< λµ≤ 3.39 1 The (0,0) equil. is a saddle point 1 It is a repelling periodic solution (θ1,θ2) 3.39< λµ < 4.39 1 The (0,0) equil. is a saddle point 1 It is a repelling periodic solution Every soln. converges to (0,0) λµ= 4.39 1 The (0,0) equil. is a saddle point 1 It is a repelling periodic solution Every soln. converges to (0,0) 4.39≤ λµ < 6.38 1 The (0,0) equil. is a saddle point 1 It is a repelling periodic solution Every soln. converges to (0,0) λµ= 6.38 1 (0,0) nonhyperbolic, Neimark-Sacker bifurcation occurs 0 −−− Every soln. converges to (0,0) λµ > 6.38 1 (0,0) is a repelling equilibrium 0 −−− Every soln. escapes to∞ REFERENCES 460 References [1] S. 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