Asymptotic stability and bifurcations of a perturbed McMillan map

This paper presents various bifurcations of the McMillan map under perturbations of its coeﬃcients, such as period-doubling, pitchfork, and hysteresis bifurcation. The associated existence regions are located. Using the quasi-Lyapunov function method, the existence of asymptotically stable ﬁxed point is also demonstrated.


Introduction
McMillan first introduced a symplectic map to describe the motion of a particle through a periodically repeated focusing system containing lumped nonlinear impulses [1].McMillan map is not only a model of accelerator lattice, but is at the core of general symplectic dynamics of the plane.Zolkin et al. considered the McMillan sextupole and octupole integrable mappings and provided complete description of all stable trajectories including parametrization of invariant curves [2]; the second one is sometimes referred to as canonical McMillan map.Both of them are the natural extensions of the optical function formalism used in accelerator physics.Gubser et al. obtained new analytic solutions describing motions of closed segmented strings in AdS 3 in terms of elliptic functions, which exactly solve instances of the McMillan map [3].Danilov et al. deduced a generalization of the McMillan map to N -body nonlinear integrable system, which can be realized in particle accelerators [4].The McMillan map exhibits plentiful dynamical behaviors such as bifurcations, asymptotic stability, and various kinds of chaotic attractors depending on the choice of its coefficients [5].The standard form of the McMillan map is a rational integrable mapping of the form H(x, y) = (y, -x + α + βy 1y 2 ), where α and β are constants.The map possesses the following biquadratic integral: x 2 y 2 -(x 2 + y 2 ) + βxy + α(x + y) = K.
Here, K is a parameter that indicates each invariant curve in a two-dimensional phase plane.
The McMillan map is also known as the autonomous discrete Painlevé II equation.Discrete versions of the Painlevé equations occur frequently in problems in mathematical physics.For example, the discrete Painlevé I equation has been obtained in the theory of orthogonal polynomials [6].The Bäcklund transformation of the continuous Painlevé IV equation led to corresponding discrete Painlevé IV equation [7].Similarity reduction [8] of the modified KdV equation led to the discrete Painlevé II equation given by This map also has been found from unitary matrix models of quantum gravity [9].When ζ n+1 is a constant and not a function of variable n, (1) is referred to as the autonomous discrete Painlevé II equation.It admits a two-parameter family of finite-order meromorphic solutions, which is an indicator of integrability in difference equations [10].In this paper, we consider the following second order difference equation where - Here, we investigate the asymptotic stability of equilibrium of the perturbed McMillan map (2) by the quasi-Lyapunov function method (see [11]).The bifurcation and asymptotic stability of equilibrium of difference equations are considered in numerous papers (see [11][12][13][14] and the references cited therein).For example, Merino proved the global attractivity of a difference equation via Lyapunov function method (see [14]).

Lyapunov function, equilibria, and 2-cycle
Many researchers [2][3][4] studied various properties of solutions of rational integrable maps using the parametrization of the invariant curves.Similarly, to investigate the asymptotic stability for the mapping F of (3), we introduce a function V as follows: where -1 < γ < 1 or -3 < γ < -1, α ≥ 0, and β is a real number.It is obvious that V (x, y) = K is the biquadratic integral of the mapping H for γ = -1.Now we present some propositions of the function V (x, y) in a two-dimensional phase plane.
Note that γ = -1, then a direct computation shows the following relation: We denote by T = {(x, y)|(1γ )x(1y 2 ) = α + βy, (x 2 -1)(y 2 -1) = 0}, then the equalities V (x, y) = V (F(x, y)) and F(x, y) = (y, x) hold for (x, y) ∈ T .Now we consider the set T .For each point (x, y) ∈ T , we have To obtain the fixed points and 2-cycle of the mapping F of (3), we consider the following set of equations: ( For simplicity of presentation, we use G to denote the set of solutions of (5 Proposition 1 Let (x, y) be a point of the set G \ L. Then this point (x, y) is a fixed point or 2-cycle of the mapping F of (3).
That is, this point (x, y) is a fixed point of the mapping F. Now we consider each (x, y) ∈ G \ L with x = y.It follows that F(x, y) = (y, x) = (x, y) and It turns out that this point (x, y) is a 2-cycle of F. This completes the proof.
Next we find the elements of the set G \ L and their existence regions (see Fig. 1).
then there exist a 2-cycle and an equilibrium point of the mapping F, where the fixed point lies on : , and α = 2(1γ ), then there exist a 2-cycle and two equilibrium points of the mapping F, where the fixed points lie on .(iv) 2 , and β = ±α, then there exist a 2-cycle and three equilibrium points of the mapping F, where the fixed points lie on .
4 (1γ ), then there exist two equilibrium points of the mapping F, which lie on .
, then there exist three fixed points of the mapping F, which lie on .
Bifurcations of fixed points and 2-cycle are summarized in Fig. 1.The seven associated existence regions are also marked on the parameter plane.Theorem 2 shows that a pitchfork bifurcation occurs when β 1-γ passes through the point 1 along the line α 1-γ = 0.When β 1-γ increasingly crosses L 2 , a 2-cycle of the mapping F appears.That is, L 2 is a perioddoubling bifurcation curve.A complete hysteresis bifurcation arises as the parameter β 4 in the range from -2 to 1 2 .For parameters γ = -1, α = 0, and -2 ≤ β ≤ 2, the corresponding constant level sets V (x, y) = K of the mapping F are provided in Fig. 5 by Zolkin et al. in [2].