Abstract
By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: , where , and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.
1. Introduction and Preliminaries
The Gumowski-Mira equation [1] is given by where is one of the functions and the parameters and are positive. These equations were considered by Gumowski and Mira in a series of papers and the book [1]. System (1) implies and so satisfies the difference equation In this paper we will consider (4) with , where and the initial conditions are real numbers; that is, we consider where and the initial conditions and are real numbers.
Several authors have studied the Gumowski-Mira equation (5) and have obtained some results on the stability of the equilibrium points, the bifurcation of the global behavior of solutions, periodic and chaotic solutions, and so forth; see [1–11]. By using the methods of algebraic and projective geometry in [2, 3, 5, 6] the authors obtained very precise description of complicated global behavior which includes finding the feasible periods of all periodic solutions, proving the existence of chaotic solutions through conjugation of maps, and so forth. These methods were first used by Zeeman [12] for the study of so-called Lyness’ equation. All these methods are based on the careful algebraic study of an invariant that (5) possesses. Our method here is purely analytic and is based on the asymptotic expansions. This method can be used to obtain similar results about periodic and chaotic solutions.
Our method is based on the application of KAM theory (Kolmogorov-Arnold-Moser), which brings the considered equation to certain normal form which, in addition to investigation of stability of an equilibrium, can be used to find different periodic solutions, chaotic solutions, and so forth. This technique was used successfully in [13–16] in the case of difference equations while there exists vast literature in the case of differential equations; see [17, 18]. Computer simulations of the trajectories of (5) indicate the existence of an infinite nested family of invariant closed curves surrounding an elliptic fixed point, sequences of periodic islands in the regions between the invariant curves, and stochastic regions surrounding the periodic islands and between invariant closed curves. Furthermore, the entire structure seems to exhibit self-similar structure.
The main feature of (5) is that the corresponding map is the area-preserving map of the plane having a nondegenerate elliptic fixed point. We show that the complicated orbit structure near the elliptic fixed point is an immediate consequence of classical results from the KAM theory. Away from the elliptic fixed point, the KAM theory does not apply and one has to study the geometric structures through some other analytical, algebraic, or geometric methods, such as those in [2, 3, 5, 6, 11, 12]. Most important geometric structures of interest for (5) are periodic points, which are typically of hyperbolic or elliptic type; the invariant manifolds associated with hyperbolic periodic points; KAM invariant curves (around the elliptic fixed point or around elliptic periodic orbits); and cantori, which are remnant sets of Cantor type of destroyed invariant circles. The building blocks for these structures are the periodic orbits, as the other geometric objects can be obtained as limits of periodic orbits.
First, we present the basic results that will be used in the sequel. See [10, 13, 17–19].
Theorem 1 (Birkhoff normal form). Let be an area-preserving map (times continuously differentiable) with a fixed point at the origin whose complex-conjugate eigenvalues and are on the unit disk (elliptic fixed point).
Suppose there exists an integer such that
and suppose that the eigenvalues satisfy
Let be the integer part of .
Then there exists a smooth function that vanishes with its derivatives up to order at , and there exists a real polynomial
such that the map can be reduced to the normal form by suitable change of complex coordinates
In other words the corresponding system of difference equations
can be reduced to the form
where
Here denotes a convergent power series in and with terms of order greater than or equal to which vanishes at the origin and denotes the least integer greater than or equal to .
The numbers are called twist coefficients. Using Theorem 1 we can state the main stability result for an elliptic fixed point, known as the KAM theorem (or Kolmogorov-Arnold-Moser theorem); see [10, 18, 19].
Theorem 2 (KAM theorem). Let be an area-preserving map with an elliptic fixed point at the origin satisfying the conditions of Theorem 1. If the polynomial is not identically zero, then the origin is a stable equilibrium point. In other words if for some one has in (12), then the origin is a stable equilibrium point.
Remark 3. Consider an invariant annulus in a neighbourhood of the elliptic fixed point, for a sufficiently small positive number. Note that the linear part of normal form approximation leaves all circles invariant. The motion restricted to each of these circles is a rotation by some angle. Also note that if at least one of the twist coefficients is nonzero, the angle of rotation will vary from circle to circle. A radial line through the fixed point will undergo twisting under the mapping. The KAM theorem says that, under the addition of the remainder term, most of these invariant circles will survive as invariant closed curves under the full map [17, 18]. Precisely, the following result holds see [13, 18].
Theorem 4. Assuming that is not identically zero and is sufficiently small, then the map has a set of invariant closed curves of positive Lebesgue measure close to the original invariant circles. Moreover, the relative measure of the set of surviving invariant curves approaches full measure as approaches . The surviving invariant closed curves are filled with dense irrational orbits.
The following is a consequence of Moser’s twist map theorem [18, 19].
Theorem 5. Let be an area-preserving diffeomorphism and a nondegenerate elliptic fixed point. There exist periodic points with arbitrarily large period in every neighbourhood of .
Indeed the theorem implies that arbitrarily close to the fixed point there are always infinitely many gaps between consecutive invariant curves that contain periodic points. Within these gaps, one finds, in general, orbits of hyperbolic and elliptic periodic points. These facts can hardly be seen from computer simulations since some periodic orbits can exist on very small scales.
The linearized part of (11) represents a rotation for angle and so if is rational multiple of every solution is periodic with the same period while if is irrational multiple of there will exist chaotic solutions. In this paper we will not go into detailed study of these behaviors.
2. Equilibrium Solutions and Linearized Stability Analysis
Equation (5) has at most three equilibrium points: for all values of parameter and when . The linearized equation which corresponds to (5) at any equilibrium point is and its characteristic equation is which shows that the corresponding map is area preserving.
The characteristic equation at is with characteristic roots . When characteristic roots are positive and , which shows that the zero equilibrium in the case is a saddle point. In the case characteristic roots are complex conjugate numbers lying on the unit circle, which means that the zero equilibrium is nonhyperbolic of the elliptic type; see [10, 19]. If then the characteristic roots are both equal to and the zero equilibrium is nonhyperbolic of the parabolic type. Similarly, if then the characteristic roots are both equal to and the zero equilibrium is nonhyperbolic of the parabolic type. In this paper we consider the case when .
If then and we have , , which shows that the zero equilibrium is a saddle point.
The characteristic equation at is with characteristic roots . When characteristic roots are complex conjugate numbers lying on the unit circle, which means that the non-zero equilibrium solutions are nonhyperbolic of the elliptic type see [10, 19].
3. KAM Theory Applied to (5) at Zero Equilibrium for
First, we use the substitution to transform (5) to the system The corresponding linearized system at is The characteristic equation of (19) is (15) with characteristic roots . We consider the case where in which case is nonhyperbolic equilibrium of elliptic type.
A straightforward calculation gives the following expressions for second, third, and fourth power of the characteristic root: Clearly, and for . Thus the assumptions of Theorem 1 are satisfied for and we will find the Birkhoff normal form of (18) by using the sequence of transformations described in Section 1.
3.1. First Transformation
Notice that the matrix of the linearized system (19) at the origin is given as
A straightforward calculation shows that the matrix of the corresponding eigenvectors which correspond to and of is
In order to obtain the Birkhoff normal form of system (18) we will expand the right-hand sides of the equations of system (18) at the equilibrium point up to the order . We obtain Now the change of variables or transforms system (23) into which after some simplifications becomes where
3.2. Second Transformation
The objective of second transformation is to obtain the nonlinear terms up to order in normal form. The change of variables where for and , yields Solving these equations for and one obtain By using (32) and in first equality of (31) and rescaling by replacing with , we have By using (33) in the left-hand side and (31) in the right-hand side of (27), we obtains The last relation holds if the corresponding coefficients are equal, which leads to the following set of equalities: Now This implies
3.3. Third Transformation
The objective of the third transformation consists in expressing the terms in (32) as real values. This is achieved by using the transformation Comparing the system obtained with (11) and using (12) for , we determine the twist coefficients and .
We have In order to apply the KAM theorem we have to show that . Indeed Since , this implies (Figure 1).
(a)
(b)
Thus we have proved the following result.
Theorem 6. The zero equilibrium solution of (5) is stable for .
4. KAM Theory Applied to (5) at Nonzero Equilibrium Solutions for
In this section we will apply KAM theory to establish the stability of the non-zero equilibrium solutions for (5). First, we rescale (5) and then we use the substitution to obtain the system with the corresponding equilibrium point at the origin.
The linearized system of system (42) at is whose characteristic equation is (16) and the characteristic roots are . As we mentioned earlier, when the characteristic roots are complex conjugate numbers lying on the unit circle, which means that the non-zero equilibrium is nonhyperbolic of the elliptic type and so KAM theory is a natural tool to be applied.
A straightforward calculation gives the following expressions for second, third, and fourth power of the characteristic root: It can be shown that and for and so .
4.1. First Transformation
Notice that the matrix of the linearized system (43) is given as A straightforward calculation shows that the matrix of the corresponding eigenvectors which correspond to and of is In order to obtain the Birkhoff normal form of system (42) we will expand the right hand sides of the equations of system (42) at the equilibrium point up to the order . We obtain Next we use the change of variables or and after tedious simplification we obtain the transformed system up to the terms of order three in the form where
4.2. Second Transformation
Similarly, as in the case of the zero equilibrium, we obtain Now we simplify the right-hand sides of these expressions because the coefficient plays the crucial role in determining stability of the equilibrium. We have The real part of last expression is
4.3. Third Transformation
Using (39) we have So, we have that for all .
The computation for the other non-zero equilibrium solution is similar.
Thus we have proved the following result.
Theorem 7. The equilibrium solution of (5) is stable for .
Remark 8. The cases and can be treated by the method of local Lyapunov function as in [6, 9, 10, 20]. To do so one needs the corresponding invariant which assumes a minimum value at the isolated equilibrium point , and by Morse’s lemma, see [19, 20], the level sets are diffeomorphic to the circles in the neighborhood of . This method can be extended further to give some global results on the dynamics of (5) as it was done in [6, 21] (Figures 2 and 3).
(a)
(b)
Remark 9. Theorems 6 and 7 show that (5) undergoes a bifurcation as the parameter passes through 1. Precisely, as the parameter passes through the zero equilibrium changes its local character from a non-hyperbolic equilibrium point of elliptic type, when , to a saddle point, when , where, at the critical value , zero equilibrium is a non-hyperbolic equilibrium point of parabolic type. The positive equilibrium solutions are always the non-hyperbolic equilibrium points of elliptic type. The global change of behavior is that zero equilibrium loses its stability as the parameter passes through and its stability is picked up by the positive equilibrium. So this bifurcation can be described as the exchange of stability bifurcation. The remaining case is dynamics at in which case the zero equilibrium, which is unique, is a non-hyperbolic equilibrium point of parabolic type. Finally, if (5) reduces to which has the unique zero equilibrium and all solutions are periodic of period two in which case we have some trivial stability of equilibrium.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.