Existence of homoclinic connections in continuous piecewise linear systems.

Numerical methods are often used to put in evidence the existence of global connections in differential systems. The principal reason is that the corresponding analytical proofs are usually very complicated. In this work we give an analytical proof of the existence of a pair of homoclinic connections in a continuous piecewise linear system, which can be considered to be a version of the widely studied Michelson system. Although the computations developed in this proof are specific to the system, the techniques can be extended to other piecewise linear systems.

computations developed in this proof are specific to the system, the techniques can be extended to other piecewise linear systems.

Introduction
Recently, in work [Carmona et al., 2008], the proof of the existence of a reversible T-point heteroclinic cycle has been given in a continuous piecewise linear system. The used methods are based on the explicit integration of the flow in each linear region of the space of variables and the construction of a system of equations and inequalities that have to be fullfilled by such kind of global bifurcation.
The system studied in work [Carmona et al., 2008], where c > 0, can be considered as a continuous piecewise linear version of the well known Michelson system [Freire et al., 2002;Kuramoto et al., 1976;Michelson, 1986;Webster et al., 2003]. In fact, the equations of (1.1) can be obtained from the Michelson system performing a simple linear change of variables followed by the change of function x 2 → |x|. Moreover, both systems are volume-preserving and time-reversible with respect to the involution R(x, y, z) = (−x, y, −z). Some other dynamical aspects of the Michelson system also remain in its piecewise linear version [Carmona et al., 2008].
System (1.1) is formed by two linear systems separated by the plane {x = 0}, called separation plane, and it can be written in a matricial form aṡ In the half-space {x < 0}, the system has exactly one equilibrium point p − = (−1/c, 0, 0) T which is a saddle-focus point. Let λ > 0 and α ± i β be the eigenvalues of the Jacobian matrix at p − . This clearly implies that By the reversibility with respect to R, there exists exactly one saddle-focus equilibrium p + = (1/c, 0, 0) T in the half-space {x > 0} whose eigenvalues are given by −λ and −α ± i β.
Using the expression of the parameter c given in (1.3), system (1.1) can be written as (1.4) and the parameter λ > 0 can be choosen as the fundamental parameter of the family.
Homoclinic connections are orbits that are biasymptotic, for t → ±∞, to the same equilibrium point. The existence of a homoclinic connection to a saddle-focus equilibrium point usually forces a complex dynamical behaviour in a neigbourhood of such connection, see [Gonchenko et al., 1997]. For instance, the celebrate works of Shil'nikov [Shil'nikov, 1965;Shil'nikov, 1970] assure, if a certain eigenvalue ratio condition is satisfied, the existence of infinitely many periodic orbits of saddle type accumulating to the homoclinic cycle.
The proof of the existence of a homoclinic connection is generally a difficult task, even for piecewise linear systems. Some recent works [Wilczak, 2005;Wilczak, 2006] have been devoted to obtain computer-assisted proofs of the existence of global connections in Michelson system. Regarding piecewise linear systems, there are a lot of works about the existence of homoclinic cycles. In many of them [Arneodo et al., 1981;Chua et al., 1986;Coullet et al., 1979;Matsumoto et al., 1985;Matsumoto et al., 1988;Medrano et al., 2005;Medrano et al., 2006] authors require numerical arguments to show that existence. In others [Llibre et al., 2007], authors start from a degenerate situation to avoid any numerical dependence. In the present work we consider a different strategy which can be also used in a generic case.
In the particular case of piecewise linear systems with two zones, homoclinic connections can be classified attending to the number of intersections with the separation plane. It is obvious that the number of intersections between any homoclinic connection of (1.4) and the separation plane {x = 0} has to be greater than one. So, we say that a homoclinic connection of system (1.4) is direct if it intersects the separation plane {x = 0} at exactly two points.
The analytical proof of the existence of a pair of direct homoclinic connections will be the main goal of this work, as it is summarized in the following theorem. On the other hand, the proof of Theorem 1.1 is partially based on some results of [Carmona et al., 2008] where the boundary value 1/2, which does not have any dynamical meaning, was choosen for the sake of simplicity of the handmade calculations. In fact, some numerical computations allow to obtain λ h ≈ 0.660759953.
In Figure 1 the pair of homoclinic connections of (1.4) given by Theorem 1.1 are shown.
The rest of the paper is organized as follows. In section 2 we describe the basic geometric elements of the problem. Section 3 is devoted to the proof of Theorem 1.1, which is divided into two parts. In section 4 we deal with other global connections and show some numerical results. For every point p = (x p , y p , z p ) T ∈ R 3 we denote by x p (t; λ) = (x p (t; λ) , y p (t; λ) , z p (t; λ)) T the solution of the system (1.4) with parameter λ and initial condition x p (0; λ) = p. The corresponding orbit is denoted by γ p .
If x p = 0 and y p > 0, then the orbit γ p crosses transversaly the plane {x = 0} with x p (−t; λ) < 0 and x p (t; λ) > 0 for t > 0 small enough. If x p (t; λ) vanishes in (0, +∞), then we define the flying time t + p as the positive value such that In such a case, we define the Poincaré map Π + at the point p as Π + (p) = x p (t + p ; λ).
Note that the Poincaré map Π + only depends on the linear systemẋ = A + x + e 3 given in (1.2).
If x p = 0 and y p < 0, then the orbit γ p crosses transversaly the plane {x = 0} with x p (−t; λ) > 0 and x p (t; λ) < 0 for t > 0 small enough. If x p (t; λ) vanish in (0, +∞), then we define the flying time t − p as the positive value such that x p t − p ; λ = 0 and x p (t; λ) < 0 in In such a case, we define the Poincaré map Π − at the point p as

This map only depends on the linear systemẋ =
If p belongs to the z-axis; i.e. x p = 0 and y p = 0, then p is called a contact point of the flow of system (1.4) with the plane {x = 0} because the vector field at this point is tangent to the plane. Following [Llibre et al., 2004], the first coordinate of the Taylor expansion of Now we describe the basic elements of the linear dynamics in the half-space {x < 0}, all this information is summarized in Figure 2. The elements in the other half-space can be obtained using the involution R.
The half-line and the plane {x = 0} intersect at the point The stable two-dimensional manifold W s (p − ) is locally contained in the half-plane which is called the focal half-plane of p − . This half-plane is obtained from the eigenvectors associated to the complex eigenvalues of A − . The half-plane P − and the separation plane {x = 0} intersect along the straight-line Let us emphasize that not every point in D − belongs to the stable manifold W s (p − ). The 3 Existence of a direct homoclinic connection to p − .
A direct homoclinic orbit to p − has to intersect the plane {x = 0} at m − , since it corresponds to the linear one-dimensional manifold of p − . On the other side, this orbit also has to belong to the two-dimensional manifold of p − , that is, it has to intersect segment S − . Thus, when the condition Π + (m − ) ∈ S − holds, a direct homoclinic connection to p − exists in system (1.4). In fact, the existence of such homoclinic connection can be derived from conditions where q = (0, −1/λ 2 , 0) is the intersection point of the straight lines D − and D + , see Figure 2.
In Figure 4   (3.10) Proof. The first part of the equivalence, that is, the proof that a solution (t, λ) of system (3.4) with t > 0 and λ > 0 also satisfies the system (3.10), is direct.
For the other implication, let us consider the system which represents the intersection in coordinates (X, Y ) of a straight line with positive slope containing the origin and the unit circle. Obviously, system (3.11) has a unique solution which negative second coordinate.
Note that is a solution of system (3.11) whose second coordinate is negative for t > 0 and λ > 0.
On the other hand, if (t, λ) is a solution of system (3.4) with t > 0 and λ > 0, then is also a solution of system (3.11) whose second coordinate is negative.
Now let us proof that system (3.10) has at least a solution.
Proof. From lemma 3.1 it is known that systems (3.3) and (3.10) are equivalent for t > 0 and λ > 0.
Since the third condition of (3.10) is satisfied for every (t, λ) ∈ Ω it is only necessary to has solution in Ω. This is, as it is going to be proved, a consequence of Poincaré-Miranda theorem [Kulpa, 1997], which can be considered as a n-dimensional extension of Bolzano theorem.
At this moment we have proved that there exists a point (t h , λ h ) ∈ Ω such that x m − (t h , λ h ) ∈ D − . For condition (3.2) to be fulfilled it is also necessary to prove that x m − (t, λ h ) > 0 for every t ∈ (0, t h ). The next result deals with this inequality.
Proof. According to the equations of system (1.4), the derivative with respect to t of function x m − (t, λ h ) is given by y m − (t, λ h ). By integrating this system for x > 0, we obtaiṅ (3.16) where On one hand, note that x m − (0, λ h ) = 0 andẋ m − (0, λ h ) > 0. On the other hand, let us assume that (t h , λ h ) ∈ Ω is a solution of system (3.10). Therefore, Let us also assume that there exists a valuet ∈ (0, t h ) such that x m − (t, λ h ) = 0. Then, y m − (t, λ h ) must vanish in at least three values in (0, t h ), that is, the equation which is obtained from y m − (t, λ h ) = 0, has to vanish in at least three values in (0, 2π).

Other global connections
In the previous sections, the existence of a pair of direct homoclinic connections, which are symmetric respect to the involution R, has been proved for λ = λ h ≈ 0.660759953. The first step of this proof is the analysis of the solutions of the system (3.10). Those solutions are the intersections of the solid and dashed curves of Figure 4 which lie in the shadow regions. Besides the first intersection, which corresponds to the value λ h , we can observe that other intersections exist.
The second intersection point corresponds to (t H , λ H ) ≈ (10.15402101, 0.43391236). It can be also proved that a pair of direct homoclinic connections, which are symmetric respect to the involution R, exist for λ H . Remember that the existence of a intersection point is not the only condition that has to be fulfilled to assure the existence of a homoclinic connection; it is also necessary to check that the orbit with initial condition m − does not intersect the separation plane for t ∈ (0, t H ) and x m − (t H , λ H ) belongs to S − . As a comparison with the first pair of homoclinic orbits, these second homoclinic connections give an extra loop around the one-dimensional manifold of the other equilibrium. The homoclinic connection to p − is shown in Figure 5. Regarding to the remainder intersection points in Figure 4, they do not correspond with real direct homoclinic connections: although each one of them is a solution (t ′ , λ ′ ) of system (3.10), the orbit with initial condition m − intersects the separation plane for values of t ∈ (0, t ′ ).
This behavior is similar for reversible T-point heteroclinic cycles in system (1.4) . In [Carmona et al., 2008], the existence of a "direct" reversible T-point heteroclinic cycle was proved for λ ≈ 0.65153556. This cycle is called direct in the sense that its heteroclinic orbit corresponding to the one-dimensional manifolds has exactly three intersections with the separation plane (which is the minimum possible number of intersections) while the heteroclinic orbit corresponding to the two-dimensional manifolds has only one intersection. Moreover, the existence of another direct reversible T-point heteroclinic cycle can be proved for λ ≈ 0.43327834. This cycle has two extra loops around the one-dimensional manifolds of the equilibria, see Figure 6.
A first step in the proof of the existence of these reversible T-point heteroclinic cycles is the analysis of the existence of solution of a system analogous to (3.10) (given by equations (4.3) and (4.6) in [Carmona et al., 2008]). Besides the values of λ given in the previous paragraph, there exist other solutions of the system which, as the homoclinic case, do not correspond with real reversible T-point heteroclinic cycles. P. Coullet, C. Tresser and A. Arneodo, "Transition to stochasticity for a class of forced oscillators", Phys. Lett. A 72, 268-270 (1979).