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In this paper, a methodology for the numerical location of a global point-to-point (P2P for short) homoclinic asymptotically connecting orbit is applied to a modified version of Shimizu-Morioka system, which models a semiconductor laser. This type of global bifurcation can be considered as a stylized mathematical description of self-pulsation in this laser type, associ-ated with saturation. The location is achieved by use of a custom algorithm based on the method of orthogonal collocation on finite elements with fourth order boundary conditions, constructed through scale order approximations. The effectiveness of the algorithm and the superiority of high-order boundary conditions over the widely used first order ones are justified throughout the obtained graphical results.

The numerical location of global asymptotic orbits is often proved to be a computationally demanding task, even in low dimensional systems. However, the recent improvement of hardware capabilities and symbolic mathematical software allows researchers to locate global asymptotic orbits numerically more easily. Homoclinic point-to-point (P2P for short) connecting orbits arise in various applications where hysteresis and saturation phenomena are present. These orbits are considered to be separatrices in the nonlinear state space in 2D conservative ODEs (ordinary differential equations), since they divide the phase space into qualitatively different regions of motion; one region of periodic solutions and one of non-periodic ones, respectively. In that sense, homoclinic orbits can be perceived as the limit of periodic solutions, which is a periodic orbit, the fundamental period of which tends to infinity, while the orbit itself remains bounded. From a different perspective, a homoclinic P2P orbit can be regarded as the result of the collision of a limit cycle and a fixed point.

In the present paper, an algorithm for the numerical computation of global homoclinic P2P orbits is applied to a three-dimensional differential dynamical system, be it a modified Shimizu-Morioka system [

The respective procedure involves the evaluation of high order boundary conditions (BC from now on). Note that both the well-known and widely used techniques of projection BC and the method of eigenvectors (see [

the truncation interval, which in turn, however, can increase the computational time (mainly in ordinary PCs with low to moderate CPU power such as an Intel Core i7 870). So, the use of high order BC can be proved useful in cases like that. An appropriate combination of the multiple scales approximation method with that of successive approximations leads to the construction of a technique for the determination of high order BC. The idea for this combination comes from Deprit and Henrard [

The theoretical background of the computational procedure is given in [

The method of orthogonal collocation on finite elements with fourth order BC has been applied to a modified version of Shimizu-Morioka system [

x ˙ = y y ˙ = x − q y − x z z ˙ = − b z + b x 2 (1)

and the connection between its parameters and the Lorenz ones is presented in [

x ˙ = y y ˙ = ( a + 1 ) x − a y − ( a + 1 ) x z z ˙ = − z + x 2 (2)

where dot denotes differentiation with respect to τ and we have resubstituted τ with t and Y with y. The state variables x , y , z describe the small amplitude dynamics of the Laser with Saturable Absorber (LSA) [

The LSA lasers exhibit sustained laser oscillations consisting of pulse trains of really short high intensity and high frequency laser output (also known as passive Q-switching or self-pulsing behaviour). Such behaviour has been both theoretically and experimentally confirmed for CO_{2} lasers, microchip solid state lasers etc. Semiconductor lasers exhibit rates of high repetition ranging from hundreds of MHz to almost 1/10 GHz, while they have useful applications in telecommunication and in optical data storage using CD and DVD systems. Some other applications include the optical feedback noise reduction in semiconductor injection lasers and optical timing extraction by injection locking of self-pulsing optical oscillators. The phenomenon of self-pulsation is a result of the nonlinear interaction of the slowly responding amplifying and absorbing media and the fast response of the electric field in lasers driven by a constant pumping power. Actually, the basic mechanism responsible for the generation of these oscillations starts as the laser is turned on and the amplifying medium, the gain, is excited to a sufficiently high level via some type of pumping process. The absorber absorbs the free photons in the laser and thus the intensity of the electric field remains low and the saturation of the gain goes on. As soon as the absorbing medium saturates, the usual laser process starts with a strongly excited gain causing a high electric field intensity and thus truly enhanced output power. During this process both the gain and the absorber return to ground state and the process starts all over again. In semiconductor lasers, this can produce a pulse train with a typical frequency of the order of several GHz. The homoclinic point-to-point connecting orbit arising in the system of interest, can be considered as a mathematical representation of the aforementioned high power self-pulsation. A characteristic of lasers with obvious practical importance is that by changing either the material or the pump power, the qualitative behaviour of the laser beam can change dramatically, that is a large variety of local and global bifurcations can occur.

However, the presence of quite different time scales makes numerical simulations of lasers both challenging and time consuming, so the use of appropriate time scaling transformations can be useful, sometimes.

By means of (2) we easily find the three equilibria of the system, be them E 0 ( 0 , 0 , 0 ) , E ± ( ± 1 , 0 , 1 ) . Also, the Equation (2) are invariant under the transformation x → − x , y → − y , so its orbits are symmetric with respect to z-axis and we can restrict our analysis to x > 0 . Moreover, the Jacobian matrix associated with (2), is

J = ( 0 1 0 a + 1 − ( a + 1 ) z 0 − a − ( a + 1 ) x 0 2 x 0 0 − 1 ) (3)

where ( x 0 , y 0 , z 0 ) represents the equilibrium under consideration (the nonlinearity condition for higher order terms is obviously valid). Thus, for E 0 we have

J | E 0 = ( 0 1 0 a + 1 − a 0 0 0 − 1 ) (4)

giving the eigenvalues λ 1 = − 1 , λ 2 = 1 , λ 3 = − a − 1 , hence E 0 is a saddle. Furthermore, with regard to E + the Jacobian matrix becomes

J | E + = ( 0 1 0 0 − a − a − 1 2 0 − 1 ) (5)

with characteristic equation

λ 3 + ( a + 1 ) λ 2 + a λ + 2 ( a + 1 ) = 0 (6)

By setting p 0 = 2 ( a + 1 ) , p 1 = a , p 2 = a + 1 , then according to Routh-Hurwitz criterion, as long as the following relations:

p 0 > 0 ⇒ a + 1 > 0 p 1 > 0 ⇒ a > 0 D 2 = p 1 p 2 − p 0 > 0 ⇒ ( a − 2 ) ( a + 1 ) > 0 (7)

hold together, that is for α > 2 (only positive values of the parameter have a physical meaning for our model), the equilibrium E + ( 1 , 0 , 1 ) is locally asymptotically stable, since in that case the eigenvalues have negative real part. Also, according to Liu criterion [

p 1 p 2 − p 0 = 0 ⇒ ( a − 2 ) ( a + 1 ) = 0 ⇒ a = a 0 = 2 (8)

determines a Hopf bifurcation, if in addition d D 2 ( a 0 ) / d a ≠ 0 also holds, which is equivalent to the transversality condition. Since the critical first Lyapunov coefficient is evaluated l 1 ( a 0 ) ≃ − 5.183052 × 10 − 2 (by means of the corresponding normal form formulae included in the algorithm presented in [

Let us now apply the aforementioned high order BC. Assuming the solutions of the differential system under consideration can be approximated by

x ( t ) ≈ ∑ i = 1 k ε i x i ( t ) , y ( t ) ≈ ∑ i = 1 k ε i y i ( t ) , z ( t ) ≈ ∑ i = 1 k ε i z i ( t ) (9)

where ε is the orbital parameter and k is the maximal order of approximation. Then, by substituting the expressions of (9) into (2) (the Taylor expansion with respect to E 0 coincides with the system itself) and equating the same order terms, we obtain the respective linear (with respect to the variables corresponding to the j-order, j = 1 , ⋯ , k ) systems. In terms of the present analysis the maximal order of approximation has been chosen to be k = 4 . Then, each system is solved and the solution is substituted in the subsequent, higher order systems (method of successive approximations). Let us present these systems.

1st order of approximation

x ˙ 1 = y 1 y ˙ 1 = ( a + 1 ) x 1 − a y 1 z ˙ 1 = − z 1 (10)

2nd order of approximation

x ˙ 2 = y 2 y ˙ 2 = ( a + 1 ) x 2 − a y 2 − ( a + 1 ) x 1 z 1 z ˙ 2 = − z 2 + x 1 2 (11)

3rd order of approximation

x ˙ 3 = y 3 y ˙ 3 = ( a + 1 ) x 3 − a y 3 − ( a + 1 ) ( x 1 z 2 + x 2 z 1 ) z ˙ 3 = − z 3 + 2 x 1 x 2 (12)

4th order of approximation

x ˙ 4 = y 4 y ˙ 4 = ( a + 1 ) x 4 − a y 4 − ( a + 1 ) ( x 1 z 3 + x 2 z 2 + x 3 z 1 ) z ˙ 3 = − z 3 + 2 x 1 x 3 + x 2 2 (13)

By means of the procedure described in [

1st order approximation

x 1 ( t ) = c 1 e t + c 2 e − ( a + 1 ) t y 1 ( t ) = c 1 e t + c 2 ( − a − 1 ) e − ( α + 1 ) t z 1 ( t ) = c 3 e − t (14)

where c 1 , c 2 , c 3 denote the integration constants (from now on c i , i = 1 , 2 , 3 , 4 , ⋯ will denote integration constants unless stated otherwise). By setting c 2 = c 3 = 0 we get the first order approximation of the outgoing solution vector

x 1 o u t = c 1 e t y 1 o u t = c 1 e t z 1 o u t = 0 (15)

2nd order approximation

x 2 ( t ) = c 4 e t + c 5 e − ( a + 1 ) t y 2 ( t ) = c 4 e t − c 5 ( a + 1 ) e − ( α + 1 ) t z 2 ( t ) = c 6 e − t + 1 3 c 1 2 e t (16)

and setting c 4 = c 5 = c 6 = 0 we take the second order approximation

x 2 o u t ( t ) = 0 y 2 o u t ( t ) = 0 z 2 o u t ( t ) = 1 3 c 1 2 e − t e 2 t (17)

3rd order approximation

x 3 ( t ) = c 7 e t + c 8 e − ( a + 1 ) t − a + 1 6 ( a + 4 ) c 1 3 e 3 t y 3 ( t ) = c 7 e t − c 8 ( a + 1 ) e − ( a + 1 ) t − a + 1 2 ( a + 4 ) c 1 3 e 3 t z 3 ( t ) = c 9 e − t (18)

The third order approximation is obtained by setting c 7 = c 8 = c 9 = 0 ,

x 3 o u t = − a + 1 6 ( a + 4 ) c 1 3 e 3 t y 3 o u t = − a + 1 2 ( a + 4 ) c 1 3 e 3 t z 3 o u t = 0 (19)

4th order approximation

x 4 ( t ) = c 10 e t + c 11 e − ( a + 1 ) t y 4 ( t ) = c 10 e t − ( a + 1 ) c 11 e − ( a + 1 ) t z 4 ( t ) = c 12 e − t − a + 1 15 ( a + 4 ) c 1 4 e 4 t (20)

and setting c 10 = c 11 = c 12 = 0 we get the fourth order approximation of the outgoing solution

x 4 o u t ( t ) = 0 y 4 o u t ( t ) = 0 z 4 o u t ( t ) = − a + 1 15 ( a + 4 ) c 1 4 e 4 t (21)

So, by substituting the aforementioned formulae of ( x j o u t , y j o u t , z j o u t ) , j = 1 , 2 , 3 , 4 for ( x j , y j , z j ) in (9), we arrive at the outgoing solution up to the fourth order:

x o u t = ε c 1 e t − a + 1 6 ( a + 4 ) c 1 3 e 3 t y o u t = ε c 1 e t − a + 1 6 ( a + 4 ) c 1 3 e 3 t z o u t = 1 3 c 1 2 e − t e 2 t − 1 15 a + 1 a + 4 c 1 4 e 4 t (22)

where c 1 can be user defined. Similarly, the expressions of the incoming vector can be set, as well. There, the integration constants associated with the “unstable eigenvalues” must be set equal to zero. The effectiveness of high order BC implemented compared to the classic first order ones, often encountered in bibliography, is presented in

Via the method of orthogonal collocation on finite elements combined with the aforementioned fourth order BC, the homoclinic connecting orbit of interest (i.e. at the origin) has been computed inside the truncated time interval [ − 23.5 , 23.5 ] , which has been determined with the aid of the well-known Beyn’s method [

An efficient custom algorithm of orthogonal collocation on finite elements implemented in Mathworks Matlab has been applied to a laser model based on a modified version of Shimizu-Morioka system for the numerical location of a homoclinic orbit. An initial approximation of this orbit has become available through a numerical continuation of limit cycles, bifurcated from a Hopf bifurcation, up to a high value of the fundamental period. The efficiency of the algorithm lies in the fact that all the required equations, be them the collocation equations, the continuity equations, the boundary conditions and the phase condition, are converted to Matlab functions automatically, so that integrated, sophisticated Matlab routines used for solving systems of nonlinear algebraic equations, as well as optimization routines or any other relevant, user-defined routines can be applied directly for the solution of the aforementioned system of nonlinear algebraic equations. Furthermore, the high order boundary conditions defined and used herein can be useful when ordinary PCs of low to moderate computational power are used for the location of homoclinic orbits, as they do not require the increase of the length of the truncation interval in order to achieve the precision sought for the computation. The effectiveness of the high

order BC is evident in

Last, the physical meaning of the homoclinic orbit of the laser computed above has already been mentioned and it concerns the description of the saturation phenomenon present in this laser type. More precisely, the homoclinic point-to-point connecting orbit computed can be considered as a stylized representation of a high power self-pulsation associated with the aforementioned phenomenon (see

P. S. Douris is pleased to acknowledge his financial support from “Andreas Mentzelopoulos Scholarships for the University of Patras”.

The authors declare that there is no conflict of interest regarding the publication of this paper.

Douris, P.S. and Markakis, M.P. (2018) Computation of a Point-to-Point Homoclinic Orbit for a Semiconductor Laser Model. Applied Mathematics, 9, 1258-1269. https://doi.org/10.4236/am.2018.911082