Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations

Abstract

Analytical solution of the homoclinic orbit of a two-dimensional system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment is described.

1. Introduction

A homoclinic orbit is the trajectory of a flow of dynamical system that joins a saddle equilibrium point to itself; that is, the homoclinic trajectory ℎ(𝑡) converges to the equilibrium point as 𝑡→±∞ [1].

The analytical solutions of homoclinic orbits are very important for many applications as in the use of the homoclinic Melnikov function, in order to prove the existence of transversal homoclinic orbits and chaotic behavior.

In what follows, we find the analytical solution of the homoclinic orbit of a one-degree-of-freedom system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment [2].

The aim of the work in [2] was to study the asymptotic behavior of the system. Specifically, the initial dissipative system composed of two linear and one nonlinear oscillators was reduced to a nonautonomous damped strongly nonlinear second-order differential equation. With the use of the complexification-averaging technique (CX-A), we obtained the slow flow of the system, that is, a system of two, first-order, differential equations governed by slow time 𝑎^′=−𝜖𝜆2𝑏𝑎+2−3𝐶8𝑎²+𝑏²𝐽𝑏−2+𝐵2𝜖sin𝐵𝜔₂₀𝑡,𝑏^′=−𝜖𝜆2𝑎𝑏−2+3𝐶8𝑎²+𝑏²𝐴𝑎−2−𝐵2𝜖cos𝐵𝜔₂₀𝑡, (1) where 𝑎, 𝑏 are the variables and 𝐴, 𝐵, 𝐽, 𝜖, 𝐵, 𝜔₂₀ are parameters.

From the study of the dynamics of the slow flow [3], we concluded that the slow flow may do regular or chaotic oscillations. The computation of the analytical solution of the homoclinic orbit of the unperturbed problem is the first step in order to investigate the chaotic behavior, of the above system, with the use of the homoclinic Melnikov function.

2. Main Results

The unperturbed part of the above system is 𝑎^′=𝑏2−3𝐶8𝑎²+𝑏²𝐽𝑏−2+𝐵2𝑏sin𝛾,_′𝑎=−2+3𝐶8𝑎²+𝑏²𝐴𝑎−2−𝐵2cos𝛾, (2) and 𝛾 is a parameter. The equilibrium points are found considering 𝑎^′=0, 𝑏^′=0. After some simple algebraic manipulations, we have 𝑎=𝐴+𝐵cos𝛾−𝐽+𝐵sin𝛾𝑏, (3) and the third-order equation 𝑏−3𝐶4𝐴+𝐵cos𝛾²+𝐽−𝐵sin𝛾²𝐽−𝐵sin𝛾²𝑏³−𝐽+𝐵sin𝛾=0. (4) When (4) has three real roots, we have three equilibrium points. It is well known that, in order (4) to have three real roots, it must hold that the determinant 𝑞²+4𝑝³<0, where 𝑝=(3𝑎𝑐−𝑏²)/(9𝑎²), 𝑞=(2𝑏³−9𝑎𝑏𝑐+27𝑎²𝑑)/(27𝑎³), 𝑎=−(3𝐶/4)(((𝐴+𝐵cos𝛾)²+(𝐽−𝐵sin𝛾)²)/(𝐽−𝐵sin𝛾)²), 𝑏=0, 𝑐=1, and 𝑑=−𝐽+𝐵sin𝛾. This holds for 𝐷<16,81𝐶 (5) where 𝐷=(𝐴+𝐵cos𝛾)²+(𝐽−𝐵sin𝛾)².

The equilibrium points are 𝑎₁4=−𝐴+𝐵cos𝛾3√𝜛𝐶𝐷cos3,𝑏₁=4𝐽−𝐵sin𝛾3√𝜛𝐶𝐷cos3,𝑎₂4=−𝐴+𝐵cos𝛾3√𝜛𝐶𝐷cos3+2𝜋3,𝑏₂=4𝐽−𝐵sin𝛾3√𝜛𝐶𝐷cos3+2𝜋3,𝑎₃4=−𝐴+𝐵cos𝛾3√𝜛𝐶𝐷cos3+4𝜋3,𝑏₃=4𝐽−𝐵sin𝛾3√𝜛𝐶𝐷cos3+4𝜋3, (6) where 𝜛=(1/3)cos⁻¹((−9√𝐶𝐷)/4).

The Hamiltonian of system (2) is given by 𝑎ℎ=²+𝑏²4−3𝐶𝑎32²+𝑏²²+𝑏𝐵sin𝛾−𝐽2+𝑎2𝐴+𝐵cos𝛾. (7) We perform the canonical transformation 𝑎=2𝜌cos𝜗, 𝑏=2𝜌sin𝜗, and the hamiltonian becomes 𝜌ℎ=2−3𝐶8𝜌²+2𝜌2,𝜌sin𝜗𝐵sin𝛾−𝐽+cos𝜗𝐴+𝐵cos𝛾_′=2𝜌2𝐵sin𝛾−𝐽cos𝜗−sin𝜗𝐴+𝐵cos𝛾. (8)(9) From the square of (9) by adding in both sides of the equation the quantity ((2𝜌/2)((𝐵sin𝛾−𝐽)sin𝜗+cos𝜗(𝐴+𝐵cos𝛾)))², we have 𝜌^′2+2𝜌2𝐵sin𝛾−𝐽sin𝜗+cos𝜗𝐴+𝐵cos𝛾²=𝜌2𝐷. (10) From the hamiltonian (8), (10) becomes 𝜌^′2=𝜌𝐷√2−𝜌ℎ−2+3𝐶8𝜌²×𝜌𝐷√2+𝜌ℎ−2+3𝐶8𝜌². (11) We denote by 𝜌^∗, 𝜗^∗ the unstable equilibrium point ((𝜌^∗)^′=0), and (10) becomes 𝐵sin𝛾−𝐽sin𝜗^∗+cos𝜗^∗𝐴+𝐵cos𝛾=±√𝐷. (12) By substituting (for the case −√𝐷) in the hamiltonian, we have 𝜌ℎ=^∗2−3𝐶8𝜌^∗2−𝜌^∗𝐷2, (13) and (11) becomes 𝜌^′2=𝜌𝐷√2−𝜌^∗2+3𝐶8𝜌^∗2−𝜌^∗𝐷2−𝜌2+3𝐶8𝜌²×𝜌𝐷√2+𝜌^∗2+3𝐶8𝜌^∗2−𝜌^∗𝐷2−𝜌2+3𝐶8𝜌². (14) For the right hand side of the previous equation after some simple algebra manipulations, we have 𝜌𝐷√2−𝜌^∗2+3𝐶8𝜌^∗2−𝜌^∗𝐷2−𝜌2+3𝐶8𝜌²×𝜌𝐷√2+𝜌^∗2+3𝐶8𝜌^∗2−𝜌^∗𝐷2−𝜌2+3𝐶8𝜌²=𝜌−𝜌^∗𝐷2𝐷2−𝜌^∗+3𝐶2𝜌^∗3+3𝐶4𝜌^∗𝜌−3𝐶4𝜌^∗𝜌^∗−𝜌−𝜌^∗4+3𝐶8𝜌−𝜌^∗𝜌+𝜌^∗−9𝐶²64𝜌−𝜌^∗𝜌+𝜌^∗2. (15) We calculate 𝑏𝑎^′−𝑎𝑏^′ and derive −2𝜗^′𝜌=𝜌−3𝐶2𝜌²+𝜌2sin𝜗𝐵sin𝛾−𝐽+cos𝜗𝐴+𝐵cos𝛾, (16) and for the equilibrium point ((𝜗^∗)^′=0) we have 𝜌^∗−3𝐶2𝜌^∗2+𝜌^∗2sin𝜗^∗𝐵sin𝛾−𝐽+cos𝜗^∗𝐴+𝐵cos𝛾=0. (17) Using (12) and the above equality, (15) becomes 𝜌𝐷√2−𝜌^∗2+3𝐶8𝜌^∗2−𝜌^∗𝐷2−𝜌2+3𝐶8𝜌²×𝜌𝐷√2+𝜌^∗2+3𝐶8𝜌^∗2−𝜌^∗𝐷2−𝜌2+3𝐶8𝜌²=𝜌−𝜌^∗2𝐷23𝐶4𝜌^∗−14+3𝐶8𝜌+𝜌^∗−9𝐶²64𝜌+𝜌^∗2. (18) Then from (14) and (18), we have 𝑑𝜌𝜌−𝜌^∗(𝐷/2)(3𝐶/4)𝜌^∗−(1/4)+(3𝐶/8)𝜌+𝜌^∗−(9𝐶²)/64𝜌+𝜌^∗21/2−1=𝑑𝑡, (19) which is our main differential equation and is easily solved [4].

As it is seen in Figure 1 when our system has three equilibrium points, then depending on the parameters, we may have two homoclinic orbits. In our analysis, this result is given by the absolute value in (19).

Figure 1 

Phase space of the hamiltonian for 𝐴=0.1, 𝐵=0.001, 𝐽=0.00001, 𝛾=(𝜋/9), and 𝐶=2.0.

For the case 𝜌^∗>𝜌, the homoclinic solution is 𝜌(𝑡)=128𝑒^𝑄(𝑡)/2+27𝐶³2𝐷𝜌^∗𝜌^∗−16𝑒^𝑄(𝑡)/43𝐶6𝜌^∗−3𝐶𝜌^∗2+42𝐷𝜌^∗×−8128𝑒^𝑄(𝑡)/2+27𝐶³2𝐷𝜌^∗−48𝑒^𝑄(𝑡)/43𝐶𝜌^∗−2⁻¹, (20)where 𝑄(𝑡)=𝑡−4+3𝐶(22𝐷𝜌^∗+𝜌^∗(4−3𝐶𝜌^∗)). After substituting the solution (20) in (16) and integrate we derive 𝜗=𝑊₁tan⁻¹𝑔₁+𝑔₂𝑒^𝑄(𝑡)/4+𝑊₂tan⁻¹𝑔₃+𝑔₄𝑒^𝑄(𝑡)/4+𝑊₃𝑡, (21) where 𝑊₁=3𝐶8−6(1+3𝐶)𝜌^∗+9𝐶(1+𝐶)𝜌^∗−12𝐶2𝐷𝜌^∗×−9𝐶²𝜌^∗−23𝐶²+6𝐶³2𝐷𝜌^∗×−9𝐶²𝜌²−23𝐶²+6𝐶2𝐷𝜌^∗−1,𝑊₂=ℎ𝜌^∗−9𝐶²𝜌^∗−(2/3𝐶)²+6𝐶³2𝐷𝜌^∗×6(3𝐶−1)𝜌^∗−9𝐶(𝐶−1)𝜌^∗+12𝐶2𝐷𝜌^∗×−854𝐶³𝜌^∗22𝐷𝜌^∗−8+3𝐶3𝜌^∗𝐶𝜌^∗−2−42𝐷𝜌^∗2−1/2,𝑊₃=9𝐶𝜌^∗2−16𝜌^∗−8ℎ64𝜌^∗,𝑔₁=2−3𝐶𝜌^∗−9𝐶²𝜌^∗−(2/3𝐶)²+6𝐶³2𝐷𝜌^∗,𝑔₂=163−9𝐶²𝜌^∗−(2/3𝐶)²+6𝐶³2𝐷𝜌^∗,𝑔₃=8+9𝐶²𝜌^∗2−6𝐶3𝜌^∗+22𝐷𝜌^∗×54𝐶³𝜌^∗22𝐷𝜌^∗−8+3𝐶3𝜌^∗𝐶𝜌^∗−2−42𝐷𝜌^∗2−1/2,𝑔₄=16𝜌^∗×54𝐶³𝜌^∗22𝐷𝜌^∗−8+3𝐶3𝜌^∗𝐶𝜌^∗−2−42𝐷𝜌^∗2−1/2. (22) For the case 𝜌>𝜌^∗, the homoclinic solution is 𝜌(𝑡)=128𝑒^{−𝑄(𝑡)/2}+27𝐶³2𝐷𝜌^∗𝜌^∗−16𝑒^{−𝑄(𝑡)/4}3𝐶6𝜌^∗−3𝐶𝜌^∗2+42𝐷𝜌^∗×−8128𝑒^{−𝑄(𝑡)/2}+27𝐶³2𝐷𝜌^∗−48𝑒^{−𝑄(𝑡)/4}3𝐶𝜌^∗−2⁻¹,𝜗=𝑊₃𝑡−𝑊₁tan⁻¹𝑔₁+𝑔₂𝑒^{−𝑄(𝑡)/4}−𝑊₂tan⁻¹𝑔₃+𝑔₄𝑒^{−𝑄(𝑡)/4}. (23)

Conflict of Interests

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