# Boundary Value Problems for Liénard-Type Equations with Quadratic Dependence on the “Velocity”

Authors: A. Kirichuka; F. Sadyrbaev
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

The estimates were obtained for the number of solutions for the Neumann and Dirichlet boundary value problems associated with the Liénard equation with a quadratic dependence on the “velocity.” Sabatini’s transformation is used to reduce this equation to a conservative one, which does not contain the derivative of an unknown function. Despite the one-to-one correspondence between the equilibria, the topological structure of the phase portraits of both equations can differ significantly.


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## Body

## 1. Introduction

Nonlinear boundary value problems (BVP) for ordinary differential equations (ODE) are an integral part of many mathematical models of real phenomena. When calculating these models, it is important to know if a solution exists. In studying complex phenomena, multiple solutions can appear. To compute them, additional information is needed, such as the location, initial conditions, and oscillatory properties. The important problem of determining the number of solutions has attracted the attention of researchers for a long time. The nonlinear oscillation in physics and applied mathematics has been intensively studied in many articles. Such papers as [1, 2] presented analytical approximations to the periodic solutions and, in particular, to periodic solutions for oscillators described by ordinary differential equations with the odd-degree nonlinearity.However, insufficient attention has been paid to a differential equation with even-degree nonlinearities. Although, for example, equations with quadratic nonlinearities have practical applications. As written in the work [3], this equation “has been used as a mathematical model of human eardrum oscillations.” This fact motivated the search for an exact solution to a differential equation with quadratic nonlinearity. Equations with quadratic nonlinearities were studied in [4, 5] also.In the articles [6, 7] the authors have studied the boundary value problems of the form
(1)x′′+ax−btxn=0,x0=xT=0orx′0=x′T=0,where n is a positive integer (most results concern the cases n=2, n=3, and n=5). The exact estimates of the number of solutions were obtained for autonomous equations of the form x′′+ax−bxn=0, and some results (mostly of computational nature) were stated for the case of bt being piece-wise constant function (in the articles, [7, 8]). The phase plane method was used extensively.In this paper, our goal is different. We study the equation with quadratic term(2)x′′+fxx′2+gx=0,together with the two-point boundary conditions.Equations of the form(3)x′′+fxx′+gx=0are a classical object for investigation. The Liénard and Van-der-Pol equations fall into this class. Both arose from practice. Equations of the form (3) are rich in oscillatory behaviors. They are known to have (under suitable conditions) isolated periodic solutions. The problem of estimating the number of limit cycles for the case of polynomial functions fx and gx has attracted the attention of prominent researchers. In contrast, equation (2) can be reduced to a conservative equation. Our goal is to compare equation (2) with that for shorter equation
(4)x′′+gx=0.We focus on the casegx=ax−bx3, a>0, b>0, and consider two-point boundary conditions for both equations. In particular, we would like to compare the number of solutions to the respective BVPs.For this, we make use of the special change of variables resulting in eliminating the middle term in (2). This technique was proposed by Sabatini ([9]) when studying isochronous problems. An equation in new variables has a simpler form and can be (formally) integrated. This transformation keeps the trivial solution. This is important because in various sources devoted to the study of multiple solutions of BVP, the following idea was exploited. Imagine that the oscillatory behavior of solutions can be measured around the trivial solution. If a comparison can be made with solutions far away from the trivial one, some conclusions can be made about the number of solutions for two-point boundary value problems. After reduction of the equation (2) to the form (4) using the above-mentioned variable change, another comparison can be made, namely, the equation in question versus the reduced equation. This approach will be considered in the next sections.


## 2. Reduction to Equation of the Formu′′+hu=0

Consider equation (2). We suppose that f and g are continuous functions. Introduce the variable u by the transformation
(5)u≔∫0xeFsds,where Fx=∫0xfξdξ. The primitive Fx can change sign arbitrarily, since there are no sign restrictions on fx, but F0=0. Therefore eFx is a positive valued function with no restrictions on monotonicity. In particular, the integral ∫0+∞eFsds or ∫0−∞eFsds may be convergent, and the range of values of a new variable u can be bounded. Since du/dx=eFx>0, the function u=ux is monotonically increasing, passing through zero, u0=0. The inverse function xu exists, but it can be defined on a bounded or a semi-bounded interval only.The equation (2) takes the form
(6)u′′+hu=0,where
(7)hu=gxueFxu.The equation (6) can be considered instead of (2). There is a one-to-one correspondence between x and u.


## 3. Critical Points and Linearization

In the study of multiplicity of solutions to equationsx′′+ax−bx3=0 and/or x′′+ax−bx2=0, a significant role was played by the linearization at the trivial solution xt≡0. For instance, linearization of x′′+ax−bx3=0 at the trivial solution yields the linear equation y′′+ay=0, and estimations of the number of solutions of the Dirichlet and the Neumann problem were made in terms of this a.Let us clarify this question in a general setting. Consider the conservative equation (4), where gx=0 in several points xi, including x0=0. Then there is the trivial solution of the Dirichlet and the Neumann problems with zero boundary conditions. Linearization around the trivial solution yields
(8)y′′+gx0y=0.Notice thatgx0 need not be zero.Consider now perturbed equation with the quadratic term (2): x′′+fxx′2+gx=0, provided the same assumptions on gx. The critical points for an equivalent system x′=y, and y′=−fxy2−gx are given by
(9)y=0,fxy2+gx=0,which reduces to gx=0. So the critical points for equations (2) and (4) are the same.Linearization of (2) at the trivial solution gives y′′+fxx′2+gxx≡0y+2x′fx≡0y′=0 or y′′+gx0y=0, which is the same as (8).The estimates for the equationx′′+ax−bx3=0 (and/or x′′+ax−bx2=0) were obtained by considering solutions of the Cauchy problems with small initial values and passing to a heteroclinic solution, which is slow.


## 4. Case 1: Equationx′′+μx′2+gx=0

Consider the equation(10)x′′+μx′2+gx=0,where gx is given. We will assume that gx is a particular cubic polynomial. To apply the Sabatini transformation, we denote Fx=∫0xμdξ=μx. Introduce the new variable
(11)u≔∫0xeFsds=∫0xeμsds=1/μ∫0xeμsdμs=1/μeμx−1.Comparison will be made with the equationx′′+gx=0. For μ positive (we assume this), u=ux is the function, monotonically increasing from −1/μ to +∞. The inverse function x=xu=1/μlnμu+1 exists.The equation (6) takes the form
(12)u′′+gxueFxu=u′′+gxueμxu=u′′+gxueμ1/μlnμu+1==u′′+gxuμu+1=0.In Figure1, the functions ux (solid) are depicted (blue for μ=1 and red for μ=5) together with the inverse functions xu (dashed).Figure 1
The graphs ofux=1/μeμx−1 (solid), xu=1/μlnμu+1 (dashed), blue for μ=1, and red for μ=5.Ifgx=ax−bx3, the equation (12) turns to
(13)u′′+1μlnμu+1a−b/μ2ln2μu+1μu+1=0,where−1/μ<u<+∞.


### 4.1. Comparison ofx′′+ax−bx3=0 and x′′+μx′2+ax−bx3=0

#### 4.1.1. Number of Solution for the Neumann Problem

Consider the Neumann problem(14)x′′+μx′2+gx=0,(15)x′0=x′1=0.The equation (14) turns to the equation
(16)u′′+gxuμu+1=0,wheregx=ax−bx3,or takes the form
(17)u′′+1/μlnμu+1a−b/μ2ln2μu+1μu+1=0.Sincedx/dt=dx/dudu/dt=1/du/dxdu/dt=e−Fxdu/dt, e−Fx>0, the boundary conditions (15) for ut are
(18)u′0=u′1=0.Any solutionxt of problems (14) and (15) corresponds to a solution ut=∫0xteFsds of problems (17) and (18). Therefore, we can estimate the number of solutions of problems (17) and (18). Let us linearize equation (17) at the critical point u=0. One has
(19)dgxuμu+1du=dgxuduμu+1+dμu+1dugxu=dgxdxdxduμu+1+μgxu=dax−bx3dx1du/dxμu+1+μgxu=a−3bx2u1eFxμu+1+μgxu=a−3bx2ue−Fxμu+1+μgxu.Evaluating (19) at u=0 and taking into account that xuu=0=0,F0=0, and g0=0, one gets that the right side in (19) is a and the linearized equation for (17) at u=0 is y′′+ay=0. This corresponds to the linearization of equation
(20)x′′=−ax+bx3,a>0,b>0at x=0.Hence the following statement is valid.Theorem 1.
The numberN of solutions of problems (14) and (15) is not less than the number n of solutions of problems (15) and (20).
The number of nontrivial solutions of the BVP(21)x′′=−ax+bx3,x′0=0,x′1=0is exactly 2n, where n is the number of extrema of a solution xt;α of the Cauchy problem x0=0,x′0=α in the open interval 0,1 for α>0 and arbitrarily small. This follows from the monotonocity ([7]) of the period Tα, when α changes from zero to αheteroclinic, where the last value corresponds to the heteroclinic solution.


#### 4.1.2. Phase Portraits

The comparison differential equation(22)x′′+gx=x′′+ax−bx3=0,and the equivalent system
(23)x′=y,y′=−ax+bx3have three critical points at −a/b,0,0,0, and a/b,0.The equation (16) and the equivalent system
(24)u′=v,v′=−gxuμu+1have also three critical points at u1,0,0,0, and u3,0, where u1 and u3 are such that xu1=−a/b, and xu3=a/b. The behavior of solutions of (23) in the region between two heteroclinic trajectories and the behavior of solutions of (24) in the region inside the homoclinic solution are similar. All of them are periodic, and periods are defined for α small by the same linearization.The phase portraits for systems (23) and (24), a=50,b=25, and μ=1, are depicted in Figures 2 and 3.Figure 2
The phase portrait of system (23), a=50, b=25.Figure 3
The phase portrait of system (24).


#### 4.1.3. Solutions for the Neumann Problem

The number of solutions of the Neumann problem(25)x′′+μx′2+ax−bx3=0,(15) is not less than that for problems (15) and (20).Consider the Cauchy problem (20),
(26)x0=xα,x′0=0,−ab<xα<ab,xα≠0.Theorem 2.
Leti be a positive integer such that
(27)iπ<a<i+1π.
The Neumann problems (15) and (20) have exactly 2i nontrivial solutions such that x0=xα, x′0=0,−a/b<xα<a/b, xα≠0.The proof of Theorem2 can be found in the articles [6].Consider the Cauchy problem (25)
(28)x0=xα,x′0=0,−x1<xα<x2,xα≠0,where x1 and x2 are critical points of equation (25).We know from Theorem1 that the number of (nontrivial) solution of the BVP (15) and (25) is at least 2i. We wish to illustrate this.Example 1.
Consider equation (20), where a=50 and b=25:
(29)x′′=−50x+25x3,with the initial conditions x0=xα, x′0=0, and 0<xα<2, and then the number of nontrivial solutions satisfying the boundary conditions (15) is two, and for initial conditions x0=xα, x′0=0, and −2<xα<0, there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 2. This is the case for i=2 (namely, 2π<50<3π) in the inequality (27).Consider equation (25), where a=50, b=25, and μ=1:
(30)x′′=−x′2−50x+25x3with the initial conditions xα>0, and then the number of nontrivial solutions satisfying the boundary conditions (15) is two. Graphs xt for solutions of problems (15) and (30), where xα>0, are depicted in Figure 4, but graphs x′t are depicted in Figure 5. For initial conditions xα<0, there are also two solutions to the problem. Graphs xt for solutions of problems (12) and (30), where xα<0, are depicted in Figure 6, but graphs x′t are depicted in Figure 7. Totally the Neumann problems (15) and (30) have four solutions. This is in accordance with Theorem 1.Figure 4
Graphsxt for solutions of problems (15) and (30), xα≈0.352, xα≈0.40228 (dashed).Figure 5
Graphsx′t for solutions of problems (15) and (30), xα≈0.352, xα≈0.40228 (dashed).Figure 6
Graphsxt for solutions of problems (15) and (30) xα≈−0.74, xα≈−1.4092 (dashed).Figure 7
Graphsx′t for solutions of problems (15) and (30) xα≈−0.74, xα≈−1.4092 (dashed).


#### 4.1.4. Number of Solution for the Dirichlet Problem

Question. As we see in Figure 8, two heteroclinic solutions have disappeared and were replaced by a homoclinic solution. The number of solutions for the Neumann problems for equations x′′+ax−bx3=0 and x′′+μx′2+ax−bx3=0 is, probably, the same. As to the Dirichlet problem with zero boundary conditions, extra solution(s) can appear outside of the region, bounded by homoclinic trajectory. Is this the case for the example in Figure 8 (and generally)?Figure 8
The phase portrait ofx′′+μx′2+ax−bx3=0, a=50, b=25, and μ=1.Consider the Dirichlet problem (25),
(31)x0=x1=0.Theorem 3.
Leti be a positive integer such that
(32)iπ<a<i+1π.
The Dirichlet problems (20) and (31) has exactly 2i nontrivial solutions such that x0=0, x′0=α, and −αmax<α<αmax, where αmax≕a/2b, α≠0.The proof of Theorem3 can be found in the articles [6].Example 2.
Consider equation (29), with the initial conditions x0=0, x′0=α, and 0<α<52, and then the number of solutions satisfying the boundary conditions (31) is two, and for initial conditions x0=0, x′0=α, and −52<α<0, there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 3 for the case for i=2 (namely, 2π<50<3π) in the inequality (32).Consider equation (30) with the initial conditions 0<α<αhomoclinic, then the number of solutions satisfying the boundary conditions (31) is one. The reason for this difference comparing with “short” equation (29) is that the phase portrait (depicted in Figure 8) is not symmetric with respect to the vertical axis, and the rotation is faster in the left half-plane. The graph xt for a solution of problems (30) and (31), where 0<α<αhomoclinicα≈3.1, is depicted in Figure 9. In the same picture, an additional solution of the problem with αhomoclinic<α<α0α≈16.4802 is shown. The value α0 corresponds to the intersection point of a trajectory going to the critical point (right of the homoclinic region) with the x′ axis. For the negative initial conditions −αhomoclinic<α<0, there are two solutions to the problem. The graphs xt for solutions of problems (30) and (31), where α<0, are depicted in Figure 10. Totally the Dirichlet problems (30) and (31) has four solutions. Of them three solutions are in the region bounded by the homoclinic trajectory, and one solution lies outside this region.Theorem 4.
The Dirichlet problems (25) and (31) have at least 2i−1 nontrivial solutions for the initial conditions in the region bounded with homoclinic solutions. The extra solution(s) can appear outside of this region.Figure 9
Graphsxt for solutions of problems (30) and (31), α≈16.4802, α≈3.1 (dashed).Figure 10
Graphsxt for solutions of problems (30) and (31), α≈−3.11, α≈−3.6605 (dashed).The phase portrait is similar to that for a quadratic equationx′′=−ax+bx2. Therefore, the proof about number of solutions is similar also.Proof.
Consider the phase portrait for equation (25) depicted in Figure 8. The trajectory of any nontrivial solution of problems (25) and (31) for 0<α<αhomoclinic is located inside the region bounded by homoclinic orbits. Introduce the notation G for this region.
Consider solutions of the Cauchy problem (25),
(33)x0=0,x′0=α,−αhomoclinic<α<αhomoclinic,αhomoclinic≠0.
Solutions forα small enough behave like solutions of the equation of variations y′′=−ay around the trivial solution. Due to the assumption iπ≤a<i+1π, solutions yt along with solutions xt;α (for small enough α) have exactly i zeros in the interval 0,1, but for α large, the first zero t1α tends to zero and stays in the interval. Therefore, for 0<α<αhomoclinic, there exist exactly i−1 solutions.
Consider−αhomoclinic<α<0. Solutions for α small enough have i zeros. These zeros move monotonically to the right as α increases in modulus and negative. Solutions xt;α with −αhomoclinic<α and close enough to −αhomoclinic have no zeros in 0,1 since the respective trajectories are close to the homoclinic, therefore, exactly i solutions of problems (25) and (31).
The extra solution(s) can appear outside of this region if initial condition isαhomoclinic<α<α0. The value α0 is the intersection point of the phase trajectory, which starts at the critical point, with the x′ axis.


## 4.1. Comparison ofx′′+ax−bx3=0 and x′′+μx′2+ax−bx3=0

### 4.1.1. Number of Solution for the Neumann Problem

Consider the Neumann problem(14)x′′+μx′2+gx=0,(15)x′0=x′1=0.The equation (14) turns to the equation
(16)u′′+gxuμu+1=0,wheregx=ax−bx3,or takes the form
(17)u′′+1/μlnμu+1a−b/μ2ln2μu+1μu+1=0.Sincedx/dt=dx/dudu/dt=1/du/dxdu/dt=e−Fxdu/dt, e−Fx>0, the boundary conditions (15) for ut are
(18)u′0=u′1=0.Any solutionxt of problems (14) and (15) corresponds to a solution ut=∫0xteFsds of problems (17) and (18). Therefore, we can estimate the number of solutions of problems (17) and (18). Let us linearize equation (17) at the critical point u=0. One has
(19)dgxuμu+1du=dgxuduμu+1+dμu+1dugxu=dgxdxdxduμu+1+μgxu=dax−bx3dx1du/dxμu+1+μgxu=a−3bx2u1eFxμu+1+μgxu=a−3bx2ue−Fxμu+1+μgxu.Evaluating (19) at u=0 and taking into account that xuu=0=0,F0=0, and g0=0, one gets that the right side in (19) is a and the linearized equation for (17) at u=0 is y′′+ay=0. This corresponds to the linearization of equation
(20)x′′=−ax+bx3,a>0,b>0at x=0.Hence the following statement is valid.Theorem 1.
The numberN of solutions of problems (14) and (15) is not less than the number n of solutions of problems (15) and (20).
The number of nontrivial solutions of the BVP(21)x′′=−ax+bx3,x′0=0,x′1=0is exactly 2n, where n is the number of extrema of a solution xt;α of the Cauchy problem x0=0,x′0=α in the open interval 0,1 for α>0 and arbitrarily small. This follows from the monotonocity ([7]) of the period Tα, when α changes from zero to αheteroclinic, where the last value corresponds to the heteroclinic solution.


### 4.1.2. Phase Portraits

The comparison differential equation(22)x′′+gx=x′′+ax−bx3=0,and the equivalent system
(23)x′=y,y′=−ax+bx3have three critical points at −a/b,0,0,0, and a/b,0.The equation (16) and the equivalent system
(24)u′=v,v′=−gxuμu+1have also three critical points at u1,0,0,0, and u3,0, where u1 and u3 are such that xu1=−a/b, and xu3=a/b. The behavior of solutions of (23) in the region between two heteroclinic trajectories and the behavior of solutions of (24) in the region inside the homoclinic solution are similar. All of them are periodic, and periods are defined for α small by the same linearization.The phase portraits for systems (23) and (24), a=50,b=25, and μ=1, are depicted in Figures 2 and 3.Figure 2
The phase portrait of system (23), a=50, b=25.Figure 3
The phase portrait of system (24).


### 4.1.3. Solutions for the Neumann Problem

The number of solutions of the Neumann problem(25)x′′+μx′2+ax−bx3=0,(15) is not less than that for problems (15) and (20).Consider the Cauchy problem (20),
(26)x0=xα,x′0=0,−ab<xα<ab,xα≠0.Theorem 2.
Leti be a positive integer such that
(27)iπ<a<i+1π.
The Neumann problems (15) and (20) have exactly 2i nontrivial solutions such that x0=xα, x′0=0,−a/b<xα<a/b, xα≠0.The proof of Theorem2 can be found in the articles [6].Consider the Cauchy problem (25)
(28)x0=xα,x′0=0,−x1<xα<x2,xα≠0,where x1 and x2 are critical points of equation (25).We know from Theorem1 that the number of (nontrivial) solution of the BVP (15) and (25) is at least 2i. We wish to illustrate this.Example 1.
Consider equation (20), where a=50 and b=25:
(29)x′′=−50x+25x3,with the initial conditions x0=xα, x′0=0, and 0<xα<2, and then the number of nontrivial solutions satisfying the boundary conditions (15) is two, and for initial conditions x0=xα, x′0=0, and −2<xα<0, there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 2. This is the case for i=2 (namely, 2π<50<3π) in the inequality (27).Consider equation (25), where a=50, b=25, and μ=1:
(30)x′′=−x′2−50x+25x3with the initial conditions xα>0, and then the number of nontrivial solutions satisfying the boundary conditions (15) is two. Graphs xt for solutions of problems (15) and (30), where xα>0, are depicted in Figure 4, but graphs x′t are depicted in Figure 5. For initial conditions xα<0, there are also two solutions to the problem. Graphs xt for solutions of problems (12) and (30), where xα<0, are depicted in Figure 6, but graphs x′t are depicted in Figure 7. Totally the Neumann problems (15) and (30) have four solutions. This is in accordance with Theorem 1.Figure 4
Graphsxt for solutions of problems (15) and (30), xα≈0.352, xα≈0.40228 (dashed).Figure 5
Graphsx′t for solutions of problems (15) and (30), xα≈0.352, xα≈0.40228 (dashed).Figure 6
Graphsxt for solutions of problems (15) and (30) xα≈−0.74, xα≈−1.4092 (dashed).Figure 7
Graphsx′t for solutions of problems (15) and (30) xα≈−0.74, xα≈−1.4092 (dashed).


### 4.1.4. Number of Solution for the Dirichlet Problem

Question. As we see in Figure 8, two heteroclinic solutions have disappeared and were replaced by a homoclinic solution. The number of solutions for the Neumann problems for equations x′′+ax−bx3=0 and x′′+μx′2+ax−bx3=0 is, probably, the same. As to the Dirichlet problem with zero boundary conditions, extra solution(s) can appear outside of the region, bounded by homoclinic trajectory. Is this the case for the example in Figure 8 (and generally)?Figure 8
The phase portrait ofx′′+μx′2+ax−bx3=0, a=50, b=25, and μ=1.Consider the Dirichlet problem (25),
(31)x0=x1=0.Theorem 3.
Leti be a positive integer such that
(32)iπ<a<i+1π.
The Dirichlet problems (20) and (31) has exactly 2i nontrivial solutions such that x0=0, x′0=α, and −αmax<α<αmax, where αmax≕a/2b, α≠0.The proof of Theorem3 can be found in the articles [6].Example 2.
Consider equation (29), with the initial conditions x0=0, x′0=α, and 0<α<52, and then the number of solutions satisfying the boundary conditions (31) is two, and for initial conditions x0=0, x′0=α, and −52<α<0, there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 3 for the case for i=2 (namely, 2π<50<3π) in the inequality (32).Consider equation (30) with the initial conditions 0<α<αhomoclinic, then the number of solutions satisfying the boundary conditions (31) is one. The reason for this difference comparing with “short” equation (29) is that the phase portrait (depicted in Figure 8) is not symmetric with respect to the vertical axis, and the rotation is faster in the left half-plane. The graph xt for a solution of problems (30) and (31), where 0<α<αhomoclinicα≈3.1, is depicted in Figure 9. In the same picture, an additional solution of the problem with αhomoclinic<α<α0α≈16.4802 is shown. The value α0 corresponds to the intersection point of a trajectory going to the critical point (right of the homoclinic region) with the x′ axis. For the negative initial conditions −αhomoclinic<α<0, there are two solutions to the problem. The graphs xt for solutions of problems (30) and (31), where α<0, are depicted in Figure 10. Totally the Dirichlet problems (30) and (31) has four solutions. Of them three solutions are in the region bounded by the homoclinic trajectory, and one solution lies outside this region.Theorem 4.
The Dirichlet problems (25) and (31) have at least 2i−1 nontrivial solutions for the initial conditions in the region bounded with homoclinic solutions. The extra solution(s) can appear outside of this region.Figure 9
Graphsxt for solutions of problems (30) and (31), α≈16.4802, α≈3.1 (dashed).Figure 10
Graphsxt for solutions of problems (30) and (31), α≈−3.11, α≈−3.6605 (dashed).The phase portrait is similar to that for a quadratic equationx′′=−ax+bx2. Therefore, the proof about number of solutions is similar also.Proof.
Consider the phase portrait for equation (25) depicted in Figure 8. The trajectory of any nontrivial solution of problems (25) and (31) for 0<α<αhomoclinic is located inside the region bounded by homoclinic orbits. Introduce the notation G for this region.
Consider solutions of the Cauchy problem (25),
(33)x0=0,x′0=α,−αhomoclinic<α<αhomoclinic,αhomoclinic≠0.
Solutions forα small enough behave like solutions of the equation of variations y′′=−ay around the trivial solution. Due to the assumption iπ≤a<i+1π, solutions yt along with solutions xt;α (for small enough α) have exactly i zeros in the interval 0,1, but for α large, the first zero t1α tends to zero and stays in the interval. Therefore, for 0<α<αhomoclinic, there exist exactly i−1 solutions.
Consider−αhomoclinic<α<0. Solutions for α small enough have i zeros. These zeros move monotonically to the right as α increases in modulus and negative. Solutions xt;α with −αhomoclinic<α and close enough to −αhomoclinic have no zeros in 0,1 since the respective trajectories are close to the homoclinic, therefore, exactly i solutions of problems (25) and (31).
The extra solution(s) can appear outside of this region if initial condition isαhomoclinic<α<α0. The value α0 is the intersection point of the phase trajectory, which starts at the critical point, with the x′ axis.


## 4.1.1. Number of Solution for the Neumann Problem

Consider the Neumann problem(14)x′′+μx′2+gx=0,(15)x′0=x′1=0.The equation (14) turns to the equation
(16)u′′+gxuμu+1=0,wheregx=ax−bx3,or takes the form
(17)u′′+1/μlnμu+1a−b/μ2ln2μu+1μu+1=0.Sincedx/dt=dx/dudu/dt=1/du/dxdu/dt=e−Fxdu/dt, e−Fx>0, the boundary conditions (15) for ut are
(18)u′0=u′1=0.Any solutionxt of problems (14) and (15) corresponds to a solution ut=∫0xteFsds of problems (17) and (18). Therefore, we can estimate the number of solutions of problems (17) and (18). Let us linearize equation (17) at the critical point u=0. One has
(19)dgxuμu+1du=dgxuduμu+1+dμu+1dugxu=dgxdxdxduμu+1+μgxu=dax−bx3dx1du/dxμu+1+μgxu=a−3bx2u1eFxμu+1+μgxu=a−3bx2ue−Fxμu+1+μgxu.Evaluating (19) at u=0 and taking into account that xuu=0=0,F0=0, and g0=0, one gets that the right side in (19) is a and the linearized equation for (17) at u=0 is y′′+ay=0. This corresponds to the linearization of equation
(20)x′′=−ax+bx3,a>0,b>0at x=0.Hence the following statement is valid.Theorem 1.
The numberN of solutions of problems (14) and (15) is not less than the number n of solutions of problems (15) and (20).
The number of nontrivial solutions of the BVP(21)x′′=−ax+bx3,x′0=0,x′1=0is exactly 2n, where n is the number of extrema of a solution xt;α of the Cauchy problem x0=0,x′0=α in the open interval 0,1 for α>0 and arbitrarily small. This follows from the monotonocity ([7]) of the period Tα, when α changes from zero to αheteroclinic, where the last value corresponds to the heteroclinic solution.


## 4.1.2. Phase Portraits

The comparison differential equation(22)x′′+gx=x′′+ax−bx3=0,and the equivalent system
(23)x′=y,y′=−ax+bx3have three critical points at −a/b,0,0,0, and a/b,0.The equation (16) and the equivalent system
(24)u′=v,v′=−gxuμu+1have also three critical points at u1,0,0,0, and u3,0, where u1 and u3 are such that xu1=−a/b, and xu3=a/b. The behavior of solutions of (23) in the region between two heteroclinic trajectories and the behavior of solutions of (24) in the region inside the homoclinic solution are similar. All of them are periodic, and periods are defined for α small by the same linearization.The phase portraits for systems (23) and (24), a=50,b=25, and μ=1, are depicted in Figures 2 and 3.Figure 2
The phase portrait of system (23), a=50, b=25.Figure 3
The phase portrait of system (24).


## 4.1.3. Solutions for the Neumann Problem

The number of solutions of the Neumann problem(25)x′′+μx′2+ax−bx3=0,(15) is not less than that for problems (15) and (20).Consider the Cauchy problem (20),
(26)x0=xα,x′0=0,−ab<xα<ab,xα≠0.Theorem 2.
Leti be a positive integer such that
(27)iπ<a<i+1π.
The Neumann problems (15) and (20) have exactly 2i nontrivial solutions such that x0=xα, x′0=0,−a/b<xα<a/b, xα≠0.The proof of Theorem2 can be found in the articles [6].Consider the Cauchy problem (25)
(28)x0=xα,x′0=0,−x1<xα<x2,xα≠0,where x1 and x2 are critical points of equation (25).We know from Theorem1 that the number of (nontrivial) solution of the BVP (15) and (25) is at least 2i. We wish to illustrate this.Example 1.
Consider equation (20), where a=50 and b=25:
(29)x′′=−50x+25x3,with the initial conditions x0=xα, x′0=0, and 0<xα<2, and then the number of nontrivial solutions satisfying the boundary conditions (15) is two, and for initial conditions x0=xα, x′0=0, and −2<xα<0, there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 2. This is the case for i=2 (namely, 2π<50<3π) in the inequality (27).Consider equation (25), where a=50, b=25, and μ=1:
(30)x′′=−x′2−50x+25x3with the initial conditions xα>0, and then the number of nontrivial solutions satisfying the boundary conditions (15) is two. Graphs xt for solutions of problems (15) and (30), where xα>0, are depicted in Figure 4, but graphs x′t are depicted in Figure 5. For initial conditions xα<0, there are also two solutions to the problem. Graphs xt for solutions of problems (12) and (30), where xα<0, are depicted in Figure 6, but graphs x′t are depicted in Figure 7. Totally the Neumann problems (15) and (30) have four solutions. This is in accordance with Theorem 1.Figure 4
Graphsxt for solutions of problems (15) and (30), xα≈0.352, xα≈0.40228 (dashed).Figure 5
Graphsx′t for solutions of problems (15) and (30), xα≈0.352, xα≈0.40228 (dashed).Figure 6
Graphsxt for solutions of problems (15) and (30) xα≈−0.74, xα≈−1.4092 (dashed).Figure 7
Graphsx′t for solutions of problems (15) and (30) xα≈−0.74, xα≈−1.4092 (dashed).


## 4.1.4. Number of Solution for the Dirichlet Problem

Question. As we see in Figure 8, two heteroclinic solutions have disappeared and were replaced by a homoclinic solution. The number of solutions for the Neumann problems for equations x′′+ax−bx3=0 and x′′+μx′2+ax−bx3=0 is, probably, the same. As to the Dirichlet problem with zero boundary conditions, extra solution(s) can appear outside of the region, bounded by homoclinic trajectory. Is this the case for the example in Figure 8 (and generally)?Figure 8
The phase portrait ofx′′+μx′2+ax−bx3=0, a=50, b=25, and μ=1.Consider the Dirichlet problem (25),
(31)x0=x1=0.Theorem 3.
Leti be a positive integer such that
(32)iπ<a<i+1π.
The Dirichlet problems (20) and (31) has exactly 2i nontrivial solutions such that x0=0, x′0=α, and −αmax<α<αmax, where αmax≕a/2b, α≠0.The proof of Theorem3 can be found in the articles [6].Example 2.
Consider equation (29), with the initial conditions x0=0, x′0=α, and 0<α<52, and then the number of solutions satisfying the boundary conditions (31) is two, and for initial conditions x0=0, x′0=α, and −52<α<0, there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 3 for the case for i=2 (namely, 2π<50<3π) in the inequality (32).Consider equation (30) with the initial conditions 0<α<αhomoclinic, then the number of solutions satisfying the boundary conditions (31) is one. The reason for this difference comparing with “short” equation (29) is that the phase portrait (depicted in Figure 8) is not symmetric with respect to the vertical axis, and the rotation is faster in the left half-plane. The graph xt for a solution of problems (30) and (31), where 0<α<αhomoclinicα≈3.1, is depicted in Figure 9. In the same picture, an additional solution of the problem with αhomoclinic<α<α0α≈16.4802 is shown. The value α0 corresponds to the intersection point of a trajectory going to the critical point (right of the homoclinic region) with the x′ axis. For the negative initial conditions −αhomoclinic<α<0, there are two solutions to the problem. The graphs xt for solutions of problems (30) and (31), where α<0, are depicted in Figure 10. Totally the Dirichlet problems (30) and (31) has four solutions. Of them three solutions are in the region bounded by the homoclinic trajectory, and one solution lies outside this region.Theorem 4.
The Dirichlet problems (25) and (31) have at least 2i−1 nontrivial solutions for the initial conditions in the region bounded with homoclinic solutions. The extra solution(s) can appear outside of this region.Figure 9
Graphsxt for solutions of problems (30) and (31), α≈16.4802, α≈3.1 (dashed).Figure 10
Graphsxt for solutions of problems (30) and (31), α≈−3.11, α≈−3.6605 (dashed).The phase portrait is similar to that for a quadratic equationx′′=−ax+bx2. Therefore, the proof about number of solutions is similar also.Proof.
Consider the phase portrait for equation (25) depicted in Figure 8. The trajectory of any nontrivial solution of problems (25) and (31) for 0<α<αhomoclinic is located inside the region bounded by homoclinic orbits. Introduce the notation G for this region.
Consider solutions of the Cauchy problem (25),
(33)x0=0,x′0=α,−αhomoclinic<α<αhomoclinic,αhomoclinic≠0.
Solutions forα small enough behave like solutions of the equation of variations y′′=−ay around the trivial solution. Due to the assumption iπ≤a<i+1π, solutions yt along with solutions xt;α (for small enough α) have exactly i zeros in the interval 0,1, but for α large, the first zero t1α tends to zero and stays in the interval. Therefore, for 0<α<αhomoclinic, there exist exactly i−1 solutions.
Consider−αhomoclinic<α<0. Solutions for α small enough have i zeros. These zeros move monotonically to the right as α increases in modulus and negative. Solutions xt;α with −αhomoclinic<α and close enough to −αhomoclinic have no zeros in 0,1 since the respective trajectories are close to the homoclinic, therefore, exactly i solutions of problems (25) and (31).
The extra solution(s) can appear outside of this region if initial condition isαhomoclinic<α<α0. The value α0 is the intersection point of the phase trajectory, which starts at the critical point, with the x′ axis.


## 5. Case 2: Equationx′′+μx2−1x′2+gx=0

The Liénard equation is(34)x′′−μ1−x2x′+x=0.We take the coefficient of the quadratic term from (34) and consider the equation
(35)x′′+μx2−1x′2+gx=0,where gx will be specified later. One has fx=μx2−1, Fx=μ1/3x3−x, and ux=∫0xeμ1/3s3−sds.In Figure11, the functions μ1/3x3−x (blue), eμ1/3x3−x (red), and ux=∫0xeμ1/3s3−sds (green) are depicted for μ=1 (dashed) and μ=5 (solid).Figure 11
The graphs ofμ1/3x3−x (blue), eμ1/3x3−x (red), and ux=∫0xeμ1/3s3−sds (green), μ=1 (dashed), and μ=5 (solid).The functionsux have horizontal asymptote for x⟶−∞, and the inverse functions xu have vertical asymptote at u∗=∫0−∞eμ1/3s3−sds. This means that the phase plane for the respective equation u′′+Hu=0 is restricted to u∗<u<+∞. This does not affect the number of solutions to the Neumann boundary value problem for the equation (35).The phase portrait ofx′′+μx2−1x′2+ax−bx3=0, a=50, b=25, μ=1, is depicted in Figure 12.Figure 12
The phase portrait ofx′′+μx2−1x′2+ax−bx3=0, a=50, b=25, μ=1.


### 5.1. Comparison ofx′′+ax−bx3=0 and x′′+μx2−1x′2+ax−bx3=0

Consider equation(36)x′′+μx2−1x′2+gx=0,gx=ax−bx3.Generally, as in Case 1, we can say that the following is true.Theorem 5.
The numberN of solutions of problems (15) and(36) is not less than the number n of solutions of problems (15) and (20)The reason for this will be discussed in the Conclusions section.


#### 5.1.1. Number of Solution for the Neumann Problem

The number of solution of problems (15) and (20) is defined in Theorem 2. Consider equation (36), where a=50, b=25, and μ=1:
(37)x′′=−x2−1x′2−50x+25x3,with the initial conditions xα>0, and then the number of solutions satisfying the boundary conditions (15) is two. Graphs xt for solutions of problems (15) and (37), where 0<xα<xhomoclinic (in this example, xhomoclinic≈1.42), are depicted in Figure 13, but graphs x′t are depicted in Figure 14. For initial conditions −0.46<xα<0, there are also two solutions to the problem. Graphs xt for solutions of problems (15) and (37), where xα<0, are depicted in Figure 15, but graphs x′t are depicted in Figure 16. Totally the Neumann problems (15) and (37) have four solutions. This is the same number of solutions as in Theorem 2 for problems (15) and (20). Therefore the Theorem 5 is fulfilled.Figure 13
Graphsxt for solutions of problems (15) and (37), xα≈0.79, xα≈1.4115 (dashed).Figure 14
Graphsx′t for solutions of problems (15) and (37), xα≈0.79, xα≈1.4115 (dashed).Figure 15
Graphsxt for solutions of problems (15) and (30), xα≈−0.375, xα≈−0.45568 (dashed).Figure 16
Graphsx′t for solutions of problems (15) and (30), xα≈−0.375, xα≈−0.45568 (dashed).


#### 5.1.2. Number of Solution for the Dirichlet Problem

For Case 2, the number of solution for Dirichlet problem is the same as defined in Theorem4 for Case 1: equation x′′+μx′2+gx=0, gx=ax−bx3. The Dirichlet problems (31) and (37) have three solutions in region bounded with homoclinic trajectory and one solution outside this region (see in Figures 17 and 18).Figure 17
Graphsxt for solutions of problem (31) and (37), α≈3.367, α≈4.214 (dashed).Figure 18
Graphsxt for solutions of problem (31) and (37), α≈−3.378, α≈−12.3317 (dashed).


## 5.1. Comparison ofx′′+ax−bx3=0 and x′′+μx2−1x′2+ax−bx3=0

Consider equation(36)x′′+μx2−1x′2+gx=0,gx=ax−bx3.Generally, as in Case 1, we can say that the following is true.Theorem 5.
The numberN of solutions of problems (15) and(36) is not less than the number n of solutions of problems (15) and (20)The reason for this will be discussed in the Conclusions section.


### 5.1.1. Number of Solution for the Neumann Problem

The number of solution of problems (15) and (20) is defined in Theorem 2. Consider equation (36), where a=50, b=25, and μ=1:
(37)x′′=−x2−1x′2−50x+25x3,with the initial conditions xα>0, and then the number of solutions satisfying the boundary conditions (15) is two. Graphs xt for solutions of problems (15) and (37), where 0<xα<xhomoclinic (in this example, xhomoclinic≈1.42), are depicted in Figure 13, but graphs x′t are depicted in Figure 14. For initial conditions −0.46<xα<0, there are also two solutions to the problem. Graphs xt for solutions of problems (15) and (37), where xα<0, are depicted in Figure 15, but graphs x′t are depicted in Figure 16. Totally the Neumann problems (15) and (37) have four solutions. This is the same number of solutions as in Theorem 2 for problems (15) and (20). Therefore the Theorem 5 is fulfilled.Figure 13
Graphsxt for solutions of problems (15) and (37), xα≈0.79, xα≈1.4115 (dashed).Figure 14
Graphsx′t for solutions of problems (15) and (37), xα≈0.79, xα≈1.4115 (dashed).Figure 15
Graphsxt for solutions of problems (15) and (30), xα≈−0.375, xα≈−0.45568 (dashed).Figure 16
Graphsx′t for solutions of problems (15) and (30), xα≈−0.375, xα≈−0.45568 (dashed).


### 5.1.2. Number of Solution for the Dirichlet Problem

For Case 2, the number of solution for Dirichlet problem is the same as defined in Theorem4 for Case 1: equation x′′+μx′2+gx=0, gx=ax−bx3. The Dirichlet problems (31) and (37) have three solutions in region bounded with homoclinic trajectory and one solution outside this region (see in Figures 17 and 18).Figure 17
Graphsxt for solutions of problem (31) and (37), α≈3.367, α≈4.214 (dashed).Figure 18
Graphsxt for solutions of problem (31) and (37), α≈−3.378, α≈−12.3317 (dashed).


## 5.1.1. Number of Solution for the Neumann Problem

The number of solution of problems (15) and (20) is defined in Theorem 2. Consider equation (36), where a=50, b=25, and μ=1:
(37)x′′=−x2−1x′2−50x+25x3,with the initial conditions xα>0, and then the number of solutions satisfying the boundary conditions (15) is two. Graphs xt for solutions of problems (15) and (37), where 0<xα<xhomoclinic (in this example, xhomoclinic≈1.42), are depicted in Figure 13, but graphs x′t are depicted in Figure 14. For initial conditions −0.46<xα<0, there are also two solutions to the problem. Graphs xt for solutions of problems (15) and (37), where xα<0, are depicted in Figure 15, but graphs x′t are depicted in Figure 16. Totally the Neumann problems (15) and (37) have four solutions. This is the same number of solutions as in Theorem 2 for problems (15) and (20). Therefore the Theorem 5 is fulfilled.Figure 13
Graphsxt for solutions of problems (15) and (37), xα≈0.79, xα≈1.4115 (dashed).Figure 14
Graphsx′t for solutions of problems (15) and (37), xα≈0.79, xα≈1.4115 (dashed).Figure 15
Graphsxt for solutions of problems (15) and (30), xα≈−0.375, xα≈−0.45568 (dashed).Figure 16
Graphsx′t for solutions of problems (15) and (30), xα≈−0.375, xα≈−0.45568 (dashed).


## 5.1.2. Number of Solution for the Dirichlet Problem

For Case 2, the number of solution for Dirichlet problem is the same as defined in Theorem4 for Case 1: equation x′′+μx′2+gx=0, gx=ax−bx3. The Dirichlet problems (31) and (37) have three solutions in region bounded with homoclinic trajectory and one solution outside this region (see in Figures 17 and 18).Figure 17
Graphsxt for solutions of problem (31) and (37), α≈3.367, α≈4.214 (dashed).Figure 18
Graphsxt for solutions of problem (31) and (37), α≈−3.378, α≈−12.3317 (dashed).


## 6. Case 3: Equationx′′+x/1−x2x′2+gx=0

### 6.1. Transformation

The equation(38)x′′+x1−x2x′2+gx=0,x<1was considered in the paper [10], where the oscillatory behaviors of solutions were compared for equation (38) and shortened equation x′′+gx=0. The following result was provided.Proposition 6.
The equation (38) by Sabatini’s transformation turns to equation
(39)u′′+Hu=0,Hu=gsinucosu,u∈−π2,π2.The following motivation was used.(40)fx=x1−x2,Fx=∫0xfsds=ln11−x2,eFx=11−x2,u=Φx=∫0xeFsds=∫0x11−s2ds=arcsinx,x<1,x=xu=sinu,u<π2.Letgx=−ax+bx3. Consider the respective equation
(41)u′′+asinu−bsin3ucosu=0.


### 6.2. Number of Solutions

It is an easy matter to check that the linearized equation for the equation (39) around the trivial solution ut≡0 is v′′+av=0. Situation differs, however, from cases 1 and 2. The function gx can have zeros outside the interval −1,1, where equation (38) is defined. This is the case for gx=−50x+25x3. This function was used for comparison of results in the previous cases 1 and 2. Now the critical points ±2 are located outside the interval −1,1.In Figures19 and 20 the central parts of the phase portraits of the equations (38) and (41) are depicted. The results of calculations, visualized in Figures 21 and 22, show that the angular speeds for the trajectories both for the equations (38) and (41) are not monotone with respect to x′0 and u′0. Therefore, periods of periodic solutions are not monotone functions of the initial data, and Theorem 3 and Theorem 4 are not applicable.Figure 19
The central part of the phase portrait of the equation (38), where gx=−50x+25x3.Figure 20
The central part of the phase portrait of the equation (41), where a=50, and b=25.Figure 21
Solutions of the Cauchy problems (38), x0=0, x′0=i, and i=1,2,⋯,30 on the interval 0,π/7.Figure 22
Solutions of the Cauchy problems (41), u0=0, u′0=i, and i=2,4,⋯,15 on the interval 0,π/7.The following, however, is true.Theorem 7.
Leta/b>1 and π/a<1.
Then the Dirichlet problem(42)x′′+x1−x2x′2+ax−bx3=0,x<1,x0=0,x1=0,and the Neumann problem
(43)x′′+x1−x2x′2+ax−bx3=0,x<1,x′0=0,x′1=0have nontrivial solutions.


## 6.1. Transformation

The equation(38)x′′+x1−x2x′2+gx=0,x<1was considered in the paper [10], where the oscillatory behaviors of solutions were compared for equation (38) and shortened equation x′′+gx=0. The following result was provided.Proposition 6.
The equation (38) by Sabatini’s transformation turns to equation
(39)u′′+Hu=0,Hu=gsinucosu,u∈−π2,π2.The following motivation was used.(40)fx=x1−x2,Fx=∫0xfsds=ln11−x2,eFx=11−x2,u=Φx=∫0xeFsds=∫0x11−s2ds=arcsinx,x<1,x=xu=sinu,u<π2.Letgx=−ax+bx3. Consider the respective equation
(41)u′′+asinu−bsin3ucosu=0.


## 6.2. Number of Solutions

It is an easy matter to check that the linearized equation for the equation (39) around the trivial solution ut≡0 is v′′+av=0. Situation differs, however, from cases 1 and 2. The function gx can have zeros outside the interval −1,1, where equation (38) is defined. This is the case for gx=−50x+25x3. This function was used for comparison of results in the previous cases 1 and 2. Now the critical points ±2 are located outside the interval −1,1.In Figures19 and 20 the central parts of the phase portraits of the equations (38) and (41) are depicted. The results of calculations, visualized in Figures 21 and 22, show that the angular speeds for the trajectories both for the equations (38) and (41) are not monotone with respect to x′0 and u′0. Therefore, periods of periodic solutions are not monotone functions of the initial data, and Theorem 3 and Theorem 4 are not applicable.Figure 19
The central part of the phase portrait of the equation (38), where gx=−50x+25x3.Figure 20
The central part of the phase portrait of the equation (41), where a=50, and b=25.Figure 21
Solutions of the Cauchy problems (38), x0=0, x′0=i, and i=1,2,⋯,30 on the interval 0,π/7.Figure 22
Solutions of the Cauchy problems (41), u0=0, u′0=i, and i=2,4,⋯,15 on the interval 0,π/7.The following, however, is true.Theorem 7.
Leta/b>1 and π/a<1.
Then the Dirichlet problem(42)x′′+x1−x2x′2+ax−bx3=0,x<1,x0=0,x1=0,and the Neumann problem
(43)x′′+x1−x2x′2+ax−bx3=0,x<1,x′0=0,x′1=0have nontrivial solutions.


## 7. Conclusions

The Neumann (and generally the Dirichlet) boundary value problems for the equation(44)x′′+gx=0,gx=ax−bx3,a,b>0are well studied. The exact number of solutions is known and it depends on the coefficient a only. The equation
(45)x′′+fxx′2+gx=0,gx=ax−bx3,a,b>0can be significantly more complicated, due to the presence of fxx′2. It appears nevertheless, that the number and the type of critical points for (44) and (45) coincide. The analysis of the equation (45) can be made easier by passing to the Newtonian equation
(46)u′′+Hu=0,Hu=axu−bx3ueFxu,where xu is the inverse function to u=∫0xeFsds, Fx is the primitive of fx. The critical points of (46) and their types are the same as the ones for the equations (44) and (45). It can be shown, that the linearized equation for the trivial solution of (46) is also v′′+av=0.The exact number of solutions to the Neumann problem for (44) was established, using the monotonicity of the period function. This cannot be guaranteed for the equation (46).If the variableu=∫0xeFx has asymptotes, the inverse variable xu can be defined on a semibounded or bounded interval. The phase space for the modified equation u′′+Hu=0 can be a strip or a half-plane.It may happen that the region between two heteroclinic trajectories (as in equationx′′=−ax+bx3) becomes a region inside a homoclinic loop for the modified equation. Therefore, the phase potraits for the equations x′′+gx=0 and u′′+Hu=0 generally are not topologically equivalent, despite of the fact that critical points are in one-to one correspondence and the characteristic types coincide.For the Dirichlet problem, the number of solutions for the equationsx′′+gx=0 and u′′+Hu=0 may be the same, but their nature in phase plane can be quite different. The symmetry with respect to the vertical axis may be broken in the equation u′′+Hu=0.Introducing the quadratic term into the equationx′′+gx=0 may heavily influence the behavior of solutions and the structure of the phase plane.



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