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.
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 where is a positive integer (most results concern the cases , , and ). The exact estimates of the number of solutions were obtained for autonomous equations of the form , and some results (mostly of computational nature) were stated for the case of 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 together with the two-point boundary conditions.
Equations of the form are 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 and 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
We focus on the case , , , 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 Form
Consider equation (2). We suppose that and are continuous functions. Introduce the variable by the transformation where The primitive can change sign arbitrarily, since there are no sign restrictions on but Therefore is a positive valued function with no restrictions on monotonicity. In particular, the integral or may be convergent, and the range of values of a new variable can be bounded. Since the function is monotonically increasing, passing through zero, The inverse function exists, but it can be defined on a bounded or a semi-bounded interval only.
The equation (2) takes the form where
The equation (6) can be considered instead of (2). There is a one-to-one correspondence between and
3. Critical Points and Linearization
In the study of multiplicity of solutions to equations and/or , a significant role was played by the linearization at the trivial solution For instance, linearization of at the trivial solution yields the linear equation , and estimations of the number of solutions of the Dirichlet and the Neumann problem were made in terms of this
Let us clarify this question in a general setting. Consider the conservative equation (4), where in several points including Then there is the trivial solution of the Dirichlet and the Neumann problems with zero boundary conditions. Linearization around the trivial solution yields
Notice that need not be zero.
Consider now perturbed equation with the quadratic term (2): provided the same assumptions on The critical points for an equivalent system and are given by which reduces to . So the critical points for equations (2) and (4) are the same.
Linearization of (2) at the trivial solution gives or which is the same as (8).
The estimates for the equation (and/or ) 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: Equation
Consider the equation where is given. We will assume that is a particular cubic polynomial. To apply the Sabatini transformation, we denote Introduce the new variable
Comparison will be made with the equation . For positive (we assume this), is the function, monotonically increasing from to The inverse function exists.
The equation (6) takes the form
In Figure 1, the functions (solid) are depicted (blue for and red for ) together with the inverse functions (dashed).
If the equation (12) turns to
4.1. Comparison of and
4.1.1. Number of Solution for the Neumann Problem
Consider the Neumann problem
The equation (14) turns to the equation or takes the form
Since , the boundary conditions (15) for are
Any solution of problems (14) and (15) corresponds to a solution 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 One has
Evaluating (19) at and taking into account that and one gets that the right side in (19) is and the linearized equation for (17) at is This corresponds to the linearization of equation at Hence the following statement is valid.
Theorem 1. The number of solutions of problems (14) and (15) is not less than the number of solutions of problems (15) and (20).
The number of nontrivial solutions of the BVP
is exactly , where is the number of extrema of a solution of the Cauchy problem in the open interval for and arbitrarily small. This follows from the monotonocity ([7]) of the period , when changes from zero to , where the last value corresponds to the heteroclinic solution.
4.1.2. Phase Portraits
The comparison differential equation and the equivalent system have three critical points at , and
The equation (16) and the equivalent system have also three critical points at , and where and are such that and 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 small by the same linearization.
The phase portraits for systems (23) and (24), and , are depicted in Figures 2 and 3.
4.1.3. Solutions for the Neumann Problem
The number of solutions of the Neumann problem (15) is not less than that for problems (15) and (20).
Consider the Cauchy problem (20),
Theorem 2. Let be a positive integer such that The Neumann problems (15) and (20) have exactly nontrivial solutions such that , , .
The proof of Theorem 2 can be found in the articles [6].
Consider the Cauchy problem (25) where and are critical points of equation (25).
We know from Theorem 1 that the number of (nontrivial) solution of the BVP (15) and (25) is at least . We wish to illustrate this.
Example 1. Consider equation (20), where and : with the initial conditions , , and , and then the number of nontrivial solutions satisfying the boundary conditions (15) is two, and for initial conditions , , and , there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 2. This is the case for (namely, ) in the inequality (27).
Consider equation (25), where , , and : with the initial conditions , and then the number of nontrivial solutions satisfying the boundary conditions (15) is two. Graphs for solutions of problems (15) and (30), where , are depicted in Figure 4, but graphs are depicted in Figure 5. For initial conditions , there are also two solutions to the problem. Graphs for solutions of problems (12) and (30), where , are depicted in Figure 6, but graphs are depicted in Figure 7. Totally the Neumann problems (15) and (30) have four solutions. This is in accordance with Theorem 1.
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 and 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)?
Consider the Dirichlet problem (25),
Theorem 3. Let be a positive integer such that The Dirichlet problems (20) and (31) has exactly nontrivial solutions such that , and , where , .
The proof of Theorem 3 can be found in the articles [6].
Example 2. Consider equation (29), with the initial conditions , and , and then the number of solutions satisfying the boundary conditions (31) is two, and for initial conditions , and , there are also two solutions to the problem, totally four solutions. This is in accordance with Theorem 3 for the case for (namely, ) in the inequality (32).
Consider equation (30) with the initial conditions , 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 for a solution of problems (30) and (31), where , is depicted in Figure 9. In the same picture, an additional solution of the problem with is shown. The value corresponds to the intersection point of a trajectory going to the critical point (right of the homoclinic region) with the axis. For the negative initial conditions , there are two solutions to the problem. The graphs for solutions of problems (30) and (31), where , 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 nontrivial solutions for the initial conditions in the region bounded with homoclinic solutions. The extra solution(s) can appear outside of this region.
The phase portrait is similar to that for a quadratic equation . 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 is located inside the region bounded by homoclinic orbits. Introduce the notation for this region.
Consider solutions of the Cauchy problem (25),
Solutions for small enough behave like solutions of the equation of variations around the trivial solution. Due to the assumption , solutions along with solutions (for small enough ) have exactly zeros in the interval , but for large, the first zero tends to zero and stays in the interval. Therefore, for , there exist exactly solutions.
Consider . Solutions for small enough have zeros. These zeros move monotonically to the right as increases in modulus and negative. Solutions with and close enough to have no zeros in since the respective trajectories are close to the homoclinic, therefore, exactly solutions of problems (25) and (31).
The extra solution(s) can appear outside of this region if initial condition is . The value is the intersection point of the phase trajectory, which starts at the critical point, with the axis.
5. Case 2: Equation
The Liénard equation is
We take the coefficient of the quadratic term from (34) and consider the equation where will be specified later. One has , , and
In Figure 11, the functions (blue), (red), and (green) are depicted for (dashed) and (solid).
The functions have horizontal asymptote for , and the inverse functions have vertical asymptote at This means that the phase plane for the respective equation is restricted to This does not affect the number of solutions to the Neumann boundary value problem for the equation (35).
The phase portrait of , , , , is depicted in Figure 12.
5.1. Comparison of and
Consider equation
Generally, as in Case 1, we can say that the following is true.
Theorem 5. The number of solutions of problems (15) and(36) is not less than the number 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 , , and : with the initial conditions , and then the number of solutions satisfying the boundary conditions (15) is two. Graphs for solutions of problems (15) and (37), where (in this example, ), are depicted in Figure 13, but graphs are depicted in Figure 14. For initial conditions , there are also two solutions to the problem. Graphs for solutions of problems (15) and (37), where , are depicted in Figure 15, but graphs 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.
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 Theorem 4 for Case 1: equation , . 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).
6. Case 3: Equation
6.1. Transformation
The equation was considered in the paper [10], where the oscillatory behaviors of solutions were compared for equation (38) and shortened equation The following result was provided.
Proposition 6. The equation (38) by Sabatini’s transformation turns to equation
The following motivation was used.
Let Consider the respective equation
6.2. Number of Solutions
It is an easy matter to check that the linearized equation for the equation (39) around the trivial solution is . Situation differs, however, from cases 1 and 2. The function can have zeros outside the interval , where equation (38) is defined. This is the case for . This function was used for comparison of results in the previous cases 1 and 2. Now the critical points are located outside the interval .
In Figures 19 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 and . Therefore, periods of periodic solutions are not monotone functions of the initial data, and Theorem 3 and Theorem 4 are not applicable.
The following, however, is true.
Theorem 7. Let and .
Then the Dirichlet problem
and the Neumann problem
have nontrivial solutions.
7. Conclusions
The Neumann (and generally the Dirichlet) boundary value problems for the equation are well studied. The exact number of solutions is known and it depends on the coefficient only. The equation can be significantly more complicated, due to the presence of 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 where is the inverse function to , is the primitive of . 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 .
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 variable has asymptotes, the inverse variable can be defined on a semibounded or bounded interval. The phase space for the modified equation can be a strip or a half-plane.
It may happen that the region between two heteroclinic trajectories (as in equation ) becomes a region inside a homoclinic loop for the modified equation. Therefore, the phase potraits for the equations and 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 equations and 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
Introducing the quadratic term into the equation may heavily influence the behavior of solutions and the structure of the phase plane.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the Daugavpils University, for funding this research work.