Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation
In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus . Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like , eight periodic multiplicities have been observed. The new formulas and are constructed. We used our new formulas to find the maximum multiplicity for class . We have succeeded to determine the maximum multiplicity ten for class which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.
Recently, the bifurcation analysis has attracted the attention of many researchers because it has wild applications in dynamics system and the universal existence of bifurcation in the nature, for example, bifurcation occurs when the small smooth change of parameters leads to qualitative change of its behavior (see ), aerodynamic limit cycle oscillation, and nonlinear oscillation in power system; the homoclinic and heteroclinic branches of a limit cycle colliding with one saddle point, two, or more saddle points and multiple biological dynamical systems are also bifurcation.
As nature is changing every moment and many changes occurring in nature are periodic like weather, blood flow inside body, circadian rhythm, oceans, and even human behavior, the study of theory of periodic or almost periodic solution is gaining attention. These periodic cycles are often observed to explore the impact of environmental factors in mathematical biology, food supplies, and harvesting. One of the classical mathematics problems is the theoretical calculation of the periodic solution of planar vector field.
The question about the number of periodic solution for nonautonomous differential equation continues to attract more interest. Neto in  states that for equation (1), we are unable to have upper bound for number of periodic solutions until some coefficients are restricted. Kadry in  gives conditions that allow determination for both upper and lower bounds. Our basic focus is to acquire highest periodic solutions of any class of the type (1); our main concern in this paper is this nearby question of bifurcation. Without loss of generality for above said arguments, we are considering the differential equation of the formwhere independent variable and coefficients are real valued functions but . To find maximum number of periodic solutions, we use complexified form of equation (1) (for more details, see [2, 4–6]). The above equation is the reminiscent of the equation described in Lloyd :with the assumption that and were periodic functions having same period. It was shown that for in (2), there are three periodic solutions; also, the equation takes form as Abel’s differential equation of first kind. It is important because of its connection with the well-known Hilbert’s sixteenth problem  for differential equations with polynomial coefficients:
Here, and are polynomials in and . Then, Hilbert’s sixteenth problem is transformed to polynomial equation, where leading coefficient randomly changes its sign.
For equation (1), complexified form is used so that we can take the maximum number of periodic solutions for each class using the perturbation method. For this reason, periodic solutions cannot be destroyed by any small perturbation of the coefficients. Consider that for equation (1) there exists such that
These solutions are periodic, even if , , and are not themselves periodic. Limit cycles bifurcate out of the fine focus when the coefficients of and are slightly perturbed. By this method, the obtained limit cycles are said to be of small amplitude. The number of periodic solutions depends upon the multiplicity of solution . The multiplicity of as a solution of equation (1) is also a multiplicity of the following displacement function:
For , the method to compute multiplicity “” is explained in ; for the sake of ease, we explained it briefly here. We write for , where lies in the regions near , and use it in equation (5). For more details, see [2, 5, 6, 8] equation, which provide differential equation for , having some starting conditions and for . Therefore,
The multiplicity is “” if
However, . When and , origin is the center. We can observe from equation (1) that , where is defined as
In this way, if
Because , we are especially interested about situation when has multiple solutions. So, we consider that (9) holds by applying the following conversion:
Equation (1) takes the following form:where and
We can see that and are periodic if , and are periodic. By using Lemma in , we consider multiplicity of as periodic solution of (1); if for equation (1), , then the multiplicity of as a periodic solution of (11) is also . So, we consider that in (1). As a result, equation (1) takes the form as
Here and may be polynomials in and in and (trigonometric functions) (for more details, see [4, 9, 10]). The functions , for are calculated by utilizing the present relation:with . However, as “” increases, certain calculations become tremendously complicated because of integration by parts, used in it. Assume that ; at that point, if and for , but , and these are known as focal values. For , functions and are given in ; for , Yasmin in  had calculated and . For , we have calculated and which are given in Section 2. In equation (11), we make restriction for . We do not give the complete detailed derivation because they form a complicated web.
In Section 2, we have discussed classical formulas by which we are able to calculate the maximum multiplicity. With the help of classical results by Alwash given in , we are able to define some conditions w.r.t stopping criteria for our equation (1), and some suitable perturbations are defined in Section 3. Sections 2 and 3 are mainly concerned for calculation of focal values, which are used in Section 4. In Section 4, calculation of focal values is done for equation (13) having polynomial coefficients. In Section 5, some examples are given, and in Section 6, conclusions about periodic multiplicity are discussed.
2. Calculation of the Focal Values
By using these functions, we get Theorem 2 that enables us to find the maximum multiplicity in which integral is like and bar “” shows integral , which is definite.
Theorem 2. The solution of (13) has a multiplicity , wherever if for and where
3. Conditions for Center and Method of Perturbation
In this section, we describe some conditions for center. From Theorem 2, we find maximum value for different classes of nonautonomous equation of the form as (1). Stopping criteria are defined for calculating maximum multiplicity . We need some conditions that assure that there is no need to proceed further (). For this, suitable conditions that are sufficient for as a center are given below in the form of theorems and corollaries.
Theorem 3. Consider that there are continuous functions , defined on and differentiable function with such that
Then, the origin is a center for (13).
Theorem 4. The origin is a center for equation (13) if is a constant multiple of and .
Corollary 1. If any or is identically 0 and other has mean value zero, then the origin is a center.
After determining the maximum multiplicity , we have to make series of perturbation of the coefficients, every one of which results in one periodic solution to come out of origin. For more details, see [4, 5, 12].
For this, suppose equation of the form given below:having multiplicity (suppose). Let be in the regions near 0 in the complex plane containing no periodic solutions except . From Theorem in , the initial point which is contained in remained fixed with respect to total number of periodic solutions with the restriction that perturbation of the coefficients considered remained small enough. Our goal is to get but by perturbing and making suitable choices of and , if possible. Obviously, the most effective solutions in and are zero solution while we get periodic solutions where as nontrivial solutions. By considering the underlying fact that the complex solutions always appear in conjugate pair, we can say that is real. Let and be the neighborhood of zero and , respectively, such that and . The periodic solutions around and are preserved when we take small perturbation in the coefficients. By applying the same procedure as above, our choice is to perturb the coefficients such that for but . So, we get . By applying that procedure, we get two nontrivial real periodic solutions with zero solution having multiplicity . In this way, we end up with equation (18) having and distinct nontrivial (other than zero) real periodic solutions.
4. Polynomial Coefficients for Some Classes
For the polynomial “,” consider indicates the class of equation of the form (13) with degree and for and , respectively. We have evaluated the maximum multiplicity denoted by the symbol “” for some classes like , and that are presented below in the form of theorems. For more classes with maximum multiplicity, see .
Theorem 5. Let be class of equation of form (13), withThen, we conclude .
Proof. Using Theorem 2, we takeThus, multiplicity of is if . And multiplicity is if but . If , then by using value of “” and “,” and are as follows:And we compute as given below:If , then either orIf , then and gives that mean value of is zero. Thus, origin is a center from Corollary 1. So, consider . Now, if (24) holds, then we calculateIf , we consider which implies thatAnd by using (26), we take asIf , we consider and either orIf , then (26) gives , and (24) gives . By using values of , equations (21) and (22) take the following form:Let ; then, . Also, . So, we write above equations asThus, from Theorem 3, origin is the center withand . So, we take . If (28) holds, then we compute asIf , recalling that , then either orIf , thenFrom Theorem 3, origin is the center with and . So, consider . By using (33), we calculate aswhereNow, if , then either orbecause . If (37) holds but , then we compute asIf equation (37), , but holds, then for with and . If recalling that , we take value of asIf (39) holds, then we calculate asHere is equal to constant number which is nonzero. Hence, we conclude that multiplicity of class is 10, i.e., .
Theorem 6. Consider the equation given below:withChoose for to be nonzero and small as compared to . Then, there exist eight real nontrivial periodic solutions for equation (41).
Proof. If we substitute , and coefficients are as written above, then the multiplicity of the origin is . Choose and for ; it can be easily seen that is constant multiple of but , so the multiplicity reduces by one, and . For that reason one periodic solution bifurcates out of the origin. Now, set , and for , and we have for but results in form of with some constant multiple, so . Now, set , and for ; then, we have for but results in form of with some constant multiple. If is sufficiently small, then there are two nontrivial real periodic solutions. Further continuing the same procedure, we own eight real periodic nontrivial solutions.
Proof. Given that , , and , then (43) has eight real periodic solutions. If but small enough, then ; using the same arguments as above, we have nine periodic solutions. These are distinct and other than 0, is another such solution, and thus we take ten real periodic solutions.
Theorem 7. Consider as the class of equations of form (13) having degree 8 for and for ; here . Then, we cease with the results , for all , respectively.
Proof. We first consider the class in which degree of is ., . The classes in which degree of is less than 7 are the special cases. LetBy using Theorem 2, we calculateIf , we calculate asFrom , we cannot substitute any value to proceed further. As a result, we make restriction of the coefficients for class like in the system of equation (44).(1)Let Utilizing Theorem 2, we get Thus, multiplicity of is if . And multiplicity if but . If , then we calculate as If , then either or If , then gives ; hence, . If , it means that the mean value of is zero. By Corollary 1, origin is a center. Suppose that . By using (50), we compute If , then We have already taken . If (52) holds, then If , then either or because . If , then from (50) and (52), , respectively. By using these values, and take the form as Let ; then, ; also, . So, we can write as Utilizing Theorem 3, origin is the center with and . So, we take . If (54) holds, then is given by Now if , recalling that , then Holding (58), we have That is, constant multiple of and is also nonzero. Thus, we conclude that multiplicity of class is 8, i.e., .(2)For the class , the degree of and ; we suppose that in system of equation (44). Let By using Theorem 2, we calculate Thus, multiplicity of is if . And multiplicity if but . If , then