On the Limit Cycles for a Class of Perturbed Fifth-Order Autonomous Differential Equations

Abstract

We study the limit cycles of the fifth-order differential equation with where is a small enough real parameter, , and are real parameters, and is a nonlinear function. Using the averaging theory of first order, we provide sufficient conditions for the existence of limit cycles of this equation.

1. Introduction and Statement of the Main Results

The study of the limit cycles is one of the main topics of the qualitative theory of differential equations and dynamical systems. A limit cycle of a differential equation is an isolated periodic orbit of this equation; it means that there is no periodic orbits in the vicinity of this limit cycle. There are several theories and methods for the study of the existence, uniqueness, or number and stability of limit cycles of differential equations which have been developed in trying to answer Hilbert’s sixteenth problem posed in 1900 [1] about the maximum number of limit cycles that a planar polynomial differential system can have.

The averaging theory is one of the most important tools used actually to the study of limit cycles for second and higher order differential equations, you can see in [2–8]. More details on the averaging theory can be found in the books of Sanders and Verhulst [9] and of Verhulst [9].

In [7], the authors studied the limit cycles of the following third-order differential equationwith ; is a small real parameter; is periodic in .

In [5], the authors studied equation (1) with which is autonomous. They studied the two cases and .

In [6], the authors studied the following fourth-order differential equation:where and are real, is a small real parameter, and is periodic in .

In [4], the authors studied equation (2) with which is autonomous.

In this paper, we shall use a result of the averaging theory to study the limit cycles of the following class of fifth-order autonomous ordinary differential equations:wherewhere the dot means derivative with respect to an independent variable , is a small enough parameter, and is a nonlinear function. Here, the variable and the parameters and are real.

In [8], the authors studied equation (3) with which depends explicitly on the independent variable . Here, we study the autonomous case using a different approach. Note that our results are distinct and new.

Now, we state our main results for the limit cycles of equation (3).

For the different values of the parameters , and , we distinguish the five following cases.

Case 1. and .

Case 2. , and .

Case 3. and .

Case 4. and .

Case 5. .
For each one of these cases, we will give a theorem which provides sufficient conditions for the existence of limit cycles of equation (3) and we provide also an application.
There are two other cases ( and ) and that we cannot study because they are too much degenerated for Theorem 6.

1.1. Case 1: and

In order to state our results for this case, we define the functionwhere

Our main result for this case is the following theorem.

Theorem 1. Assume that and . For every positive simple zero of the function given by (5) there is a limit cycle of equation (3) tending to the periodic solutionofwhen .
Theorem 1 will be proved in Section 3.1.1.
An application of Theorem 1 is the following.

Corollary 1. Assume that , , , andThen, there is a limit cycle of equation (3) tending to the periodic solutionof equation (8) when .
Corollary 1 will be proved in Section 3.1.2.

1.2. Case 2: , and

We define the functionsand

Our main result for this case is the following theorem.

Theorem 2. Assume that , and . For every zero of the system where and are given by (11) such thatthere is a limit cycle of equation (3) tending to the periodic solutionofwhen .
Theorem 2 will be proved in Section 3.2.1.
An application of Theorem 2 is the following.

Corollary 2. Assume that , , , and , then there is a limit cycle of equation (3) tending to the periodic solutionof equation (15) when .
Corollary 2 will be proved in Section 3.

1.3. Case 3: and

We define the functionswhere

Our main result for this case is the following theorem.

Theorem 3. Assume that and . For every zero of the system where and are given by (17) such thatthere is a limit cycle of equation (3) tending to the periodic solutionofwhen .
Theorem 3 will be proved in Section 3.3.1.
An application of Theorem 3 is the following.

Corollary 3. Assume that , , and , then there is a limit cycle of equation (3) tending to the periodic solutionof equation (21) when .
Corollary 3 will be proved in Section 3.3.2.

1.4. Case 4: and

We define the functionwhere

Our main result for this case is the following theorem.

Theorem 4. Assume that and . For every positive simple zero of the function given by (23), there is a limit cycle of equation (3) tending to the periodic solutionofwhen .
Theorem 4 will be proved in Section 3.4.1.
An application of Theorem 4 is the following.

Corollary 4. Assume that , , , and , then there is a limit cycle of equation (3) tending to the periodic solutionof equation (26) when .
Corollary 4 will be proved in Section 3.4.2.

1.5. Case 5:

In order to state our result for this case, we define the functionwhere

Our main result for this case is the following theorem.

Theorem 5. Assume that . For every positive simple zero of the function given by (28), there is a limit cycle of equation (3) tending to the periodic solutionofwhen .
Theorem 5 will be proved in Section 3.5.1.
An application of Theorem 5 is the following.

Corollary 5. Assume that , , , and , then there is a limit cycle of equation (3) tending to the periodic solutionof equation (31) when .
Corollary 5 will be proved in Section 3.5.2.

2. The Main Tool (First-Order Averaging Theory)

In this section, we present the basic result from the averaging theory that we need for proving the main results of this article.

We consider the problem of the bifurcation of periodic solutions from the differential systemwith to sufficiently small. The functions and are functions, periodic is in the variable , and is an open subset of . We suppose that the unperturbed systemhas a -dimensional submanifold of periodic solutions.

Let be the solution of the unperturbed system (34) such that The linearisation of system (34) along the periodic solution is written as

We denote by some fundamental matrices of the linear differential system (35) and by the projection of onto its first coordinates; i.e., .

Theorem 6. Let be open and bounded and be a function. We assume that(i), and that for each , the solution of (34) is periodic.(ii) For each , there is a fundamental matrix of (35) such that the matrix has in the upper right corner the zero matrix, and in the lower right corner a matrix with . We consider the function If there exists with and , then there is a periodic solution of the system (33) such that as .
Theorem 1 goes back to [10] and [11]; for a shorter proof, see [12].
Note that the periodic orbits provided by Theorem 6 are limit cycles.

3. Proofs of the Results

3.1. Proofs of the Results in Case 1: and

3.1.1. Proof of Theorem 1

We consider equation (3) and put , , , and , then equation (3) can be written as

System (37) with has a unique singular point at the origin and the linear part of this system has the eigenvalues , , , and . Using the change of variableswe transform system (37) into the following system:where

Note that the linear part of system (39) is in the real normal Jordan form of the linear part of system (37). We pass now from the Cartesian coordinates to the cylindrical ones with , , and we obtainwhere .

After dividing by and simplifying, we find

System (42) is now of the same form as system (33) with

We shall apply Theorem 6 to system (42). System (42) with has the periodic solutions

By the notations of Theorem 6, we have that and . Let and ; we take , , and

We also take

The fundamental matrix of the linear system (42) with with respect to the periodic solution satisfying that is the identity matrix is

We havewhich satisfy the assumption (ii) of Theorem 6. Takingwe must compute the function given by (36), and we obtainand , and are given by (6). Then, by Theorem 6, for every simple zero of the function , there exists a limit cycle of system (42) such that

Going back through the change of coordinates, we obtain a limit cycle of system (41) such that

We have a limit cycle of system (39) such that

Finally, we obtain a limit cycle of equation (3) tending to the periodic solution (7) of equation (8) when .

Theorem 1 is proved.

3.1.2. Proof of Corollary 1

If , then we havewhich have the real positive simple zero with

The proof of Corollary 1 follows directly by applying Theorem 1 and (10) is obtained by substituting in (7).

3.2. Proofs of the Results in Case 2: , and

3.2.1. Proof of Theorem 2

If , equation (3) can be written as

System (56) has a unique singular point at the origin and the eigenvalues of the linear part of this system are , 0, , and . By the linear transformationwe transform system (56) into the following system:where

Note that the linear part of system (58) is in the real Jordan normal form of the linear part of system (56). We pass now from the Cartesian coordinates to the cylindrical ones with , , and we obtainwhere .

After dividing by and simplifying, we find

System (61) is now of the same form as system (33) with

We shall apply Theorem 6 to system (61). System (61) with has the periodic solutions

By the notations of Theorem 6, we have that and . Let ; we take, and

We also take

The fundamental matrix of the linear system (61) with with respect to the periodic solution satisfying that is the identity matrix is

We havewhich satisfy the assumption (ii) of Theorem 6. Takingwe must compute the function given by (36), and we obtainwhereand , and are given by (12). Then, by Theorem 6, for every simple zero of the function there exists a limit cycle of system (61) such that

Going back through the change of coordinates, we obtain a limit cycle of system (60) such that

We have a limit cycle of system (58) such that

Finally, we obtain a limit cycle of equation (3) tending to the periodic solution (14) of equation (15) when .

Theorem 2 is proved.

3.2.2. Proof of Corollary 2

If , then we have

The system has the zero such that

The proof of Corollary 2 follows directly by applying Theorem 2 and (16) is obtained by substituting in (14).

3.3. Proofs in Case 3:

3.3.1. Proof of Theorem 3

If , , then equation (3) can be written as

System (77) has a unique singular point at the origin and the eigenvalues of the linear part of this system are , 0, and . By the linear transformationwe transform system (77) into the following system:where

Note that the linear part of system (81) is in the real Jordan normal form of the linear part of system (77). We pass now from the Cartesian coordinates to the cylindrical ones with , , and we obtainwhere .

After dividing by and simplifying, we find

System (82) is now of the same form as system (33) with

We shall apply Theorem 6 to system (82). System (82) with has the periodic solutions

By the notations of Theorem 6, we have that and . Let ; we take

and

We also take

The fundamental matrix of the linear system (82) with with respect to the periodic solution satisfying that is the identity matrix is

We havewhich satisfy the assumption (ii) of Theorem 6. Takingwe must compute the function given by (36), and we obtainwhereand , and are given by (18). Then, by Theorem 6, for every simple zero of the function , there exists a limit cycle of system (82) such that

Going back through the change of coordinates, we obtain a limit cycle of system (81) such that

We have a limit cycle of system (79) such that

Finally, we obtain a limit cycle of equation (3) tending to the periodic solution (20) of equation (21) when .

Theorem 3 is proved.

3.3.2. Proof of Corollary 3

If , then we have

The system has the zero such that

The proof of Corollary 3 follows directly by applying Theorem 3 and (22) is obtained by substituting in (20).

3.4. Proofs in Case 4: and

3.4.1. Proof of Theorem 4

If and , then equation (3) can be written as

System (98) with has a unique singular point at the origin and the linear part of this system has the eigenvalues , , and . Using the change of variableswe transform the system (98) into the following system:where

Note that the linear part of system (100) is in the real normal Jordan form of the linear part of system (98). We pass now from the Cartesian coordinates to the cylindrical ones with , , and we obtainwhere .

After dividing by and simplifying, we find

System (103) is now of the same form as system (33) with

We shall apply Theorem 6 to system (103). System (103) with has the periodic solutions

By the notations of Theorem 6, we have that and . Let and ; we take , , and

We also take

The fundamental matrix of the linear system (103) with with respect to the periodic solution satisfying that is the identity matrix is

We havewhich satisfy the assumption (ii) of Theorem 6. Takingwe must compute the function given by (34), and we obtainand , and are given by (24). Then, by Theorem 6, for every simple zero of the function , there exists a limit cycle of system (103) such that

Going back through the change of coordinates, we obtain a limit cycle of system (102) such that

We have a limit cycle of system such that

Finally, we obtain a limit cycle of equation (3) tending to the periodic solution (25) of equation (26) when .

Theorem 4 is proved.

3.4.2. Proof of Corollary 4

If , then we havewhich have the real positive simple zero with

The proof of Corollary 4 follows directly by applying Theorem 4 and (27) is obtained by substituting in (25).

3.5. Proofs in Case 5:

3.5.1. Proof of Theorem 5

If , then equation (3) can be written as

System (117) with has a unique singular point at the origin and the linear part of this system has the eigenvalues and Using the change of variableswe transform system (117) into the following system:where

Note that the linear part of system (119) is in the real normal Jordan form of the linear part of system (51). We pass now from the Cartesian coordinates to the cylindrical ones with , , and we obtainwhere .

After dividing by and simplifying, we find

System (122) is now of the same form as system (33) with

We shall apply Theorem 6 to system (122). System (122) with has the periodic solutions

By the notations of Theorem 6, we have that and . Let and ; we take , , and

We also take

The fundamental matrix of the linear system (122) with with respect to the periodic solution satisfying that is the identity matrix is

We havewhich satisfy the assumption (ii) of Theorem 6. Takingwe must compute the function given by (36), and we obtainand , and are given by (29). Then, by Theorem 6, for every simple zero of the function , there exists a limit cycle of system (122) such that

Going back through the change of coordinates, we obtain a limit cycle of system (121) such that

We have a limit cycle of system (119) such that

Finally, we obtain a limit cycle of equation (3) tending to the periodic solution (30) of equation (31) when .

Theorem 5 is proved.

3.5.2. Proof of Corollary 5

If , then we havewhich have the real positive simple zero with

The proof of Corollary 5 follows directly by applying Theorem 5 and (32) is obtained by substituting in (30).

4. Conclusion

There are several theories and methods for the study of the existence, uniqueness, or number and stability of limit cycles of differential equations which have been developed in trying to answer Hilbert’s sixteenth problem posed in 1900 (see reference [1]) about the maximum number of limit cycles that a planar polynomial differential system can obtain. In this work, we study the limit cycles of the fifth-order differential equation by using the averaging theory of first order [6, 7], and we provide sufficient conditions for the existence of limit cycles of equation (12); in the next work, we will try to apply the same method on higher order differential equations.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Group Project under Grant no. (R.G.P-2/53/42).