Research Article | Open Access
Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems
We study analytic properties of the Poincaré return map and generalized focal values of analytic planar systems with a nilpotent focus or center. We use the focal values and the map to study the number of limit cycles of this kind of systems and obtain some new results on the lower and upper bounds of the maximal number of limit cycles bifurcating from the nilpotent focus or center. The main results generalize the classical Hopf bifurcation theory and establish the new bifurcation theory for the nilpotent case.
1. Introduction and Main Result
Consider an analytic system of the form: where , , and for near the origin. A Poincaré map can be defined on a cross-section with an endpoint at the origin using positive orbits of the above system and can be written in the form: where the series converges for small . A well-known fact is that for any for all imply that ; that is, only odd values of the expansion are important for determining the behavior of trajectories near the origin. The value is called the th focal value or the th Lyapunov constant. For quadratic systems, Bautin  proved that the Poincaré map can be written in the form: where . This implies that there are at most 3 limit cycles near the origin.
Suppose now that the origin is a nilpotent singular point, so the system is written in the form: where , for near the origin. The following criterion for the existence of a center or a focus at the origin of (1.4) has been established in [2–4].
Theorem 1.1 (see [2–4]). Let (1.4) have an isolated singular point at the origin. Let where is the solution to the equation satisfying . Then the origin of (1.4) is a center or a focus if and only if is negative and .
Introducing the generalized polar coordinates: where is the solution of the initial problem
Liapunov [3, 4] proposed a method to solve the center-focus problem for (1.4). Sadovskiĭ  (see also ) and Moussu  investigated the problem using Lyapunov functions and normal forms, respectively. Other few approaches for computing focal values, Lyapunov constants or equivalent values and methods for studying bifurcations of local limit cycles were suggested by Chavarriga et al. , Giacomini et al. , Álvarez and Gasull [10, 11], and Liu and Li [12–15]. From  we know that (1.4) can be formally transformed into a formal normal form: where , (system (1.8) is a generalized Liénard system). Stróżyna and Żołądek  proved that this formal normal form can be achieved through an analytic change of variables. Thus, if (1.4) has a center or focus at the origin, then it can be changed into (1.8) with
By Filippov's theorem (see, e.g., Ye et al. ) under (1.13) system (1.11) has a stable (unstable) focus at the origin if there exists an integer with such that and it has a center at the origin if for all .
Passing to the generalized polar coordinates we obtain from (1.11) the following equation:
Assuming that Álvarez and Gasull  called the constant the th generalized Lyapunov constant of (1.15) (we will see that this definition is too rough since a half of the constants cannot be used to determine the stability of the origin of (1.8) or (1.11)). They also studied the normal form (1.11) and proved the following theorem.
In the case Liu and Li  introduced different generalized polar coordinates of the form , to change (1.4) into the form: where it is assumed that the origin is a center or a focus. Let denote the solution of the -periodic system satisfying . Note that the initial value problem is well defined also for negative .
Let be a simply connected domain. Denote by the ring of analytic functions on and by the ideal in generated by the functions from . Liu and Li  found the following facts.
Theorem 1.3 (see ). Consider system (1.4). Let the conditions of Theorem 1.1 be satisfied with (or ) so that the origin is a center or a focus. Then(1).(2), where
In particular, , imply that .(3) The origin is a stable (unstable) focus if
In the latter case the origin is called a th order weak focus of (1.4).
We remark that the conclusions of Theorem 1.3 provide new and useful information on the property of the coefficients . Liu and Li  also gave some new methods to compute the focal values , or equivalent values and studied the problem of limit cycle bifurcations near the origin (using the second conclusion of the above theorem). They found a new phenomenon: a node can generate a limit cycle when its stability changes.
In this paper we study the problem of limit cycle bifurcations near the origin of the analytic system where , is a simply connected domain, and for small and .
To perform our analysis we first introduce a novel Poincaré map using a specific transversal section, study its analytical properties (Theorems 1.5 and 1.6), and then give a new definition of generalized focal values (or generalized Lyapunov constants) following . Second, using the Poincaré map together with the generalized focal values we establish new bifurcation theory of limit cycles from a nilpotent focus or center and obtain conditions for finding a lower bound and an upper bound of the maximal number of limit cycles bifurcated directly from the nilpotent point (Theorems 1.7 and 1.8). Third, we provide a new method to compute the generalized focal values using the normal form (Theorems 1.10 and 1.11). Moreover, we prove that the normal form and the original system have the same generalized focal values when the higher-order term has a sufficiently high order (Theorem 1.12 and Corollary 1.13). For polynomial systems we prove that the maximal order of a nilpotent focus is uniformly bounded (Theorem 1.14). All these results directly generalize the classical Hopf bifurcation theory and establish the new bifurcation theory in the nilpotent case.
We now state our main results more precisely. Let be the solution to the equation . We define the following two functions:
Definition 1.4. Let for all (1.23) and (1.24) be satisfied for some so that the the origin is a center or a focus of (1.21). In this case we say that (1.21) has a singular point of multiplicity at the origin.
Next, let us define a Poincaré return map for the planar system (1.21). For each and with small consider the solution of (1.21) with the initial condition . Then there is a unique least positive number such that and (see Figure 1 for the case of small ).
The map is the Poincaré map we will use for the remainder of the paper.
Obviously, the function is continuous at if (1.23) and (1.24) hold. It is easily seen that (1.21) has a periodic orbit near the origin if and only if the map has two fixed points near zero: one positive and the other one negative. Moreover, we note that the function is uniquely defined since the Poincaré section is chosen to be on the curve . This enables us to obtain some nice analytical properties of this function at , as stated in the following theorem.
Theorem 1.5. Let (1.21) satisfy (1.23) and (1.24) for all . Then there is a unique analytic function in at , satisfying and such that the displacement function has the expansion for sufficiently small, where(1)if is odd, then for all small;(2)if is even, then for all small where denotes the inverse of in .
It follows from the theorem that system (1.21) has a periodic orbit near the origin if and only if the analytic function defined by (1.26) has two zeros in near , among which one is positive and the other one is negative. The function is called the displacement function or the bifurcation function of (1.21).
The above theorem tells us that the function is analytic in at if is odd, and not analytic in at if is even unless the origin is a center (in this case is the identity). The theorem is a natural generalization of the case of elementary center or focus to the nilpotent case (), but it deals with two different cases (odd and even ), and the phenomenon in the case of even is new.
Define . Then the conclusions of the above theorem can be written uniformly as
Theorem 1.7 (Bifurcations from the Focus). Let (1.21) satisfy (1.23) and (1.24) for all . Denote that .(1) If there is an integer such that then there exists a neighborhood of the origin such that (1.21) has at most limit cycles in for all , where is any compact subset of .(2) If there is such that , then for all near (1.21) has at most limit cycles in a neighborhood of the origin. If further, then for an arbitrary sufficiently small neighborhood of the origin there are some near such that (1.21) has exactly limit cycles in the neighborhood.
Theorem 1.8 (Bifurcations from the Center). Let (1.21) satisfy (1.23) and (1.24) for all . Assume(i) there exist and an integer such that (1.30) is satisfied,(ii) the origin is a center of (1.21) if , , then there exists a neighborhood of the origin such that (1.21) has at most limit cycles in for all near , and also, for an arbitrary sufficiently small neighborhood of the origin, there are some near such that (1.21) has exactly limit cycles in the neighborhood. Hence, the cyclicity of the system at the point is equal to .
Definition 1.9. We call the generalized focal value of order of (1.21) at the origin and call the origin a focus of order if and for .
The above definition is very reasonable and natural, since by Theorem 1.7, we see that a nilpotent focus of order generates at most limit cycles under perturbations which satisfy (1.23) and (1.24).
By the above definition, condition (ii) of Theorem 1.8 means that the origin is a focus of (1.21) of order at most . This condition alone is not enough to ensure the conclusion of the theorem. For example, using Theorem 1.10 stated below one can prove that the system has exactly two limit cycles near the origin for . But, the focus at the origin has the order at most 1 for (the origin is a center for ).
The generalized focal values can be calculated using the normal form of system (1.21). We will give a different method to compute them. From a result of Stróżyna and Żołądek  we know that (1.21) has the analytic normal form:
Note that and in (1.32) may be different from the ones given by (1.22). As before, let where is a domain. Also, suppose that for small the function satisfies (1.23). Define
It is easy to see that for the equation defines a unique analytic function . Set
By Theorem 1.1, if (1.32) satisfies (1.23) and (1.24), then it has a center or focus at the origin. Thus, under (1.23) and (1.24) the Poincaré map for (1.32) is well defined near the origin.
Theorem 1.10. Let (1.32) satisfy (1.23) and (1.24) for all . Then, for small, the Poincaré map has the form: where , , , are positive constants and . Thus, Theorems 1.7 and 1.8 hold if is replaced by , .
For system (1.32) we have the following result.
Theorem 1.11. Let (1.32) satisfy (1.23), (1.34), and (1.37) for all . Assume there exist and such that
Let one of the following conditions be satisfied:(a), and (b), , , and
Then(1)for (1.32) has a stable (unstable) focus at the origin.(2)If further then for an arbitrary sufficiently small neighborhood of the origin there are some near such that (1.32) has at least limit cycles in the neighborhood.
From Theorems 1.5–1.10, it seems that under (1.23) and (1.24) we have solved the problem of limit cycle bifurcation for generic systems. Theoretically it is, but in practice it is not. The reason is that in general we do not know what is the transformation from (1.21) to its normal form (1.32). Here we give a method to solve the problem completely both theoretically and in practice. It includes three main steps described below.
Under (1.23) and (1.24) by the normal form theory (see, e.g., ) for any integer there is a change of variables of the form: where is a polynomial in of degree at most , such that it transforms (1.21) into (1.43) (called the normal form of order of (1.21), or the Takens normal form; we still use for the new variables ) where with and , and , being analytic functions satisfying . Here, we should mention that the functions and depend only on the terms of degree at most of the expansions of the functions and in (1.21) at the origin.
Truncating the series (1.43) at terms of order we obtain the polynomial system
In practice, for given system (1.21) it is not difficult to find the corresponding system (1.46). For (1.46) we can use Theorem 1.10 to find its focal values at the origin up to any large order. Let denote the Poincaré map of (1.46). We write the expansion of the displacement function as where the series converge for small .
We intend to use instead of . To this end, we have to solve the following problem. For any given find such that for . The following theorem gives a solution.
Therefore, we have the following.
In the case of elementary center or focus, the above conclusion is well-known.
Finally, we consider the following polynomial system of degree
Theorem 1.14. For any there is an integer such that an arbitrary polynomial system of the form (1.49) has a singular point of multiplicity at most at the origin (i.e., one must have if (1.49) satisfies (1.23) and (1.24), see Definition 1.4). Further, for each , if (1.49) satisfies (1.23) and (1.24), then there exists an integer such that for (1.49) the origin is a focus of order at most . Hence, the origin as a nilpotent focus of (1.49) generates at most limit cycles.
Consider system (1.21). In this section we will always suppose that (1.23) and (1.24) are satisfied. Introducing the new variable we obtain from (1.21) (reusing for ) where the functions and are given by (1.22), and are analytic functions near the origin with . In the discussion below for convenience we will often omit . As suggested by Liu and Li in  we pass in (2.1) to the generalized polar coordinates
Proof. From (2.2) we have
We solve the above equations for and and obtain (2.3) with
Then noting that and that (2.2) is invariant as is replaced by one can easily prove (2.4). The other conclusions are direct. This ends the proof.
Let denote the solution of (2.9) with the initial value . For properties of the solution we have the following.
Lemma 2.2. The solution is analytic in for small and satisfies the following:(1);(2).
Proof. Let . Then by (2.9) and (2.10) we have
This means that is also a solution to (2.9). Then the first conclusion follows from the uniqueness of the solution to the initial problem. The second one follows in the same way. This completes the proof.
Further we have the following.
Proof. First, it is easy to see that (1.21) and (2.1) have the same Poincaré map . Then, noting from (2.3) that for small , by the definition of and (2.2), we see that
for small. Now consider the case of . Let denote the solution of (2.9) satisfying . Similarly as above we have
since under (2.2) the points and on the plane correspond to the points and on the plane, respectively.
Further, by Lemma 2.2(1), we have
Noting that, by Lemma 2.2(2), if and only if , we see that (2.15) becomes
Therefore, by (2.14) and (2.16) we have for
This ends the proof.
Lemma 2.4. Let . Then there exists an analytic function convergent for small with such that for small, where .
3. Proof of the Main Results
Proof of Theorem 1.6. There are two cases to consider separately.
Case A ( odd). By (1.26) and Theorem 1.5(1), we have for all small.
By Lemma 2.4, we can suppose that where , . Substituting (3.1) and (3.2) into (2.18), we obtain
Comparing the coefficients of the terms , and on both sides yields where , , are polynomials. Thus, from the above equations we obtain
Case B ( even). By (1.26) and we find that where and each is a polynomial of degree at least 2. Now we suppose that . Then (3.1) holds by Theorem 1.5. Further, noting that by Theorem 1.5 again
Then, inserting (3.1), (3.2), (3.6), and (3.8) into (2.18) we obtain where
Finally, noting that and substituting (3.7) into (3.9) we easily see that
This ends the proof.
Proof of Theorem 1.7. For the first part, suppose that the conclusion is not true. Then there exists a sequence in such that for (1.21) has limit cycles which approach the origin as . Then by Theorem 1.6, the function has nonzero roots in which approach zero as .
Since is compact, we can assume as . By our assumption, . Thus, for some ,
Therefore, by (1.26) and Theorem 1.6, we have
Note that . It follows from Rolle's theorem (see ) that for some the function has at most nonzero roots in for all . We have proved that the function has nonzero roots which approach zero as . It then follows that , contradicting to . The first conclusion follows.
For the second one, from the above proof one can see that for all near (1.21) has at most limit cycles in a neighborhood of the origin. Then, by Theorem 1.6, the displacement function can be written as where . Like in  one can show that are series convergent in a neighborhood of (see also, e.g., [20, 21]). Further, by (1.30), we can take as free parameters, varying near zero. Precisely, if we change them such that then by (3.14) the function has exactly positive zeros in near , which give limit cycles. This finishes the proof.
Proof of Theorem 1.8. Under (1.30) the values can be taken as free parameters. Further, by our assumption, the origin is a center of (1.21) as , . It then follows that
Therefore, (3.14) can be further written in the form: where and are series convergent in a neighborhood of . Using the reasoning of Bautin  (see also e.g., [20–22]) one can easily see that the conclusion of the theorem holds. The proof is completed.
Proof of Theorem 1.10. Now we consider (1.32), where satisfies (1.23). Let
If satisfies (1.24), then the origin is a center or focus of (1.32), and where and satisfies for small. Note that (1.32) is equivalent to the following system: which has the same Poincaré return map as (1.32). Introducing the change of variables and system (3.23) becomes which is equivalent to where
The systems (3.25) and (3.26) have the same Poincaré return map, denoted by . One can see that the maps and have the relation . Hence, where is analytic. By (1.26) and (3.14), for small we have
Hence, where , .
Since satisfies and for small, we have or , where . Thus, we have
Thus, by (3.31) we have
Substituting into the above equality and comparing with (3.21) we obtain