Research Article | Open Access
Tadashi Kawanago, "Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods", Advances in Numerical Analysis, vol. 2013, Article ID 420897, 9 pages, 2013. https://doi.org/10.1155/2013/420897
Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods
We establish codimension-m bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems established by (Kawanago, 2004). As a numerical example, we treat Hopf bifurcation, which is codimension-2 bifurcation.
By the recent growth of the computer power, we can observe numerically bifurcation phenomena of solutions without difficulty for a lot of differential equations and systems. It is in general difficult, however, to analyze rigorously such phenomena by the use of pure analytical methods. Actually, it seems impossible at least at present to analyze by the use of pure analytical methods the Hopf bifurcation phenomena in the Brusselator model treated in Section 4. We need some computer-assisted analysis to treat it. We now have various excellent bifurcation theorems from the theoretical point of view. It needs in general, however, some particular devices to apply them to a given concrete dynamical system since we are usually not able to check some conditions in such theorems directly by numerical methods.
Another important approach to computer-assisted analysis for bifurcation problems is to establish new bifurcation theorems applicable directly to numerical verification methods. This approach is our theme in this paper. It is useful from the practical and applied mathematical point of view. In , the author established some codimension-1 bifurcation theorems applicable directly to numerical verification methods. Using a symmetry-breaking bifurcation theorem [1, Theorem 3.1] and the numerical verification methods, we proved the existence of a -symmetry-breaking bifurcation point for a nonlinear forced vibration system described by a wave equation in , and Nakao et al. verified some symmetry-breaking bifurcation points for two-dimensional Rayleigh-Bénard heat convection system in [3, 4].
In this paper, we establish codimension- bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems mentioned above. In Section 4, we apply our new theorem to Hopf bifurcation, which is codimension-2 bifurcation.
Here, we present our main theorem. Let and be real Banach spaces. Let and be closed subspaces of and let and be closed subspaces of . We assume that and . Here, means the direct sum. Let and have the following properties: We denote by the first row vector of the identity matrix of order . We define by Here, , and we assume that for any . We define projections and by
In what follows, we always set formally for the case: . We define by
We set and . Our main theorem is the following.
Theorem 1. In addition to the assumptions above, we assume that satisfies the following (H1) and (H2):(H1) A point is an isolated solution of the extended system .(H2) The linear operator is bijective.
Then, the point is a bifurcation point of the equation . Exactly, there exist an open neighborhood of in , , , , an open neighborhood of 0 in and such that , , and
Roughly speaking, the well-known pitchfork bifurcation theorem [5, Theorem 1.7] by Crandall and Rabinowitz is equivalent to Theorem 1 with , , and (see Section 2). We immediately obtain a -symmetry breaking bifurcation theorem [1, Theorem 3.1] by setting in Theorem 1 and by choosing the symmetric subspace of as and the anti-symmetric subspace as . We can apply Theorem 1 to Hopf bifurcation by setting and choosing an appropriate space of periodic functions as , the subspace of steady functions as , and the complementary subspace of as (see Section 4 below).
Finally, we illustrate by Hopf bifurcation as an example that known bifurcation theorems are not in general applicable directly to the numerical verification methods. We consider the next autonomous ordinary differential equation: The classical Hopf bifurcation theorem reads as follows.
Theorem 2. The point is a Hopf bifurcation point of , that is, a branch of periodic solutions of bifurcates at the point from a branch of steady solutions of with the initial period , provided that the following conditions (C1)–(C4) are satisfied:(C1), (C2) ± are the simple eigenvalues of (so, by the implicit function theorem, the matrix has a pair of complex conjugate of eigenvalues , with ), (C3) (Transversality condition of eigenvalues) , (C4) is not an eigenvalue of for .
It is difficult to check rigorously by numerical methods the dynamic condition (C3), and we need a particular device for it. See Remark 15 for more detailed information. On the other hand, the condition (C3) is implied by (H1) and (H2) in Theorem 1, which are static conditions, that is, regularity conditions for linear operators.
The paper is organized as follows. In Section 2, we mainly establish a basic bifurcation theorem, which is simpler than Theorem 1. In Section 3, we prove Theorem 1. In Section 4, we present a numerical example and treat Hopf bifurcation as an application of Theorem 1 with and our numerical verification method. We present some final remarks in Section 5.
Notation. Let and be Banach spaces. (1)We denote by the set of all real numbers, by the set of all rational numbers, by the set of all integers, and by the set of all complex numbers. We write .(2)We denote by the first row vector of the identity matrix of order .(3)We denote by the norm of . If is a Hilbert space, then stands for the norm induced by the inner product of . For and , we write . The subscript will be often omitted if no possible confusion arises.(4)We denote by the space of bounded linear operators from to , and stands for . For , denotes the operator norm of . The subscript will be omitted if no possible confusion arises.(5)Let be an operator. We denote by the domain of and by the range of . We define .(6)For an open set , we denote by the space of -times continuously differentiable functions from to .(7)Let be a domain of . We denote Lebesgue spaces by with the norms: . We define the norm of the Sobolev space by (8)Let and be norm spaces. We denote their direct product space usually by and also by in Section 4.5 below. Actually, the direct product space is interchangeably called the direct sum space (see, e.g., [6, Section II.1]). We define for .
2. Basic Bifurcation Theorems
Let and be real Banach spaces and be an open neighborhood of 0 in . Let and be an open neighborhood of in . Let be a map such that and the partial Fréchet derivative exists and is continuous for . Here, for any fixed , represents the Fréchet derivative of the function at . We denote for simplicity. We define by Here, . In what follows, we often use the same notations as in Section 1. We define by We set .
Theorem 3. In addition to the assumptions above, we assume that(H) there exists such that is an isolated solution of the extended system .
Then, there exist an open neighborhood of (0,0) in , and continuous functions , such that , such that Moreover, if (), then .
For simplicity, we write , , , , and so on. We have
Proof of Theorem 3. The proof is similar to that of [1, Theorem 2.1] and [5, Theorem 1.7]. We obtain from (H) that and . Let be any element of . We set . Then, we have . Since is one to one, . So we obtain . We choose and an open neighborhood of in such that for . We define a map by By , we have . We verify that the partial derivative of with respect to at coincides with and is bijective. So, by applying the implicit function theorem to at , there exist an open neighborhood of (0,0), and continuous functions , such that , , and for any and that and imply and . Thus, includes the right-hand side of (12). From the same argument as in the proof of [5, Theorem 1.7], we can verify that (12) actually holds.
Proposition 4. The condition (H) in Theorem 3 is equivalent to the following (i) and (ii):(i) and , (ii) such thatHere, we denote .
Proof. First, we assume (H). By the proof of Theorem 3, we have and (15). Since is onto, we have . Let and satisfy . In order to prove (16), it suffices to show . We choose and such that and . We set . Then, . We have and since is an injective linear map. So, we obtain , which implies . Thus, (16) holds. Next, we show by the same discussion as above that is linearly independent. Let satisfy . Then, we have , which implies . Thus, is linearly independent. So, . By this and (16), we have . Thus, (i) and (ii) hold.
We show the inverse. We assume (i) and (ii). Clearly, (15) implies and . So, it suffices to show that is bijective. Let satisfy . Then, by (13) we have and . We obtain from (16) that and . We have since (i) and (16) imply . By and (15), we have . Hence, is one to one. Finally, let be an element of . By (16), there exist and such that . We set . Then, . Thus, is onto. The proof is complete.
3. Proof of Theorem 1
We set . We use the same notations as used in Section 2. We set , , , and . Then, we have
Proof of Theorem 1. By (H2) and the implicit function theorem, there exist an open neighborhood of in and a map such that and for any . We define by Then, we have (9), , and It follows from (9) and (H2) that is uniquely determined by the equation: We define by (10), and set . Then, we have (13). In view of Theorem 3, the proof is complete if we show that the map is bijective. Let satisfy . It follows from (2), (3), (13), and (19) that So, by (2), (H2), (17), (20), and (21), we have and . We obtain since is one to one. Hence, is one to one. Next, let be an element of . Let with and . Since is onto, there exists such that . By (H2), (17), (19), and (20), we have , , and . We set . Then, by (19), we have . Therefore, is onto. The proof is complete.
4. A Numerical Example
In this section, we present a numerical example. We treat Hopf bifurcation as an application of Theorem 1 with . This section is organized as follows. In Section 4.1, we present a partial differential system we consider. In Section 4.2, we rewrite our problem under appropriate setting of functional spaces. In Section 4.3, we describe our numerical verification result on the existence of a Hopf bifurcation point. In Section 4.4, we present the principal abstract results in our numerical verification methods. In Section 4.5, we describe the outline of derivation of our verification result.
4.1. Brusselator Model
We consider the following Brusselator model: Here, , , , , and are parameters of the problem. In what follows, we set , , , and and consider as the bifurcation parameter. By some numerical methods, Kubíček and Holodniok  found some period-doubling bifurcation points for an ordinary differential system related closely to . On the other hand, we verified the existence of a Hopf bifurcation point for .
4.2. The Setting of Functional Spaces
Let , , and . We define a real Hilbert space by with the inner product for . Then, we easily verify that is continuously embedded in and that for . We set . We denote by (resp., ) the subspace of (resp., ) consisting of steady functions, that is, We set , , , and .
We set . We define and by Namely, (resp., ) equals to the Fourier coefficient of (resp., ) for . Let be defined by (4) with . We define by Then, (H1) is equivalent to the following (K1) since and :(K1) A point is an isolated solution of the extended system .
Similarly, (H2) is equivalent to the following (K2): (K2) The linear operator is bijective.
4.3. A Verification Result
By our numerical verification method described in Section 4.5, we verified the existence of a Hopf bifurcation point of satisfying Here, is an isolated solution of the extended system , and is its numerical approximation. The initial period of bifurcating periodic solutions is . We have , which is defined by (42) below. We set and . The functions and have the forms with . We have Here, .
Remark 5. In the process of deriving the above result, we often used computer arithmetic with double precision, without taking into account the effects of the round-off errors, though (31) holds rigorously. It is sufficient, however, for our purpose, which is to check that Theorem 1 is applicable to the numerical verification method. We can make the above result completely rigorous if we always use the interval arithmetic or the computational method described in [2, Section 2.1] in the process of deriving the conclusion.
4.4. Principle of Our Numerical Verification Method
In this subsection, we present two important abstract results in our numerical verification method.
Let and be Banach spaces in what follows in this subsection. We assume . We can immediately obtain the following result from [8, Theorem 3.2].
Theorem 6 (the convergence theorem of simplified Newton method). Let and be bijective. We define a map by Let be a constant and be a non-decreasing function such that Let be a constant such that We assume that there exist constants and such that , Then, the equation has an isolated solution . Moreover, the solution of is unique in .
Theorem 6 plays a central role in our numerical verification method.
Remark 7. Theorem 6 is a corollary of [9, Theorem 1.1]. Though the latter is in general stronger than the former, the former is simpler. The former works well for our present problem since the residue of the approximate solution is very small (see (31)).
Let and be closed subspaces of such that . Let and be closed subspaces of such that . We denote by the projection defined by for with . We denote by the projection defined by for with . Let and be a closed linear operator. We set . We assume We define . We denote by (resp., ) the identity operator on (resp., ) and simply by the operators and when there is no ambiguity.
Proposition 8. In addition to the above assumptions, we assume that is bijective and that satisfies . Then, is bijective. Moreover, we have
Finally, we describe how Proposition 8 works well for our problem. In our present context, we can assume in addition the following (A1)–(A3):(A1) and are Hilbert spaces, and is embedded continuously and densely in ,(A2) there is a CONS of such that is an eigenfunction of for any . Let and be subspaces of and , respectively, such that both and are equal to as sets. Let (resp., ) be the orthogonal projection on (resp., ) onto (resp., ), (A3) and .
We show that Proposition 8 works well if we take and for sufficiently large . Let .
Proposition 9. Under the assumptions above we have the following. (i)If is bijective, then is bijective for sufficiently large , and we have(ii)If is bijective andthen is bijective with the estimate . (iii)We haveHere, we formally define (resp., ) when (resp., ) is not bijective.
Proof. We set . Then, on . We obtain from (A3) that as .
(i) Let be bijective. Then, for sufficiently large . Therefore, by [2, Corollary ], is bijective for sufficiently large , and we obtain (38).
(ii) We immediately obtain the desired conclusion from [2, Corollary ].
(iii) If is bijective, (i) implies (40). So, we consider the case where is not bijective. We proceed by contradiction. We assume that . Then, . So, there exists a large number such that . So, (ii) implies that is bijective. This contradicts to our assumption. Therefore, (40) holds.
We consider the case where the condition of Proposition 8 is satisfied. Let . The estimate (41) below is useful when we apply Theorem 6 to concrete problems. We denote by the operator with , , , , and replaced by , , , , and , respectively ( = 1, 2). Then, we easily verify that and that the right-hand side of (41) converges to as in view of (A3). So, the right-hand side of (41) is a sharp upper bounds of when is sufficiently large.
Remark 10. We denote by the operator defined in Proposition 8 with , , , and replaced by , , and , respectively. Though (39) is a sufficient condition for the existence of , another sufficient condition is in general more efficient. The reason is that converges to () more rapidly in general than .
4.5. Derivation of Our Verification Result
Here, we describe the outline of derivation of our verification result in Section 4.3. We define closed subspaces of : We similarly define closed subspaces of : Clearly, , and . We set and . Here, we replace the symbol of direct product by the direct sum (see the notation (8) just after Section 1), which helps us to describe the decomposition of . Then, we can represent by the direct sum of maps: . Since , the condition (K1) is equivalent to the following (L1) and (L2):(L1) A point is an isolated solution of the extended system . (L2) The linear operator is bijective.
By the above discussion and Theorem 1, it suffices to verify (L1), (L2), and (H2) in order to show that is a Hopf bifurcation point of . We verified (L1) and (29) by applying Theorem 6 with and , and (L2), (H2) by Proposition 8. We need the following embedding inequality to check the conditions in Theorem 6 and Proposition 9. For simplicity, we denote .
Proposition 11. We have where .
Proof. The proof is similar to that of [2, Proposition ].
First, let . In a similar way to the proof of [2, Lemma (ii)], we easily verify that We easily verify that We set and . It suffices to prove (44) for any since is dense in . Let . It follows from (45) and [2, Lemma (ii)] that Let . We substitute to (47). Then, we obtain from (46) and Schwarz inequality that By (48) and (49), we have (44) with replaced by . The proof is complete.
Remark 12. Though our numerical verification method is not difficult, the verification process consists of a lot of steps and is very complicated. So, we omit its details here. They are much similar to the discussions in [2, 9]. In particular, the verification process is described fully in detail in .
5. Final Remarks
Remark 13. Our main result Theorem 1 is new and useful from the practical and applied mathematical point of view. It does not mean, however, that it is new from the theoretical point of view, though there is no known result equivalent to our main result with as far as the author knows. In view of Proposition 4, Theorem 1 is theoretically considered as a corollary of [5, Theorem 1.7] for the case and is generalization of it for the case . We refer to  by Hale and  by López-Gómez where some related bifurcation theorems were obtained. They are generalization of [5, Theorem 1.7] different from our main result. Actually, Theorem 1 implies Theorem 2, as shown in our work mentioned in Remark 14 below. On the other hand, the bifurcation theorems in [10, 11] do not imply Theorem 2.
Remark 14. In a near future work, we will study the equivalent relations between conditions in Theorem 1 with and those in Theorem 2 and show that Theorem 1 with is stronger than the existence part of [12, Theorem 1.11].
Remark 15. Nishida et al. studied problems of stability and bifurcation of solutions for some fluid equations in [13, 14]. We can apply the same technique as in [13, 14] to check rigorously the condition (C3) in Theorem 2 by the numerical method.
Let , , and . Then, we can obtain in the same way as in  and [14, Section 4] that Here, , , and satisfies . We can check the condition (C3) since we can find by the numerical verification method an accurate approximate value of the right-hand side of (50) with a rigorous error bound.
The author would like to express his sincere gratitude to Professor Atsushi Yagi and Professor Takaaki Nishida for their encouragement. The author is grateful to the referees for constructive comments.
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