Homoclinic orbits and Lie rotated vector fields

International Journal of Mathematics and Mathematical Sciences, Aug 2018

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

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Homoclinic orbits and Lie rotated vector fields

Internat. J. Math. & Math. Sci. Vol. HOMOCLINIC ORBITS AND LIE ROTATED VECTOR FIELDS JIE WANG CHEN CHEN Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well. 1. Introduction. The rotated vector fields have been considered as a very important tool which is efficient in the study of the numbers of limit cycles and the distribution of homoclinic orbits. In this paper, we continue with the investigation of the Lie rotated vector fields [6]. Based on the definition of Lie rotated vector fields [6], we give the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. As application of Lie rotated vector fields, we prove that the definition of Lie rotated vector has nothing to do with the change of coordinate. Afterwards, we require the singular points of X(?) not to be moved as parameter ? is changed. We study the motion of homoclinic orbits that pass through saddle points on Lie rotated vector fields and the change of a family of periodic orbits that are in the inner neighborhood of homoclinic orbit. Of course, we give some examples to illustrate the concept and notion of Lie rotated vector fields. and phrases; Lie rotated vector fields; Lie bracket; one parameter group; homoclinic orbits - (2.1) use the same method, under the transform ?, Y becomes Y? d=ef ??Y (y) = D? ??1(y) ? X ??1(y) . For X?(?) and the corresponding Y?, and ?y = ?(x) ? R2, we only need to prove the condition to set up (2.3) (2.4) (2.5) (2.6) (2.7) where [ , ] is Lie bracket, which expresses the action to the variable y. Since L(0) = X?(0) ? X?? (0) + [X?(0), Y?] > 0 (< 0), X?(0) ? [X?(0), Y?] = D?(x) ? X(x, 0) ? [X(x, 0), Y (x)], X?(0) ? X?? (0) = D?(x) ? X(x, 0) ? X? (x, 0), where x = ??1(y). From formulae (2.5) and (2.6), we can find that L(0) = D?(x) ? L(0), where x = ??1(y). But ? is the C2 differential topological transform, D?(x) = 0, and L(0) > 0 (or < 0), it follows that L?(0) > 0 (or < 0). Theorem 2.1 illustrates that Lie rotated vector fields have nothing to do with the choice of coordinates, and the symbol of L?(0) in formula (2.4) is decided by the symbol of L(0) and the formula of Jacobi on the transform ?. Theorem 2.2. Let X(?) be a Lie rotated vector field, X(0)|p0 = 0. ( 1 ) If the index J0(p0) of singular point p0 of X(0) is not equals to zero, then Y |p0 = 0. ( 2 ) If the index J0(p0) of singular point p0 of X(0) is changed as ? is changed, then Y |p0 = 0. Proof. ( 1 ) Using reduction to absurdity, we let Y |p0 = 0. From [6, Lemma 2.1], we know that ???X(?) |?? (p0) = 0 (? = 0), the proof is similar to [6, part ( 2 ) of Theorem 3.2]. For ?? > 0, such that ?? (p0) ? S?(p0), |? | < ?? < ?, ?? is given by [6, Theorem 3.2]. Since ???X(?) |p0 = 0, it is noted that J? ? (p0) = 0 (? = 0) to ?S?(p0) about ???X(?) , but J0(p0) = 0 to ?S?(p0) about X(0). From [6, Lemma 2.3], we know that this is a contradiction, thus Y |p0 = 0. ( 2 ) The proof is similar to ( 1 ). In fact, we might as well let Y |p0 = 0, then ???X(?) |p0 = 0. Let J0(p0) = r0 about X(0), but J? (p0) = r? (r? = r0, ? = 0) about X(?) . From [6, Lemma 2.2], we know that J?? (p0) = r? . Again from [6, Lemma 2.3], when |? | < ?? < ?, J0(p0) = J?? (p0) = r0, i.e., r0 = r? , and by the supposition r0 = r? , this is a contradiction, thus Y |p0 = 0. Corollary 2.3. Let X(?) be a Lie rotated vector field, if p01, p02, . . . , p0m are m elementary singular points of X(0), then the corresponding vector field Y , certainly set up Y |p0j = 0 (1 ? j ? m). (2.8) 3. The motion of homoclinic orbit. From the theory of structural stability on twodimensional mainfolds, we know that the systems which have homoclinic orbit passing through the saddle points are structurally unstable, this orbit which links saddle points can extremely be burst under trouble, thus can change the topological structure of the orbit. When X(?) is Lie rotated vector field, and if ? = 0, X(0) has homoclinic orbit passing through saddle points, we consider the change of topological structure of the orbit of X(?) when ? = 0. Theorem 3.1. Let X(?) be Lie rotated vector field, ?0 = {q(t), t ? R} p0 is homoclinic orbit passing through the hyperbolic saddle point p0 on limited region. If ?0 = div X(0)|p0 = 0, then when ? is changed toward suitable direction, ?0 disappears, but it will produce unique limit cycle of X(?) in the neighborhood of ?0, and the limit cycle is stable or unstable; but when ? is changed towards other direction, ?0 disappears, and it will not produce any limit cycle of X(?) in the neighborhood of ?0. Proof. We might as well let ?0 positively oriented, ?0 < 0. From [5, Theorem 1], we know that ?0 is inner stable. Since p0 is a hyperbolic saddle point of X(0), from Corollary 2.3, we know that it certainly has Y |p0 = 0, thus it has ???X(?) |p0 = 0 (? = 0). ???X(?) has homoclinic orbit ?0 passing through a hyperbolic saddle point p0 when ? = 0. By [6, Lemma 2.3], when 0 < |? | < ?, ?0 is burst in ???X(?) (? = 0), yet from the topological equivalence of orbital structure of X(?) and orbital structure of ???X(?) , we know that ?0 of X(?) is burst when ? = 0, i.e., when 0 < ? < ? , the homoclinic orbit ?0 passing through hyperbolic saddle point p0 is burst into stable manifold S? and unstable manifold U? . Since ?0 is inner stable, by continuous dependence of solution to the parameter ? and using the Poincar?-Bendixson annular regional theorem, we prove that there is a limit cycle in the neighborhood of ?0 when ? > 0, yet by ?0 < 0, we know that there is only a unique stable limit cycle in the neighborhood of ?0. But when ? < 0, |? | < ?, let there be a limit cycle in the neighborhood of ?0, this is the same discussion as above, we know that it is sure to have semi-stable limit cycle or unstable limit cycle, this is contradiction with ?0 < 0. Using Theorems 2.2 and 3.1, we can easily prove the following corollary. Corollary 3.2. Let X(?) be a Lie rotated vector field, ?0 = {q(t), t ? R} p0 is homoclinic orbit passing through the saddle point p0 of X(0) on the limited region, ?0 = div X(0)|p0 = 0. If J0(p0) = 0 (or J0(p0) = 0, but Y |p0 = 0), then when ? is changed towards a suitable direction, ?0 disappears, but it will produce a unique limit cycle of X(?) in the neighborhood of ?0, and the limit cycle is stable or unstable; but when ? is changed towards the other direction, ?0 disappears, and it will not produce any limit cycle of X(?) in the neighborhood of ?0. Using [1, Theorems 45 and 49], [4, Theorem 1.2], and Theorem 2.2 of this paper, we have the following corollary. Corollary 3.3. Let X(?) be a Lie rotated vector field, ?0 = {q(t), y ? R} p0 is homoclinic orbit passing through saddle point p0 of X(0) on the limited region, ?? = div X(?) |p0 = 0 (|? | 1 and |? | < ?), I0 = ?0 ?0dt = 0. If J0(p0) = 0 (or J0(p0) = 0, but Y |p0 = 0), then when ? is changed towards a suitable direction, ?0 disappears, but it will produce a unique limit cycle of X(?) in the neighborhood of ?0, and the limit cycle is stable or unstable; but when ? is changed towards other direction, ?0 disappears, and it will not produce any limit cycle of X(?) in the neighborhood of ?0. Example 3.4. Let X(?) = 2x2, 2x1 ? 3x13 ? x2 x13 ? x12 + x22 + ?x 23 , when ? = 0, p0 = (0, 0), p1 = (2/3, 0) are elementary singular points of X(0), where p0 is saddle 3 2 2 point, p1 is an unstable focal point. Since ?0 : x1 ? x1 + x2 = 0 is homoclinic orbit passing through the hyperbolic saddle point p0 of X(0) (this example is shown in [5]). By Corollary 2.3, we can take Y = 0, ? 2x1 ? 3x12 , where ? ? R. For, ?? > 0, ? is taken small enough, we make open neighborhoods S?(p0) and S?(p1) of p0 and p1, respectively, then there is a limited region D ? R2, ?0 ? D, order ? = ?3 > 0, at the ordinary point of X(0) of D \ {S?(p0) S?(p1)}, we have L(0) = 2? 2x1 ? 3x12 + x23 2 + 2? 4x22 1 ? 3x1 ? x26 + 2x24 > 0, (3.1) i.e., X(?) constitutes Lie rotated vector field on D. Take |? | 1 and |? | < ?, note ?? = div X(?) |p0 = 0, again from [5], we can know that I0 = ?0 ?0dt < 0. By Corollary 3.3, when ? < 0, ?0 disappears, but it will produce a unique stable limit cycle of X(?) in the neighborhood of ?0; but when ? > 0, ?0 disappears, and it will not produce any limit cycle in the neighborhood of ?0. Theorem 3.1 and Corollary 3.2 require ?0 = 0, Corollary 3.3 requires ?? = 0 (0 ? |? | < ? and |? | 1), using [3, Lemmas 8 and 9] and the proven method of Theorem 3.1 in this paper, we have the following corollary. Corollary 3.5. Let X(?) be a Lie rotated vector field, ?0 = {q(t), t ? R} p0 is homoclinic orbit passing through the saddle point p0 of X(0) on the limited region, ?0 is stable (unstable). If J0(p0) = 0 (or J0(p0) = 0, but Y |p0 = 0), then when ? is changed towards a suitable direction (towards the other direction), ?0 disappears, but it at least produces a limit cycle in the neighborhood of ?0, the limit cycle is stable (unstable); when ? is changed towards the other direction (towards a suitable direction), ?0 disappears, but it will not produce any limit cycle in the neighborhood of ?0. Example 3.6. Let X(?) = 2x2, 2x1 ?3x12 ?x2 x13 ?x12 +x22 +?x 2 , when ? = 0, the state of X(0) is same as Example 3.4, we yet take Y = (0, ?(2x1 ? 3x12)). For, ?? > 0, ? is sufficiently small, order ? = ? > 0, then there is a region D ? R2, ?0 ? D, at the ordinary point of X(0), we have L(0) = 2? 2x1 ? 3x12 + x23 2 + 2? 4x22 1 ? 3x1 ? x26 + 2x22 > 0, (3.2) i.e., X(?) constitutes a Lie rotated vector field on D. Take |? | 1, |? | < ?, by ?0 = div X(0)|p0 = 0, ?? = div X(?) |p0 = ? (? = 0), and I0 = ?0 ?0dt < 0, we know that ?0 is inner stable, J0(p0) = ?1 = 0. From Corollary 3.5, when ? < 0, in the neighborhood of ?0, at least produces a limit cycle; when ? > 0, in the neighborhood of ?0, it does not produce any limit cycle. Now we consider that X(?) is a Lie rotated vector field. If ? = 0, ?0 is homoclinic orbit passing through saddle point p0 of X(0) on the limited region, and the inner neighborhood of ?0 is imbued a family of periodic orbits, where ?0 = I0 = 0. where h ? (a, b). Proof. Using the formula of Theorem 2.3 of Chapter 3 in ?2 of [2], we have X ? [X, Y ] = X, ?(X ? Y ) ? (X ? Y ) ? div X. Both sides of (3.4) are multiplied by the factor exp ? 0t div Xdt , and both sides are integrated along the circuit of Lh, let the period of Lh be ?(h), using [7, formula (2.16)], we have Lemma 3.7. Let X be a C1 vector field, the limited region ? 1 ? R2 is imbued a family of periodic orbits Lh for X, h ? (a, b) ? R, then for all C1 vector fields Y , we have t A0(h) = Lh (X ? [X, Y ]) ? exp ? 0 div Xdt dt = 0, X, ?(X, Y ) ? exp ? 0t div Xdt dt + Lh (X ? Y ) ? ddt exp ? 0t div Xdt X, ?(X, Y ) ? exp ? 0t div Xdt dt ? Lh ddt (X ? Y ) ? exp ? 0t div Xdt dt dt Lh = Lh = Lh t (X ? [X, Y ]) ? exp ? 0 div Xdt dt (3.3) (3.4) (3.5) (3.6) (3.7) t + (X ? Y ) ? exp ? 0 div Xdt ?(h) 0 Theorem 3.8. Let X(?) be a Lie rotated vector field, ?0 = {q(t), t ? R} p0 is a homoclinic orbit passing through the saddle point p0 of X(0) on limited region. If the inner neighborhood ? ? R2 of ?0 is imbued a family of periodic orbits Lh, h ? (a, b) ? R, then when ? = 0, the inner neighborhood ? of ?0 will not produce any closed orbit of X(?) . Proof. By [7, Theorem 4.10], when ? = 0, on the undisappeared Lh0 , we have t A1 h0 = Lh0 X(0) ? X? (0) ? exp ? 0 div X(0)dt dt = 0, where h0 ? (a, b). But from Lemma 3.7, we have t A1 h0 = A1 h0 + A0 h0 = Lh0 L(0) ? exp ? 0 div X(0)dt dt = 0. It follows that the proof is completed. Example 3.9. Let X(?) = 2x2, 2x1 ? 3x12 + ?x 2 , when ? = 0, p0 = (0, 0) is hyper3 2 2 bolic saddle point of X(0), p1 = (2/3, 0) is the center of X(0), ?0 : x1 ? x1 + x2 = 0 is homoclinic orbit passing through p0 of X(0), yet take Y = 0, ? 2x1 ? 3x12 . When ? is taken arbitrary sufficient small, 0 < ? all ordinary points of X(0) on D, we have 1, then there is region D ? R2, ?0 ? D, at i.e., X(?) constitutes a Lie rotated vector field on D. From Theorem 3.8, we can know that there is not any closed orbit in the inner neighborhood of ?0 when ? = 0. Acknowledgement. The authors express grateful thanks to Professor Yaoxian Wang for his help and direction in the work. 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Jie Wang, Chen Chen. Homoclinic orbits and Lie rotated vector fields, International Journal of Mathematics and Mathematical Sciences, DOI: 10.1155/S0161171200010061