Difference between revisions of "Bifurcation"
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620040.png" /> is a non-isolated solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620042.png" /> is a bifurcation point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620043.png" />. It was shown by a variational method [[#References|[1]]], [[#References|[2]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620044.png" /> is a non-linear completely-continuous operator in a Hilbert space which is the gradient of a weakly continuous functional, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620045.png" /> is a completely-continuous self-adjoint operator, then all characteristic values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620046.png" /> are bifurcation points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620047.png" />. The concept of a bifurcation point is also modified in the case of large solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620048.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620049.png" />. The importance of these concepts and results consists in the fact that, subject to relatively weak limitations, the branching of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620050.png" /> can be established; in particular, it is possible to prove that the solution of the non-linear problem is not unique. The methods of the theory of [[Branching of solutions|branching of solutions]] of non-linear equations often yield more accurate information [[#References|[5]]]. | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620040.png" /> is a non-isolated solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620042.png" /> is a bifurcation point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620043.png" />. It was shown by a variational method [[#References|[1]]], [[#References|[2]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620044.png" /> is a non-linear completely-continuous operator in a Hilbert space which is the gradient of a weakly continuous functional, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620045.png" /> is a completely-continuous self-adjoint operator, then all characteristic values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620046.png" /> are bifurcation points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620047.png" />. The concept of a bifurcation point is also modified in the case of large solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620048.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620049.png" />. The importance of these concepts and results consists in the fact that, subject to relatively weak limitations, the branching of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620050.png" /> can be established; in particular, it is possible to prove that the solution of the non-linear problem is not unique. The methods of the theory of [[Branching of solutions|branching of solutions]] of non-linear equations often yield more accurate information [[#References|[5]]]. | ||
− | 2) The theory of smooth dynamical systems studies one-parameter (and sometimes also two-parameter [[#References|[6]]]) families of flows (and cascades; only the former will be considered here), and the conditions under which the bifurcation is | + | 2) The theory of smooth dynamical systems studies one-parameter (and sometimes also two-parameter [[#References|[6]]]) families of flows (and cascades; only the former will be considered here), and the conditions under which the bifurcation is "typical" , i.e. preserves its character under a small change of the family in question [[#References|[9]]]; both variants A) and B) above are useful. In the second variant two flows are considered equivalent if there exists a homeomorphism of the phase space which converts the trajectories of the one into trajectories of the other while preserving the direction of motion. There exists a completely satisfactory theory of bifurcations of one-parameter families of flows with a two-dimensional phase space [[#References|[7]]], [[#References|[9]]], and a local variant referring to a neighbourhood of an equilibrium position or of a periodic solution in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620051.png" />-dimensional case [[#References|[6]]]. |
− | In variant A) the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620052.png" /> under study, which are associated to the given dynamical system, are the positions of an equilibrium or periodic solution, and occasionally certain invariant manifolds (mainly tori) or hyperbolic invariant sets. The | + | In variant A) the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620052.png" /> under study, which are associated to the given dynamical system, are the positions of an equilibrium or periodic solution, and occasionally certain invariant manifolds (mainly tori) or hyperbolic invariant sets. The "growth" of these objects — taking place "locally" near an equilibrium position, or near a periodic solution, or "semi-locally" in a neighbourhood of a "closed contour" formed by several trajectories which tend to an equilibrium position or to a periodic solution — is considered. A case of bifurcation which is connected with a similar contour in a certain sense, but which takes place (with a change of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620053.png" />) prior to its arrival, is also possible . The growth of periodic solutions is often conveniently considered by rewriting the differential equation and the periodicity condition in the form of an integral equation to which suitable methods are applied [[#References|[5]]]. |
− | 3) Various bifurcations of different objects (both the initial objects and those associated to them) are encountered in the theory of singularities of mappings. As a result, this term (or rather a term derived from it) is used in several different ways [[#References|[10]]], [[#References|[6]]], [[#References|[11]]], but it is more common to assign independent names to the corresponding concepts. They include, for instance, versal families (or deformations) [[#References|[6]]], [[#References|[11]]], [[#References|[12]]], which describe, in a certain sense, all possible bifurcations which may take place under a small deformation of the object under consideration. These include, in particular, seven elementary catastrophes [[#References|[12]]], which are represented by | + | 3) Various bifurcations of different objects (both the initial objects and those associated to them) are encountered in the theory of singularities of mappings. As a result, this term (or rather a term derived from it) is used in several different ways [[#References|[10]]], [[#References|[6]]], [[#References|[11]]], but it is more common to assign independent names to the corresponding concepts. They include, for instance, versal families (or deformations) [[#References|[6]]], [[#References|[11]]], [[#References|[12]]], which describe, in a certain sense, all possible bifurcations which may take place under a small deformation of the object under consideration. These include, in particular, seven elementary catastrophes [[#References|[12]]], which are represented by "typical" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620054.png" />-parametric (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016200/b01620055.png" />) families of functions which include a function with a degenerate critical point and which are defined in a neighbourhood of that point; they accordingly describe the corresponding bifurcation. In non-Soviet literature on the theory of singularities, the term "catastrophes" is often employed for "bifurcations" . |
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian) {{MR|}} {{ZBL|0111.30303}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.G. Krein (ed.) , ''Functional analysis'' , Wolters-Noordhoff (1972) (Translated from Russian) {{MR|0390693}} {{MR|0352920}} {{ZBL|0128.10302}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian) {{MR|}} {{ZBL|0121.10604}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M.M. Vainberg, "Variational methods for the study of nonlinear operators" , Holden-Day (1964) (Translated from Russian) {{MR|0176364}} {{ZBL|0122.35501}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian) {{MR|0344960}} {{ZBL|0274.47033}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> V.I. Arnol'd, "Lectures on bifurcations and versal families" ''Russian Math. Surveys'' , '''27''' : 5 (1972) pp. 54–123 ''Uspekhi Mat. Nauk'' , '''27''' : 5 (1972) pp. 119–184</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl. (1971) (Translated from Russian) {{MR|0344606}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8a]</TD> <TD valign="top"> N.K. Gavrilov, L.P. Shil'nikov, "On three-dimensional systems close to systems with a structurally unstable homoclinic curve I" ''Math. USSR-Sb.'' , '''17''' : 4 (1972) pp. 467–485 ''Mat. Sb.'' , '''88 (130)''' : 4 (1972) pp. 475–492 {{MR|}} {{ZBL|0255.58006}} </TD></TR><TR><TD valign="top">[8b]</TD> <TD valign="top"> N.K. Gavrilov, L.P. Shil'nikov, "On three-dimensional systems close to systems with a structurally unstable homoclinic curve II" ''Math. USSR-Sb.'' , '''19''' : 1 (1973) pp. 139–156 ''Mat. Sb.'' , '''90 (132)''' : 1 (1973) pp. 139–156 {{MR|}} {{ZBL|0273.58009}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> M.M. Peixoto, "On bifurcations of dynamical systems" R. James (ed.) , ''Proc. Internat. Congress Mathematicians (Vancouver, 1974)'' , '''2''' , Canad. Math. Congress (1975) pp. 315–319 {{MR|0428359}} {{ZBL|0344.58014}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R. Thom, "The bifurcation subset of a space of maps" , ''Manifolds (Amsterdam, 1970)'' , Springer (1971) pp. 202–208 {{MR|0281222}} {{ZBL|0216.45901}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> V.I. Arnol'd, "Critical points of smooth functions and their normal form" ''Russian Math. Surveys'' , '''30''' : 5 (1975) pp. 1–75 ''Uspekhi Mat. Nauk'' , '''30''' : 5 (1975) pp. 3–65</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) {{MR|0494220}} {{ZBL|0302.58006}} </TD></TR></table> |
Line 26: | Line 26: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S.-N. Chow, J.K. Hale, "Methods of bifurcation theory" , Springer (1982) {{MR|0660633}} {{ZBL|0487.47039}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.H. Sallinger, "Branching in the presence of symmetry" , SIAM (1983)</TD></TR></table> |
Revision as of 16:55, 15 April 2012
A term in certain branches of mathematics, applied to situations in which some object depends on a parameter
(which is not necessarily scalar) and which is such that in any neighbourhood of a certain value
of that parameter (a bifurcation value or a bifurcation point) the qualitative properties under consideration of the object
are not the same for all
. The corresponding rigorous definitions vary in different cases, but mainly follow (in a more or less modified form) the two variants below:
A) The qualitative properties of the object being studied are the existence of other objects
, associated with it in some manner. The bifurcation is characterized by the fact that when
is varied, the objects
appear or disappear (in particular, they may coincide or else one object may generate several ones). See below (Paragraph 1)).
B) The first step is to determine under what circumstances two objects are considered to be equivalent. (This definition must be such that all the qualitative properties of interest be identical for equivalent objects.) A change in the qualitative properties of
in a neighbourhood of a bifurcation point means, by definition, that
-values with non-equivalent
are found near this point. See below (Paragraph 2)).
1) In the theory of operators the initial object is a non-linear operator
in a real Banach space, depending on a real parameter
, defined in a neighbourhood of the point
and such that
. For each fixed
one associates new objects
to this
; they are the solutions
of the non-linear operator equation
. A bifurcation point is a point at which a new, non-trivial, solution of this equation is generated. It is in fact a point
such that for any
there exists a
,
, for which the equation
has a solution
satisfying the conditions
. If
, where
is a linear completely-continuous operator, the concept of a bifurcation point coincides with the concept of a characteristic value of
.
If is a non-linear completely-continuous operator which is continuously Fréchet differentiable and is such that
, then only the characteristic values of
can be bifurcation points of
. It was found by a topological method [1], [2] that each odd-fold (in particular, simple) characteristic value of
is a bifurcation point of
. An analogous sufficient condition for the case of even-fold characteristic values is formulated using the concept of the rotation of a vector field.
If is a non-isolated solution of the equation
, then
is a bifurcation point of
. It was shown by a variational method [1], [2] that if
is a non-linear completely-continuous operator in a Hilbert space which is the gradient of a weakly continuous functional, while
is a completely-continuous self-adjoint operator, then all characteristic values of
are bifurcation points of
. The concept of a bifurcation point is also modified in the case of large solutions
for
. The importance of these concepts and results consists in the fact that, subject to relatively weak limitations, the branching of the solution
can be established; in particular, it is possible to prove that the solution of the non-linear problem is not unique. The methods of the theory of branching of solutions of non-linear equations often yield more accurate information [5].
2) The theory of smooth dynamical systems studies one-parameter (and sometimes also two-parameter [6]) families of flows (and cascades; only the former will be considered here), and the conditions under which the bifurcation is "typical" , i.e. preserves its character under a small change of the family in question [9]; both variants A) and B) above are useful. In the second variant two flows are considered equivalent if there exists a homeomorphism of the phase space which converts the trajectories of the one into trajectories of the other while preserving the direction of motion. There exists a completely satisfactory theory of bifurcations of one-parameter families of flows with a two-dimensional phase space [7], [9], and a local variant referring to a neighbourhood of an equilibrium position or of a periodic solution in the -dimensional case [6].
In variant A) the objects under study, which are associated to the given dynamical system, are the positions of an equilibrium or periodic solution, and occasionally certain invariant manifolds (mainly tori) or hyperbolic invariant sets. The "growth" of these objects — taking place "locally" near an equilibrium position, or near a periodic solution, or "semi-locally" in a neighbourhood of a "closed contour" formed by several trajectories which tend to an equilibrium position or to a periodic solution — is considered. A case of bifurcation which is connected with a similar contour in a certain sense, but which takes place (with a change of the parameter
) prior to its arrival, is also possible . The growth of periodic solutions is often conveniently considered by rewriting the differential equation and the periodicity condition in the form of an integral equation to which suitable methods are applied [5].
3) Various bifurcations of different objects (both the initial objects and those associated to them) are encountered in the theory of singularities of mappings. As a result, this term (or rather a term derived from it) is used in several different ways [10], [6], [11], but it is more common to assign independent names to the corresponding concepts. They include, for instance, versal families (or deformations) [6], [11], [12], which describe, in a certain sense, all possible bifurcations which may take place under a small deformation of the object under consideration. These include, in particular, seven elementary catastrophes [12], which are represented by "typical" -parametric (
) families of functions which include a function with a degenerate critical point and which are defined in a neighbourhood of that point; they accordingly describe the corresponding bifurcation. In non-Soviet literature on the theory of singularities, the term "catastrophes" is often employed for "bifurcations" .
References
[1] | M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian) Zbl 0111.30303 |
[2] | S.G. Krein (ed.) , Functional analysis , Wolters-Noordhoff (1972) (Translated from Russian) MR0390693 MR0352920 Zbl 0128.10302 |
[3] | M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian) Zbl 0121.10604 |
[4] | M.M. Vainberg, "Variational methods for the study of nonlinear operators" , Holden-Day (1964) (Translated from Russian) MR0176364 Zbl 0122.35501 |
[5] | M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian) MR0344960 Zbl 0274.47033 |
[6] | V.I. Arnol'd, "Lectures on bifurcations and versal families" Russian Math. Surveys , 27 : 5 (1972) pp. 54–123 Uspekhi Mat. Nauk , 27 : 5 (1972) pp. 119–184 |
[7] | A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl. (1971) (Translated from Russian) MR0344606 |
[8a] | N.K. Gavrilov, L.P. Shil'nikov, "On three-dimensional systems close to systems with a structurally unstable homoclinic curve I" Math. USSR-Sb. , 17 : 4 (1972) pp. 467–485 Mat. Sb. , 88 (130) : 4 (1972) pp. 475–492 Zbl 0255.58006 |
[8b] | N.K. Gavrilov, L.P. Shil'nikov, "On three-dimensional systems close to systems with a structurally unstable homoclinic curve II" Math. USSR-Sb. , 19 : 1 (1973) pp. 139–156 Mat. Sb. , 90 (132) : 1 (1973) pp. 139–156 Zbl 0273.58009 |
[9] | M.M. Peixoto, "On bifurcations of dynamical systems" R. James (ed.) , Proc. Internat. Congress Mathematicians (Vancouver, 1974) , 2 , Canad. Math. Congress (1975) pp. 315–319 MR0428359 Zbl 0344.58014 |
[10] | R. Thom, "The bifurcation subset of a space of maps" , Manifolds (Amsterdam, 1970) , Springer (1971) pp. 202–208 MR0281222 Zbl 0216.45901 |
[11] | V.I. Arnol'd, "Critical points of smooth functions and their normal form" Russian Math. Surveys , 30 : 5 (1975) pp. 1–75 Uspekhi Mat. Nauk , 30 : 5 (1975) pp. 3–65 |
[12] | P. Bröcker, L. Lander, "Differentiable germs and catastrophes" , Cambridge Univ. Press (1975) MR0494220 Zbl 0302.58006 |
Comments
A standard reference for bifurcation theory is [a1]. The presence of symmetry in a bifurcation problem can often be a powerful tool [a2]. For more precise details in the case of bifurcations of solutions to equations cf. Branching of solutions.
References
[a1] | S.-N. Chow, J.K. Hale, "Methods of bifurcation theory" , Springer (1982) MR0660633 Zbl 0487.47039 |
[a2] | D.H. Sallinger, "Branching in the presence of symmetry" , SIAM (1983) |
Bifurcation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bifurcation&oldid=18957