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Topological stability from Gromov-Hausdorff viewpoint
Sliding Hopf bifurcation in interval systems
The University of Texas at Dallas, 800 W Campbell Road, Richardson, Tx 75080, USA |
Abstract. In this paper, the equivariant degree theory is used to analyze the occurrence of the Hopf bifurcation under effectively verifiable mild conditions. We combine the abstract result with standard interval polynomial techniques based on Kharitonov's theorem to show the existence of a branch of periodic solutions emanating from the equilibrium in the settings relevant to robust control. The results are illustrated with a number of examples.
References:
[1] |
J. C. Alexander and J. Yorke,
Global bifurcations of periodic orbits, Amer. J. Math., 100 (1978), 263-292.
doi: 10.2307/2373851. |
[2] |
Z. Balanov and W. Krawcewicz,
Symmetric Hopf Bifurcation: Twisted Degree Approach, In Battelli, F., and Feckan, M. (eds.), Handbook of Differential Equations, Handbook of Differential Equations, Ordinary Differential Equations, 4, Elsevier/North-Holland, Amsterdam, (2008), 1-131.
doi: 10.1016/S1874-5725(08)80006-5. |
[3] |
Z. Balanov, W. Krawcewicz, D. Rachinskii and A. Zhezherun,
Hopf bifurcation in symmetric networks of coupled oscillators with hysteresis, J. Dynam. Differential Equations, (2012), 713-759.
doi: 10.1007/s10884-012-9271-4. |
[4] |
Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. |
[5] |
S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, 1995. |
[6] |
T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985.
doi: 10.1007/978-3-662-12918-0. |
[7] |
D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. |
[8] |
A. R. Champneys, G. W. Hunt and J. M. T. Thompson (eds.), Localization and Solitary Waves in Solid Mechanics, World Scientific Pub, 1999.
doi: 10.1142/9789812814876. |
[9] |
S. N. Chow, J. Mallet-Paret and J. A. Yorke,
Global Hopf bifurcation from a multiple eigenvalue, Nonlinear Anal., 2 (1978), 753-763.
doi: 10.1016/0362-546X(78)90017-2. |
[10] |
E. N. Dancer and J. F. Toland,
The index change and global bifurcation for flows with first integrals, Proc. London Math. Soc., 66 (1993), 539-567.
doi: 10.1112/plms/s3-66.3.539. |
[11] |
A. Dold, Lectures on Algebraic Topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-3-642-67821-9. |
[12] |
B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Springer, 1988.
doi: 10.1007/BFb0082943. |
[13] |
B. Fiedler, Global Hopf bifurcation in porous catalysts, in Equadiff 82 (Würzburg, 1982), Lecture Notes in Math., Springer, Berlin, 1017 (1983), 177–184.
doi: 10.1007/BFb0103250. |
[14] |
F. B. Fuller,
An index of fixed point type for periodic orbits, American Journal of Mathematics, 89 (1967), 133-148.
doi: 10.2307/2373103. |
[15] |
K. Geba and W. Marzantowicz,
Global bifurcation of periodic solutions, Topol. Methods Nonlinear Anal., 1 (1993), 67-93.
|
[16] |
M. Golubitsky and W. F. Langford,
Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41 (1981), 375-415.
doi: 10.1016/0022-0396(81)90045-0. |
[17] |
M. Golubitsky and I. N. Stewart,
The Symmetry Perspective, Basel-Boston-Berlin: Birkhäuser, (2002).
doi: 10.1007/978-3-0348-8167-8. |
[18] |
M. Golubitsky, I. N. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. Ⅱ. Applied Mathematical Sciences 69, Springer, Berlin - New York, 1988.
doi: 10.1007/978-1-4612-4574-2. |
[19] |
J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988. |
[20] |
E. Hopf,
Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems. (German), Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 95 (1943), 3-22.
|
[21] |
J. Ize,
Obstruction theory and multiparameter Hopf bifurcation, Trans. Amer. Math. Soc., 289 (1985), 757-792.
doi: 10.1090/S0002-9947-1985-0784013-2. |
[22] |
J. Ize,
Equivariant degree, Handbook of topological fixed point theory, Springer, Dordrecht, 337 (2005), 301-337.
doi: 10.1007/1-4020-3222-6_9. |
[23] |
J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, 8, 2003.
doi: 10.1515/9783110200027. |
[24] |
T. Kalmar-Nagy, G. Stepan and F. C. Moon,
Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics, 26 (2001), 121-142.
doi: 10.1023/A:1012990608060. |
[25] |
V. L. Kharitonov,
Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differensial'nye Uravnenya, 14 (1978), 2086-2088.
|
[26] |
H. Kielhöfer,
Hopf bifurcation from a differentiable viewpoint, J. Differential Equations, 97 (1992), 189-232.
doi: 10.1016/0022-0396(92)90070-4. |
[27] |
A. Krasnosel'skii M. and D. I. Rachinskii,
On existence of cycles in autonomous systems, Doklady Math., 65 (2002), 344-349.
|
[28] |
Krasnosel'skii A. and D. Rachinskii,
On continua of cycles in systems with hysteresis, Doklady Math., 63 (2001), 339-344.
|
[29] |
W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1997. |
[30] |
P. Krejčı, J.P. O'Kane, A. Pokrovskii and D. Rachinskii,
Properties of solutions to a class of differential models incorporating Preisach hysteresis operator, Physica D: Nonlinear Phenomena, 241 (2012), 2010-2028.
doi: 10.1016/j.physd.2011.05.005. |
[31] |
V. S. Kozyakin and M. A. Krasnosel'skii,
The method of parameter functionalization in the Hopf bifurcation problem, Nonlinear Anal., 11 (1987), 149-161.
doi: 10.1016/0362-546X(87)90095-2. |
[32] |
K. Kuratowski, Topology, Vol. Ⅱ, Academic Press, New York-London; PWN -Polish Scientific Publishers, Warsaw, 1968. |
[33] |
A. Kushkuley and Z. Balanov, Geometric Methods in Degree Theory for Equivariant Maps, Lecture Notes in Math. , 1632, Springer-Verlag, Berlin, 1996.
doi: 10.1007/BFb0092822. |
[34] |
J. Mallet-Paret and J. A. Yorke,
Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations, 43 (1982), 419-450.
doi: 10.1016/0022-0396(82)90085-7. |
[35] |
J. Marsden and M. McCracken, Hopf Bifurcation and its Applications, Applied Mathematical Sciiences, Springer, New York, 1976. |
[36] |
C. Ning and H. Haken,
Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations, Phys. Rev. A, 41 (1990), 3826-3837.
doi: 10.1103/PhysRevA.41.3826. |
[37] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, 1986.
doi: 10.1090/cbms/065. |
[38] |
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, 26, Oxford University Press, Oxford, 2002. |
[39] |
J. Wu,
Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |
show all references
References:
[1] |
J. C. Alexander and J. Yorke,
Global bifurcations of periodic orbits, Amer. J. Math., 100 (1978), 263-292.
doi: 10.2307/2373851. |
[2] |
Z. Balanov and W. Krawcewicz,
Symmetric Hopf Bifurcation: Twisted Degree Approach, In Battelli, F., and Feckan, M. (eds.), Handbook of Differential Equations, Handbook of Differential Equations, Ordinary Differential Equations, 4, Elsevier/North-Holland, Amsterdam, (2008), 1-131.
doi: 10.1016/S1874-5725(08)80006-5. |
[3] |
Z. Balanov, W. Krawcewicz, D. Rachinskii and A. Zhezherun,
Hopf bifurcation in symmetric networks of coupled oscillators with hysteresis, J. Dynam. Differential Equations, (2012), 713-759.
doi: 10.1007/s10884-012-9271-4. |
[4] |
Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. |
[5] |
S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, 1995. |
[6] |
T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985.
doi: 10.1007/978-3-662-12918-0. |
[7] |
D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. |
[8] |
A. R. Champneys, G. W. Hunt and J. M. T. Thompson (eds.), Localization and Solitary Waves in Solid Mechanics, World Scientific Pub, 1999.
doi: 10.1142/9789812814876. |
[9] |
S. N. Chow, J. Mallet-Paret and J. A. Yorke,
Global Hopf bifurcation from a multiple eigenvalue, Nonlinear Anal., 2 (1978), 753-763.
doi: 10.1016/0362-546X(78)90017-2. |
[10] |
E. N. Dancer and J. F. Toland,
The index change and global bifurcation for flows with first integrals, Proc. London Math. Soc., 66 (1993), 539-567.
doi: 10.1112/plms/s3-66.3.539. |
[11] |
A. Dold, Lectures on Algebraic Topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-3-642-67821-9. |
[12] |
B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Springer, 1988.
doi: 10.1007/BFb0082943. |
[13] |
B. Fiedler, Global Hopf bifurcation in porous catalysts, in Equadiff 82 (Würzburg, 1982), Lecture Notes in Math., Springer, Berlin, 1017 (1983), 177–184.
doi: 10.1007/BFb0103250. |
[14] |
F. B. Fuller,
An index of fixed point type for periodic orbits, American Journal of Mathematics, 89 (1967), 133-148.
doi: 10.2307/2373103. |
[15] |
K. Geba and W. Marzantowicz,
Global bifurcation of periodic solutions, Topol. Methods Nonlinear Anal., 1 (1993), 67-93.
|
[16] |
M. Golubitsky and W. F. Langford,
Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41 (1981), 375-415.
doi: 10.1016/0022-0396(81)90045-0. |
[17] |
M. Golubitsky and I. N. Stewart,
The Symmetry Perspective, Basel-Boston-Berlin: Birkhäuser, (2002).
doi: 10.1007/978-3-0348-8167-8. |
[18] |
M. Golubitsky, I. N. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. Ⅱ. Applied Mathematical Sciences 69, Springer, Berlin - New York, 1988.
doi: 10.1007/978-1-4612-4574-2. |
[19] |
J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988. |
[20] |
E. Hopf,
Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems. (German), Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 95 (1943), 3-22.
|
[21] |
J. Ize,
Obstruction theory and multiparameter Hopf bifurcation, Trans. Amer. Math. Soc., 289 (1985), 757-792.
doi: 10.1090/S0002-9947-1985-0784013-2. |
[22] |
J. Ize,
Equivariant degree, Handbook of topological fixed point theory, Springer, Dordrecht, 337 (2005), 301-337.
doi: 10.1007/1-4020-3222-6_9. |
[23] |
J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, 8, 2003.
doi: 10.1515/9783110200027. |
[24] |
T. Kalmar-Nagy, G. Stepan and F. C. Moon,
Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynamics, 26 (2001), 121-142.
doi: 10.1023/A:1012990608060. |
[25] |
V. L. Kharitonov,
Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differensial'nye Uravnenya, 14 (1978), 2086-2088.
|
[26] |
H. Kielhöfer,
Hopf bifurcation from a differentiable viewpoint, J. Differential Equations, 97 (1992), 189-232.
doi: 10.1016/0022-0396(92)90070-4. |
[27] |
A. Krasnosel'skii M. and D. I. Rachinskii,
On existence of cycles in autonomous systems, Doklady Math., 65 (2002), 344-349.
|
[28] |
Krasnosel'skii A. and D. Rachinskii,
On continua of cycles in systems with hysteresis, Doklady Math., 63 (2001), 339-344.
|
[29] |
W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1997. |
[30] |
P. Krejčı, J.P. O'Kane, A. Pokrovskii and D. Rachinskii,
Properties of solutions to a class of differential models incorporating Preisach hysteresis operator, Physica D: Nonlinear Phenomena, 241 (2012), 2010-2028.
doi: 10.1016/j.physd.2011.05.005. |
[31] |
V. S. Kozyakin and M. A. Krasnosel'skii,
The method of parameter functionalization in the Hopf bifurcation problem, Nonlinear Anal., 11 (1987), 149-161.
doi: 10.1016/0362-546X(87)90095-2. |
[32] |
K. Kuratowski, Topology, Vol. Ⅱ, Academic Press, New York-London; PWN -Polish Scientific Publishers, Warsaw, 1968. |
[33] |
A. Kushkuley and Z. Balanov, Geometric Methods in Degree Theory for Equivariant Maps, Lecture Notes in Math. , 1632, Springer-Verlag, Berlin, 1996.
doi: 10.1007/BFb0092822. |
[34] |
J. Mallet-Paret and J. A. Yorke,
Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations, 43 (1982), 419-450.
doi: 10.1016/0022-0396(82)90085-7. |
[35] |
J. Marsden and M. McCracken, Hopf Bifurcation and its Applications, Applied Mathematical Sciiences, Springer, New York, 1976. |
[36] |
C. Ning and H. Haken,
Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations, Phys. Rev. A, 41 (1990), 3826-3837.
doi: 10.1103/PhysRevA.41.3826. |
[37] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics 65, American Mathematical Society, Providence, 1986.
doi: 10.1090/cbms/065. |
[38] |
Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, 26, Oxford University Press, Oxford, 2002. |
[39] |
J. Wu,
Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |



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