Sliding Hopf bifurcation in interval systems

Hooton Edward; Balanov Zalman; Krawcewicz Wieslaw; Rachinskii Dmitrii

doi:10.3934/dcds.2017152

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August 2017, 37(7): 3545-3566. doi: 10.3934/dcds.2017152

Sliding Hopf bifurcation in interval systems

Hooton Edward , Balanov Zalman , Krawcewicz Wieslaw and Rachinskii Dmitrii

The University of Texas at Dallas, 800 W Campbell Road, Richardson, Tx 75080, USA

Received December 2015 Revised February 2017 Published April 2017

Fund Project: The authors were supported by National Science Foundation grant DMS-1413223.
The second author is thankful for the support from the Gelbart Research Institute through Bar Ilan University (Israel).
The third author was also supported by the Chutian Scholar Program at China Three Gorges University, Yichang, Hubei (China).

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.

Keywords: Hopf bifurcation, interval polynomial, non-local branch, Kharitonov's theorem, zero exclusion principle, S¹-degree.

Mathematics Subject Classification: Primary: 37G15; Secondary: 37G40.

Citation: Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

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.

Figure 1. (a) Given an $ \alpha $ -parametrized family of characteristic polynomials with unknown coefficients that are limited to some intervals, the dashed lines bound a corridor for the real part $ \tau(\alpha) = {\rm Re} \mu(\alpha) $ of an eigenvalue, while the solid line indicates a sliding scenario for some selector of the family.(b) Possible complex behavior of the branch of periodic solutions

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Figure 2. (a) The dark grey domain that consists of two connected components is the set $ \mathfrak{R} $ of purely imaginary characteristic roots $ i\beta $ of the interval polynomial (17) for Example 4.1.The two solid black curves inside the two components of $ \mathfrak{R} $ show the set of purely imaginary roots for a representative polynomial $ P(\alpha)(\cdot) $ that belongs to the family (17).This representative has two purely imaginary roots $ i\beta_1(\alpha) $ , $ i\beta_2(\alpha) $ for some interval of $ \alpha $ values $ [\alpha_1, \alpha_2]\subset (\alpha_-, \alpha_+) = (-0.4, 0.8) $ .The light grey domains are the sets $ \mathfrak{S}_2 $ and $ \mathfrak{S}_3 $ obtained from the dark grey domain $ \mathfrak{R} $ by the transformations $ (\alpha, \beta) \mapsto (\alpha, \beta/2) $ and $ (\alpha, \beta) \mapsto (\alpha, \beta/3) $ , respectively; the dashed curves inside $ \mathfrak{S}_2 $ are the images of the solid black curves in $ \mathfrak{R} $ under this transformation.The intersection of the solid curve and the dashed curve inside the smaller component of $ \mathfrak{R} $ corresponds to the $ 2:1 $ resonance $ i\beta_1(\alpha) = 2 i\beta_2(\alpha) $ .The dashed quadrangle $ \mathcal{D}_1 $ contains the set $ \mathfrak{R} $ ; its boundary does not intersect $ \mathfrak{S}_i $ in accordance with (R5′).(b) The real parts $ \tau_1(\alpha) $ , $ \tau_2(\alpha) $ of the roots of the representative polynomial $ P(\alpha, \cdot) $ that belongs to the family (17) (schematic).The sliding intervals $ \tau_1(\alpha) = 0 $ , $ \tau_2(\alpha) = 0 $ correspond to the black curves $ \beta_1(\alpha) $ , $ \beta_2(\alpha) $ shown inside the dark grey domain $ \mathfrak{R} $ on panel (a)

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Figure 3. (a) The nonintersecting sets $ \mathfrak{R} $ (dark grey) and $ \mathfrak{S} $ (light grey) for Example 4.2 with $ [\alpha_1, \alpha_2] = [-0.5, 1.2] $ .The black curve is the set of purely imaginary roots of a representative polynomial $ P(\alpha)(\cdot) $ that belongs to the family (17).At the corner point of this curve, $ P(\alpha)(\cdot) $ has a purely imaginary root of multiplicity 2.The real parts of the roots of $ P(\alpha)(\cdot) $ behave as shown on panel (b) of Figure 2. (b) Curves $ (w_1(\alpha_\pm, \cdot), w_2(\alpha_\pm, \cdot)) $ (thick lines) for Example 4.3. The thin curve $ (w_1(\alpha, \cdot), w_2(\alpha, \cdot)) $ with $ \alpha = 0.075 $ from the interior of the interval $ [\alpha_-, \alpha_+] $ intersects the negative cone $ \{(w_1, w_2) : w_1\le 0, w_2\le0\} $

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Sliding Hopf bifurcation in interval systems

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