# American Institute of Mathematical Sciences

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

## Sliding Hopf bifurcation in interval systems

 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.

Citation: Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. Google Scholar [5] S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, 1995.Google Scholar [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. Google Scholar [7] D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] A. Dold, Lectures on Algebraic Topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-67821-9. Google Scholar [12] B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Springer, 1988. doi: 10.1007/BFb0082943. Google Scholar [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. Google Scholar [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. Google Scholar [15] K. Geba and W. Marzantowicz, Global bifurcation of periodic solutions, Topol. Methods Nonlinear Anal., 1 (1993), 67-93. Google Scholar [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. Google Scholar [17] M. Golubitsky and I. N. Stewart, The Symmetry Perspective, Basel-Boston-Berlin: Birkhäuser, (2002). doi: 10.1007/978-3-0348-8167-8. Google Scholar [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. Google Scholar [19] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, 8, 2003. doi: 10.1515/9783110200027. Google Scholar [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. Google Scholar [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. Google Scholar [26] H. Kielhöfer, Hopf bifurcation from a differentiable viewpoint, J. Differential Equations, 97 (1992), 189-232. doi: 10.1016/0022-0396(92)90070-4. Google Scholar [27] A. Krasnosel'skii M. and D. I. Rachinskii, On existence of cycles in autonomous systems, Doklady Math., 65 (2002), 344-349. Google Scholar [28] Krasnosel'skii A. and D. Rachinskii, On continua of cycles in systems with hysteresis, Doklady Math., 63 (2001), 339-344. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] K. Kuratowski, Topology, Vol. Ⅱ, Academic Press, New York-London; PWN -Polish Scientific Publishers, Warsaw, 1968. Google Scholar [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. Google Scholar [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. Google Scholar [35] J. Marsden and M. McCracken, Hopf Bifurcation and its Applications, Applied Mathematical Sciiences, Springer, New York, 1976. Google Scholar [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. Google Scholar [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. Google Scholar [38] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, 26, Oxford University Press, Oxford, 2002. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [4] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. Google Scholar [5] S. P. Bhattacharyya, H. Chapellat and L. H. Keel, Robust Control: The Parametric Approach, Prentice Hall, 1995.Google Scholar [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. Google Scholar [7] D. Calegari, Foliations and the Geometry of 3-Manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] A. Dold, Lectures on Algebraic Topology, Classics in Mathematics. Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-67821-9. Google Scholar [12] B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Springer, 1988. doi: 10.1007/BFb0082943. Google Scholar [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. Google Scholar [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. Google Scholar [15] K. Geba and W. Marzantowicz, Global bifurcation of periodic solutions, Topol. Methods Nonlinear Anal., 1 (1993), 67-93. Google Scholar [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. Google Scholar [17] M. Golubitsky and I. N. Stewart, The Symmetry Perspective, Basel-Boston-Berlin: Birkhäuser, (2002). doi: 10.1007/978-3-0348-8167-8. Google Scholar [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. Google Scholar [19] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, 8, 2003. doi: 10.1515/9783110200027. Google Scholar [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. Google Scholar [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. Google Scholar [26] H. Kielhöfer, Hopf bifurcation from a differentiable viewpoint, J. Differential Equations, 97 (1992), 189-232. doi: 10.1016/0022-0396(92)90070-4. Google Scholar [27] A. Krasnosel'skii M. and D. I. Rachinskii, On existence of cycles in autonomous systems, Doklady Math., 65 (2002), 344-349. Google Scholar [28] Krasnosel'skii A. and D. Rachinskii, On continua of cycles in systems with hysteresis, Doklady Math., 63 (2001), 339-344. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [32] K. Kuratowski, Topology, Vol. Ⅱ, Academic Press, New York-London; PWN -Polish Scientific Publishers, Warsaw, 1968. Google Scholar [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. Google Scholar [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. Google Scholar [35] J. Marsden and M. McCracken, Hopf Bifurcation and its Applications, Applied Mathematical Sciiences, Springer, New York, 1976. Google Scholar [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. Google Scholar [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. Google Scholar [38] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, 26, Oxford University Press, Oxford, 2002. Google Scholar [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. Google Scholar
(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
(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)
(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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