# 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:

show all references

##### References:
(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 $\{(w_1, w_2) : w_1\le 0, w_2\le0\}$
 [1] Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $S_n$-symmetry. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823 [2] Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105 [3] Ben Green, Terence Tao, Tamar Ziegler. An inverse theorem for the Gowers $U^{s+1}[N]$-norm. Electronic Research Announcements, 2011, 18: 69-90. doi: 10.3934/era.2011.18.69 [4] Azniv Kasparian, Ivan Marinov. Duursma's reduced polynomial. Advances in Mathematics of Communications, 2017, 11 (4) : 647-669. doi: 10.3934/amc.2017048 [5] Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011 [6] John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367 [7] Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907 [8] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [9] Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731 [10] Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313 [11] Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109 [12] Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036 [13] Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511 [14] Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037 [15] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [16] Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571 [17] Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451 [18] David Brander. Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane. Electronic Research Announcements, 2008, 15: 8-16. doi: 10.3934/era.2008.15.8 [19] John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7. [20] V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413

2018 Impact Factor: 1.143