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• 1. Misiurewicz, Michal
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Fixed points for positive permutation braids2012In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 216, no 2, p. 129-146Article in journal (Refereed)

Making use of the Nielsen fixed point theory, we study a conjugacy invariant of braids, which we call the level index function. We present a simple algorithm for computing it for positive permutation cyclic braids.

• 2. Misiurewicz, Michal
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Turning numbers for periodic orbits of disk homeomorphisms2013In: Journal of Fixed Point Theory and Applications, ISSN 1661-7738, E-ISSN 1661-7746, Vol. 13, no 1, p. 241-258Article in journal (Refereed)

We study braid types of periodic orbits of orientation preserving disk homeomorphisms. If the orbit has period n, we take the closure of the nth power of the corresponding braid and consider linking numbers of the pairs of its components, which we call turning numbers. They are easy to compute and turn out to be very useful in the problem of classification of braid types, especially for small n. This provides us with a simple way of getting useful information about periodic orbits. The method works especially well for disk homeomorphisms that are small perturbations of interval maps.

• 3.
CMUP and Departamento de Matemática Pura, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal .
CMUP and Departamento de Matemática Pura, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal .
Hopf bifurcation with SN-symmetry2009In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, no 3, p. 627-666Article in journal (Refereed)

We study Hopf bifurcation with SN-symmetry for the standard absolutely irreducible action of SN obtained from the action of SN by permutation of N coordinates. Stewart (1996 Symmetry methods in collisionless many-body problems, J. Nonlinear Sci. 6 543–63) obtains a classification theorem for the C-axial subgroups of SN × S1. We use this classification to prove the existence of branches of periodic solutions with C-axial symmetry in systems of ordinary differential equations with SN-symmetry posed on a direct sum of two such SN-absolutely irreducible representations, as a result of a Hopf bifurcation occurring as a real parameter is varied. We determine the (generic) conditions on the coefficients of the fifth order SN × S1-equivariant vector field that describe the stability and criticality of those solution branches. We finish this paper with an application to the cases N = 4 and N = 5.

• 4.
CMUP and Dep. de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal.
Centro de Matemática da Universidade do Porto (CMUP) and Dep. de Matemática Pur, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
Generating functions for Hopf bifurcation with Sn-symmetry2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 25, no 3, p. 823-842Article in journal (Refereed)

Hopf bifurcation in the presence of the symmetric group $\displaystyle{S}_{{n}}$ (acting naturally by permutation of coordinates) is a problem with relevance to coupled oscillatory systems. To study this bifurcation it is important to construct the Taylor expansion of the equivariant vector field in normal form. We derive generating functions for the numbers of linearly independent invariants and equivariants of any degree, and obtain recurrence relations for these functions. This enables us to determine the number of invariants and equivariants for all $\displaystyle{n}$, and show that this number is independent of $\displaystyle{n}$ for sufficiently large $\displaystyle{n}$. We also explicitly construct the equivariants of degree three and degree five, which are valid for arbitrary $\displaystyle{n}$.

• 5.
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216, USA and CMUP, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal .
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Spain .
On the periodic orbits of Hamiltonian systems2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, no 4Article in journal (Refereed)

We show how to apply to Hamiltonian differential systems recent results for studying the periodic orbits of a differential system using the averaging theory. We have chosen two classical integrable Hamiltonian systems, one with the Hooke potential and the other with the Kepler potential, and we study the periodic orbits which bifurcate from the periodic orbits of these integrable systems, first perturbing the Hooke Hamiltonian with a nonautonomous potential, and second perturbing the Kepler problem with an autonomous potential.

• 6.
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216 – and – CMUP, Rua do Campo Alegre 687, 4169-007 Porto, Portugal .
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216.
Non-generic cusps2011In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 363, no 7, p. 3553-3572Article in journal (Refereed)

We find the order of contact of the boundaries of the cusp for two-parameter families of vector fields on the real line or diffeomorphisms of the real line, for cusp bifurcations of codimensions 1 and 2. Moreover, we create a machinery that can be used for the same problem in higher codimensions and perhaps for other, similar problems.

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