Carlos Rocha

Departamento de Matemática
Instituto Superior Técnico
Av. Rovisco Pais, 1049-001 Lisboa Codex

phone: (351) 218417102

fax: (351) 218417035

phone: (351) 218417102

email: crocha@tecnico.ulisboa.pt

Position	Professor Catedrático (Aposentado Dez 2020)
Education	Ph.D. in Applied Mathematics, 1985, Brown University
	M.Sc. in Applied Mathematics, 1983, Brown University
	Licenciado in Electr. Engineering, 1975, Instituto Superior Técnico
Research	Center for Mathematical Analysis, Geometry and Dynamical Systems

Research Interests

Dynamical Systems, Differential Equations, Bifurcation Theory, Parabolic Partial Differential Equations

Selected Publications

• C. Rocha and B. Fiedler 
  Design of Sturm global attractors 1: Meanders with three noses, and reversibility, 
  Chaos: An Interdisciplinary Journal of Nonlinear Science,  33, 083127 (2023).
• C. Rocha and B. Fiedler 
  Meanders, zero numbers and the cell structure of Sturm global attractors, 
  J. Dynam. Differential Equations,  To appear.
• B. Fiedler and C. Rocha 
  Boundary orders and geometry of the signed Thom-Smale complex for Sturm global attractors, 
  J. Dynam. Differential Equations, 34 (2022), 4: 2787-2818.
• B. Fiedler and C. Rocha,
  Sturm 3-ball global attractors 2: Design of Thom-Smale complexes, 
  J. Dynam. Differential Equations, 31 (2019), 3: 1549-1590.
• B. Fiedler and C. Rocha,
  Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes, 
  Discrete Contin. Dyn. Syst., 38 (2018), 7: 3479-3545.
• B. Fiedler and C. Rocha,
  Sturm 3-ball global attractors 1: Thom-Smale complexes and meanders, 
  São Paulo J. Math. Sci., 12 (2018), 1: 18-67.
• R. Czaja, W. Oliva and C. Rocha,
  On a definition of Morse-Smale evolution processes, 
  Discrete Contin. Dyn. Syst., 37 (2017), 7: 3601-3623.
• J. Pimentel and C. Rocha,
  A permutation related to non-compact global attractors for slowly non-dissipative systems, 
  J. Dyn. Diff. Eq. 28 (2016), 1: 1-28.
• J. Pimentel and C. Rocha,
  Non-compact global attractors for scalar reaction-diffusion equations, 
  São Paulo J. Math. Sci., 9 (2015), 299-390.
• B. Fiedler and C. Rocha,
  Schoenflies spheres as boundaries of bounded unstable manifolds in gradient Sturm systems, 
  J. Dyn. Diff. Eq. 27 (2015), 3-4: 597-626.
• B. Fiedler and C. Rocha,
  Nonlinear Sturm global attractors: unstable manifold decompositions as CW-complexes, 
  Discrete Contin. Dyn Syst., 34 (2014), 12: 5099-5122.
• B. Fiedler, C. Ragazzo and C. Rocha,
  An explicit Lyapunov function for mirror-symmetric parabolic partial differential equations on the circle, 
  Russian Math. Surveys,  69 (2014), 3: 419-433.
• C. Rocha,
  Orbit Equivalence of Global Attractors for S¹-Equivariant Parabolic Equations,
  São Paulo J. Math. Sci., 6 (2012), 2: 365-374.
• B. Fiedler, C. Rocha and M. Wolfrum,
  Sturm global attractors for S¹-equivariant parabolic equations, 
  Netw. Heterog. Media., 7 (2012), 4: 617-659.
• B. Fiedler, C. Rocha and M. Wolfrum,
  A Permutation Characterization of Sturm Global Attractors of Hamiltonian Type,
  J. Differential Equations, 252 (2012), 1: 588-623.
• W. Oliva and C. Rocha,
  Reducible Volterra and Levin-Nohel Retarded Equations with Infinite Delay,
  J. Dyn. Diff. Eq, 22 (2010), 3: 509-532.
• B. Fiedler and C. Rocha,
  Connectivity and Design of Planar Global Attractors of Sturm Type.  III: Small and Platonic Examples,
  J. Dyn. Diff. Eq, 22 (2010), 2: 121-162.
• B. Fiedler and C. Rocha,
  Connectivity and Design of Planar Global Attractors of Sturm Type.  I: Bipolar Orientations and Hamiltonian Paths,
  J. Reine Angew. Math, 635 (2009), 71-96.
• W. Oliva and C. Rocha,
  Reducible Volterra Retarded Equations with Infinite Delay,
  São Paulo J. Math. Sci., 2 (2008), 1: 37-53.
• R. Czaja and C. Rocha,
  Transversality in Scalar Reaction-Diffusion Equations on a Circle,
  J. Differential Equations, 245 (2008), 3: 692-721.
• B. Fiedler and C. Rocha,
  Connectivity and Design of Planar Global Attractors of Sturm Type.  II: Connection Graphs,
  J. Differential Equations, 244 (2008), 5: 1255-1286.
• C. Rocha,
  Realization of Period Maps of Planar Hamiltonian Systems,
  J. Dyn. Diff. Eq. 19 (2007), 3: 571-591.
• B. Fiedler, C. Rocha and M. Wolfrum,
  Heteroclinic Orbits Between Rotating Waves of Semilinear Parabolic Equations on the Circle,
  J. Differential Equations 201 (2004), 1: 99-138.
• P. Freitas and C. Rocha,
  Lyapunov Functionals and Stability for FitzHugh-Nagumo Systems,
  J. Differential Equations 169 (2001), 1: 208-227.
• B. Fiedler and C. Rocha,
  Orbit Equivalence of Global Attractors of Semilinear Parabolic Differential Equations,
  Trans. Amer. Math. Soc. 352 (2000), no. 1, 257-284.
• B. Fiedler and C. Rocha,
  Realization of Meander Permutations by Boundary Value Problems,
  J. Differential Equations 156 (1999), 2: 282-308.
• B. Fiedler and C. Rocha,
  Heteroclinic Orbits of Semilinear Parabolic Equations,
  J. Differential Equations 125 (1996), 1: 239-281.
• C. Rocha,
  Bifurcations in Discretized Reaction-Diffusion Equations,
  Resenhas IME-USP 4 (1994), 1: 403-419.
• C. Rocha,
  On the Singular Problem for the Scalar Parabolic Equation with Variable Diffusion,
  J. of Math. Analysis Appl. 183 (1994), 2: 413-428.
• C. Rocha,
  Properties of the Attractor of a Scalar Parabolic PDE,
  J. Dyn. Diff. Eq. 3 (1991), 4: 575-591.
• G. Fusco and C. Rocha,
  A Permutation Related to the Dynamics of a Scalar Parabolic PDE,
  J. Differential Equations 91 (1991), 1: 75-94.
• C. Rocha,
  Examples of Attractors in Scalar Reaction-Diffusion Equations,
  J. Differential Equations 73 (1988), 1: 178-195.
• J. K. Hale and C. Rocha,
  Interaction of Diffusion and Boundary Conditions,
  Nonlinear Analysis 11 (1987), 5: 633-649.
• J. K. Hale and C. Rocha,
  Varying Boundary Conditions with Large Diffusivity,
  J. Mathématiques Pures Appl. 66 (1987), 2: 139-158.
• C. Rocha,
  Generic Properties of Equilibria of Reaction-Diffusion Equations,
  Proc. of the Roy. Soc. Edinburgh 101A (1985), 1-2: 45-55.
• J. K. Hale and C. Rocha,
  Bifurcations in a Parabolic Equation with Variable Diffusion,
  Nonlinear Analysis 9 (1985), 5: 479-494.

Proceedings

• C. Rocha and B. Fiedler 
  Sturm global attractors of Hamiltonian type for semilinear parabolic equations, 
  in Nonlinear Partial Differential Equations, Dynamical Systems and Their Applications, M. Tanigushi (ed.),
  RIMS Kôkyûroku 1881, (2014) 139-158.
• C. Rocha,
  Transversality in Semilinear Parabolic Equations on the Circle,
  in Proceedings of the International Conference on Differential Equations - Equadiff 2003,
  F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede and S. V. Lunel (eds.)
  World Scientific, (2005) 672-677.
• B. Fiedler, C. Rocha, D. Salazar and J. Solà-Morales,
  Dynamics of Piecewise-Autonomous Bistable Parabolic Equations,
  in Differential Equations and Dynamical Systems, Galves, Hale and Rocha (eds.)
  Fields Institute Communications 31, (2002) 151-163.
• C. Rocha,
  Some Remarks on the Singular Problem for the Scalar Parabolic Equation with Variable Diffusion,
  in International Conference on Differential Equations, Equadiff 91, Perelló, Simó and Solà-Morales (eds.)
  World Scientific, (1993) 832-837.
• P. Freitas and C. Rocha,
  Bogdanov Singularity in the FitzHugh-Nagumo Equations,
  in International Conference on Differential Equations, Equadiff 91, Perelló, Simó and Solà-Morales (eds.)
  World Scientific, (1993) 496-500.
• C. Rocha,
  Heteroclinic Connections in Parabolic Equations,
  in Advanced Topics in the Theory of Dynamical Systems, Fusco, Iannelli and Salvadori (eds.)
  Academic Press, (1989) 205-211.
• C. Rocha,
  Examples of Attractors in Scalar Reaction-Diffusion Equations,
  in Dynamics of Infinite Dimensional Systems, Chow and Hale (eds.)
  Springer-Verlag, (1987) 257-272.

Books (as editor)

• Differential Equations and Dynamical Systems,
  A. Galves, J. K. Hale and C. Rocha
  Fields Institute Communications 31, Amer. Math. Soc. 2002.
• International Conference on Differential Equations, Equadiff 95,
  L. Magalhães, C. Rocha and L. Sanchez
  World Scientific, Singapore 1993.

An Introduction to Infinite Dimensional Dynamical Systems

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