Abstract
The study of nonlinear dynamical systems is of basic importance in the understanding of several natural phenomena. If a certain mathematical model is described by a nonlinear dynamical system, then it is usually difficult to predict whether or not the system will evolve towards a stationary state or it will exhibit a chaotic behavior. The sensibility to the initial conditions and to the parameters characterizing the nonlinear system show that a correct approach to study its dynamics should be more geometric. This means that the evolution system must be treated as an ordinary differential equation whose solutions (trajectories) can be viewed as curves in a suitable phase space, possibly of infinite dimension. We recall, for instance, that the chaotic behavior of some nonlinear system can be explained by the existence of a so-called strange attractor, that is, a set, usually of finite fractal dimension, which attracts uniformly any trajectory. Actually, this means that in the long run the dynamics of an infinite-dimensional system is controlled by a finite number of parameters and this can be of some help for possible numerical approximations.
Dedicated to the memory of Brunello Terreni
“... ben tetragono ai colpi di ventura”(Par., XVII, 24)
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References
L. Amerio, G. Prouse, Abstract Almost Periodic Functions and Functional Equations, Van Nostrand, New York (1971)
A.V. Babin, M.I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam (1992)
V. Belleri, V. Pata, Attractors for semilinear strongly damped wave equation on R 3, Discrete Contin. Dynam. Systems 7, 719–735 (2001)
S. Borini, V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal. 20, 263–277 (1999)
V.V. Chepyzhov, M.I. Vishik, N on-autonomous evolution equations and their attractors, Russian J. Math. Phys. 1, 165–190 (1993)
V.V. Chepyzhov, M.I. Vishik, Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl. 73, 279–333 (1994)
V.V. Chepyzhov, M.I. Vishik, Non-autonomous evolutionary equations with translation compact symbols and their attractor, C.R. Acad. Sci. Paris Sér. I Math. 321, 153–158 (1995)
B.D. Coleman, M.E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys. 18, 199–208 (1967)
CM. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37, 297–308 (1970)
B. Engquist, W. Schmid (Eds.), Mathematics Unlimited — 2001 and Beyond, Springer, Berlin Heidelberg (2001)
M. Fabrizio, A. Morro, Mathematical problems in linear viscoelasticity, SIAM Studies Appl. Math. 12, SIAM, Philadelphia (1992)
C. Giorgi, M. Grasselli, V. Pata, Well-posedness and longtime behavior of the phase-field model with memory in a history space setting, Quart. Appl. Math. 59, 701–736 (2001)
C. Giorgi, M. Grasselli, V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J. 48, 1395–1445 (1999)
C. Giorgi, A. Marzocchi, V. Pata, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl. 5, 333–354 (1998)
C. Giorgi, A. Marzocchi, V. Pata, Uniform attractors for a non-autonomous semi-linear heat equation with memory, Quart. Appl. Math. 58, 661–683 (2000)
M. Grasselli, V. Pata, Longtime behavior of a homogenized model in viscoelastody-namics, Discrete Contin. Dynam. Systems 4, 338–359 (1998)
M. Grasselli, V. Pata, Upper semicontinuous attractor for a hyperbolic phase-field model with memory, Indiana Univ. Math. J. 50, 1281–1308 (2001)
M. Grasselli, V. Pata, On the dissipativity of a hyperbolic phase-field system with memory, Nonlinear Anal. 47, 3157–3169 (2001)
M. Grasselli, V. Pata, On the longterm behavior of a parabolic phase-field model with memory, in “Differential Equations and Control Theory” (S. Aizicovici and N.H. Pavel, eds.), 147–157, Lecture Notes in Pure and Appl. Math. 225, Dekker, New York (2002)
M. Grasselli, V. Pata, F.M. Vegni, Longterm dynamics of a conserved phase-field system with memory, submitted
M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31, 113–126 (1968)
Y.M. Haddad, Viscoelasticity of engineering materials, Chapman & Hall, London (1985)
J.K. Hale, Asymptotic behavior of dissipative systems, Math. Surv. Monogr. n. 25, Amer. Math. Soc., Providence (1988)
A. Haraux, Systèmes dynamiques dissipatifs et applications, Coll. RMA n. 17, Masson, Paris (1990)
J. Jackie, Heat conduction and relaxation in liquids of high viscosity, Phys. A 162, 377–404 (1990)
B.M. Levitan, V.V. Zhikov, Almost periodic functions and differential equations, Cambridge University Press, Cambridge (1982)
Z. Liu, S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall/CRC Res. Notes Math. n. 398, Boca Raton (1999)
V. Pata, Attractors for a damped wave equation on R 3 with linear memory, Math. Methods Appl. Sci. 23, 633–653 (2000)
V. Pata, Hyperbolic limit of parabolic semilinear heat equation with fading memory, Z. Anal. Anwendungen 20, 359–377 (2001)
V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl. (to appear)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York (1983)
M. Renardy, W.J. Hrusa, J.A. Nohel, Mathematical problems in viscoelasticity, Longman Scientific & Technical; Harlow John Wiley & Sons, Inc., New York (1987)
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Springer, New York (1988)
M.I. Vishik, Asymptotic behaviour of solutions of evolutionary equations, Cambridge University Press, Cambridge (1992)
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Grasselli, M., Pata, V. (2002). Uniform Attractors of Nonautonomous Dynamical Systems with Memory. In: Lorenzi, A., Ruf, B. (eds) Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8221-7_9
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DOI: https://doi.org/10.1007/978-3-0348-8221-7_9
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