163 research outputs found
Linear chaos for the Quick-Thinking-Driver model
The final publication is available at Springer via http://dx.doi.org/10.1007/s00233-015-9704-6In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars (Google driverless cars, http://en.wikipedia.org/wiki/Google_driverless_car).Driving models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behaviour using the same approach as for studying chaos of death models of cell growth.The authors were supported by a grant from the FPU program of MEC and MEC Project MTM2013-47093-P.Conejero, JA.; Murillo Arcila, M.; Seoane-SepĂșlveda, JB. (2016). Linear chaos for the Quick-Thinking-Driver model. Semigroup Forum. 92(2):486-493. https://doi.org/10.1007/s00233-015-9704-6S486493922Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Equ. Appl. 18(4), 647â655 (2012)Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris SĂ©rie II 329, 439â444 (2001)Banasiak, J., Lachowicz, M.: Topological chaos for birth-and-death-type models with proliferation. Math. Models Methods Appl. Sci. 12(6), 755â775 (2002)Banasiak, J., Lachowicz, M., MoszyĆski, M.: Topological chaos: when topology meets medicine. Appl. Math. Lett. 16(3), 303â308 (2003)Banasiak, J., MoszyĆski, M.: A generalization of DeschâSchappacherâWebb criteria for chaos. Discret. Contin. Dyn. Syst. 12(5), 959â972 (2005)Banasiak, J., MoszyĆski, M.: Dynamics of birth-and-death processes with proliferationâstability and chaos. Discret. Contin. Dyn. Syst. 29(1), 67â79 (2011)Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaneyâs definition of chaos. Am. Math. Mon. 99(4), 332â334 (1992)Barrachina, X., Conejero, J.A.: Devaney chaos and distributional chaos in the solution of certain partial differential equations. Abstr. Appl. Anal. 457,019, 11 (2012)BermĂșdez, T., Bonilla, A., MartĂnez-GimĂ©nez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83â93 (2011)Brackstone, M., McDonald, M.: Car-following: a historical review. Transp. Res. Part F 2(4), 181â196 (1999)BrzeĆșniak, Z., Dawidowicz, A.L.: On periodic solutions to the von FoersterâLasota equation. Semigroup Forum 78, 118â137 (2009)Chandler, R.E., Herman, R., Montroll, E.W.: Traffic dynamics: studies in car following. Op. Res. 6, 165â184 (1958)Chung, C.C., Gartner, N.: Acceleration noise as a measure of effectiveness in the operation of traffic control systems. Operations Research Center. Massachusetts Institute of Technology. Cambridge (1973)CNN (2014) Driverless car tech gets serious at CES. http://edition.cnn.com/2014/01/09/tech/innovation/self-driving-cars-ces/ . Accessed 7 Apr 2014Conejero, J.A., Rodenas, F., Trujillo, M.: Chaos for the hyperbolic bioheat equation. Discret. Contin. Dyn. Syst. 35(2), 653â668 (2015)DARPA Grand Challenge. http://en.wikipedia.org/wiki/2005_DARPA_Grand_Challenge#2005_Grand_Challengede Laubenfels, R., Emamirad, H., Protopopescu, V.: Linear chaos and approximation. J. Approx. Theory 105(1), 176â187 (2000)Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17(4), 793â819 (1997)El Mourchid, S.: The imaginary point spectrum and hypercyclicity. Semigroup Forum 73(2), 313â316 (2006)El Mourchid, S., Metafune, G., Rhandi, A., Voigt, J.: On the chaotic behaviour of size structured cell populations. J. Math. Anal. Appl. 339(2), 918â924 (2008)El Mourchid, S., Rhandi, A., Vogt, H., Voigt, J.: A sharp condition for the chaotic behaviour of a size structured cell population. Differ. Integral Equ. 22(7â8), 797â800 (2009)Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York, 2000. With contributions by Brendle S., Campiti M., Hahn T., Metafune G., Nickel G., Pallara D., Perazzoli C., Rhandi A., Romanelli S., and Schnaubelt RGodefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229â269 (1991)Greenshields, B.D.: The photographic method of studying traffic behavior. In: Proceedings of the 13th Annual Meeting of the Highway Research Board, pp. 382â399 (1934)Greenshields, B.D.: A study of traffic capacity. In: Proceedings of the 14th Annual Meeting of the Highway Research Board, pp. 448â477 (1935)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear Chaos. Universitext. Springer, London (2011)Herman, R., Montroll, E.W., Potts, R.B., Rothery, R.W.: Traffic dynamics: analysis of stability in car following. Op. Res. 7, 86â106 (1959)Helly, W.: Simulation of Bottleneckes in Single-Lane Traffic Flow. Research Laboratories, General Motors. Elsevier, New York (1953)Li, T.: Nonlinear dynamics of traffic jams. Phys. D 207(1â2), 41â51 (2005)Lo, S.C., Cho, H.J.: Chaos and control of discrete dynamic traffic model. J. Franklin Inst. 342(7), 839â851 (2005)MartĂnez-GimĂ©nez, F., Oprocha, P., Peris, A.: Distributional chaos for backward shifts. J. Math. Anal. Appl. 351(2), 607â615 (2009)Pipes, L.A.: An operational analysis of traffic dynamics. J. Appl. Phys. 24, 274â281 (1953
Chaotic asymptotic behaviour of the solutions of the Lighthill Whitham Richards equation
[EN] The phenomenon of chaos has been exhibited in mathematical nonlinear models that describe traffic flows, see, for instance (Li and Gao in Modern Phys Lett B 18(26-27):1395-1402, 2004; Li in Phys. D Nonlinear Phenom 207(1-2):41-51, 2005). At microscopic level, Devaney chaos and distributional chaos have been exhibited for some car-following models, such as the quick-thinking-driver model and the forward and backward control model (Barrachina et al. in 2015; Conejero et al. in Semigroup Forum, 2015). We present here the existence of chaos for the macroscopic model given by the Lighthill Whitham Richards equation.The authors are supported by MEC Project MTM2013-47093-P. The second and third authors are supported by GVA, Project PROMETEOII/2013/013Conejero, JA.; MartĂnez JimĂ©nez, F.; Peris Manguillot, A.; RĂłdenas EscribĂĄ, FDA. (2016). Chaotic asymptotic behaviour of the solutions of the Lighthill Whitham Richards equation. Nonlinear Dynamics. 84(1):127-133. https://doi.org/10.1007/s11071-015-2245-4S127133841Albanese, A.A., Barrachina, X., Mangino, E.M., Peris, A.: Distributional chaos for strongly continuous semigroups of operators. Commun. Pure Appl. Anal. 12(5), 2069â2082 (2013)Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Equ. Appl. 18(4), 647â655 (2012)Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris SĂ©r. II 329, 439â444 (2001)Banasiak, J., Lachowicz, M.: Topological chaos for birth-and-death-type models with proliferation. Math. Models Methods Appl. Sci. 12(6), 755â775 (2002)Banasiak, J., MoszyĆski, M.: A generalization of DeschâSchappacherâWebb criteria for chaos. Discrete Contin. Dyn. Syst. 12(5), 959â972 (2005)Banasiak, J., MoszyĆski, M.: Dynamics of birth-and-death processes with proliferationâstability and chaos. Discrete Contin. Dyn. Syst. 29(1), 67â79 (2011)Barrachina, X., Conejero, J.A.: Devaney chaos and distributional chaos in the solution of certain partial differential equations. Abstr. Appl. Anal. Art. ID 457019, 11 (2012)Barrachina, X., Conejero, J.A., Murillo-Arcila, M., Seoane-SepĂșlveda, J.B.: Distributional chaos for the forward and backward control traffic model (2015, preprint)Bayart, F., Matheron, Ă.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bayart, F., Matheron, Ă.: Mixing operators and small subsets of the circle. J Reine Angew. Math. (2015, to appear)BermĂșdez, T., Bonilla, A., Conejero, J.A., Peris, A.: Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Stud. Math. 170(1), 57â75 (2005)BermĂșdez, T., Bonilla, A., MartĂnez-GimĂ©nez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83â93 (2011)Bernardes Jr, N.C., Bonilla, A., MĂŒller, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265(9), 2143â2163 (2013)Brackstone, M., McDonald, M.: Car-following: a historical review. Transp. Res. Part F Traffic Psychol. Behav. 2(4), 181â196 (1999)Conejero, J.A., Lizama, C., Rodenas, F.: Chaotic behaviour of the solutions of the MooreâGibsonâThompson equation. Appl. Math. Inf. Sci. 9(5), 1â6 (2015)Conejero, J.A., Mangino, E.M.: Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators. Mediterr. J. Math. 7(1), 101â109 (2010)Conejero, J.A., MĂŒller, V., Peris, A.: Hypercyclic behaviour of operators in a hypercyclic C 0 -semigroup. J. Funct. Anal. 244, 342â348 (2007)Conejero, J.A., Murillo-Arcila, M., Seoane-SepĂșlveda, J.B.: Linear chaos for the quick-thinking-driver model. Semigroup Forum (2015). doi: 10.1007/s00233-015-9704-6Conejero, J.A., Peris, A., Trujillo, M.: Chaotic asymptotic behavior of the hyperbolic heat transfer equation solutions. Int. J. Bifur. Chaos Appl. Sci. Eng. 20(9), 2943â2947 (2010)Conejero, J.A., Rodenas, F., Trujillo, M.: Chaos for the hyperbolic bioheat equation. Discrete Contin. Dyn. Syst. 35(2), 653â668 (2015)Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17(4), 793â819 (1997)Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194. Springer, New York (2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. SchnaubeltGrosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Universitext. Springer, London (2011)Herzog, G.: On a universality of the heat equation. Math. Nachr. 188, 169â171 (1997)Li, K., Gao, Z.: Nonlinear dynamics analysis of traffic time series. Modern Phys. Lett. B 18(26â27), 1395â1402 (2004)Li, T.: Nonlinear dynamics of traffic jams. Phys. D Nonlinear Phenom. 207(1â2), 41â51 (2005)Lustri, C.: Continuum Modelling of Traffic Flow. Special Topic Report. Oxford University, Oxford (2010)Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. 229, 317â345 (1955)Maerivoet, S., De Moor, B.: Cellular automata models of road traffic. Phys. Rep. 419(1), 1â64 (2005)Mangino, E.M., Peris, A.: Frequently hypercyclic semigroups. Stud. Math. 202(3), 227â242 (2011)Murillo-Arcila, M., Peris, A.: Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398, 462â465 (2013)Murillo-Arcila, M., Peris, A.: Strong mixing measures for C 0 -semigroups. Rev. R. Acad. Cienc. Exactas FĂs. Nat. Ser. A Math. RACSAM 109(1), 101â115 (2015)Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)Protopopescu, V., Azmy, Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79â90 (1992)Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42â51 (1956
Integral representation of the linear Boltzmann operator for granular gas dynamics with applications
We investigate the properties of the collision operator associated to the
linear Boltzmann equation for dissipative hard-spheres arising in granular gas
dynamics. We establish that, as in the case of non-dissipative interactions,
the gain collision operator is an integral operator whose kernel is made
explicit. One deduces from this result a complete picture of the spectrum of
the collision operator in an Hilbert space setting, generalizing results from
T. Carleman to granular gases. In the same way, we obtain from this integral
representation of the gain operator that the semigroup in L^1(\R \times \R,\d
\x \otimes \d\v) associated to the linear Boltzmann equation for dissipative
hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic
Recycling of end-of-life tyres in seismic isolation foundation systems
Over 6.3 million waste tyres are produced annually in New Zealand (Tyrewise, 2021), leading to
socioeconomic and environmental concerns. The 2010-11 Canterbury Earthquake Sequence inflicted
extensive damage to ~6,000 residential buildings, highlighting the need to improve the seismic
resilience of the residential housing sector. A cost-effective and sustainable eco-rubber geotechnical
seismic isolation (ERGSI) foundation system for new low-rise buildings was developed by the authors.
The ERGSI system integrates a horizontal geotechnical seismic isolation (GSI) layer i.e., a
deformable seismic energy dissipative filter made of granulated tyre rubber (GTR) and gravel (G) â
and a flexible rubberised concrete raft footing. Geotechnical experimental and numerical
investigations demonstrated the effectiveness of the ERGSI system in reducing the seismic demand
at the foundation level (i.e., reduced peak ground acceleration) (Hernandez et al., 2019; Tasalloti et
al., 2021). However, it is essential to ensure that the ERGSI system has minimal leaching attributes
and does not result in long-term negative impacts on the environment
Quantum Kinetic Evolution of Marginal Observables
We develop a rigorous formalism for the description of the evolution of
observables of quantum systems of particles in the mean-field scaling limit.
The corresponding asymptotics of a solution of the initial-value problem of the
dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution
of marginal observables and the evolution of quantum states described in terms
of a one-particle marginal density operator are established. Such approach
gives the alternative description of the kinetic evolution of quantum
many-particle systems to generally accepted approach on basis of kinetic
equations.Comment: 18 page
The von Neumann Hierarchy for Correlation Operators of Quantum Many-Particle Systems
The Cauchy problem for the von Neumann hierarchy of nonlinear equations is
investigated. One describes the evolution of all possible states of quantum
many-particle systems by the correlation operators. A solution of such
nonlinear equations is constructed in the form of an expansion over particle
clusters whose evolution is described by the corresponding order cumulant
(semi-invariant) of evolution operators for the von Neumann equations. For the
initial data from the space of sequences of trace class operators the existence
of a strong and a weak solution of the Cauchy problem is proved. We discuss the
relationships of this solution both with the -particle statistical
operators, which are solutions of the BBGKY hierarchy, and with the
-particle correlation operators of quantum systems.Comment: 26 page
Chaos for the Hyperbolic Bioheat Equation
The Hyperbolic Heat Transfer Equation describes heat processes
in which extremely short periods of time or extreme temperature gradients
are involved. It is already known that there are solutions of this equation
which exhibit a chaotic behaviour, in the sense of Devaney, on certain spaces
of analytic functions with certain growth control. We show that this chaotic
behaviour still appears when we add a source term to this equation, i.e. in the
Hyperbolic Bioheat Equation. These results can also be applied for the Wave
Equation and for a higher order version of the Hyperbolic Bioheat Equation.The authors are supported in part by MEC and FEDER, Projects MTM2010-14909 and MTM2013-47093-P.Conejero, JA.; RĂłdenas EscribĂĄ, FDA.; Trujillo Guillen, M. (2015). Chaos for the Hyperbolic Bioheat Equation. Discrete and Continuous Dynamical Systems - Series A. 35(2):653-668. doi:10.3934/dcds.2015.35.653S65366835
Distributionally chaotic families of operators on Fréchet spaces
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., KostiÄ, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on FrĂ©chet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on FrĂ©chet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915
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