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December  2017, 37(12): 6069-6098. doi: 10.3934/dcds.2017261

Impulsive motion on synchronized spatial temporal grids

1. 

Worcester Polytechnic Institute, 100 Worcester Road, Worcester, MA 01609, USA

2. 

Accademia Nazionale Delle Scienze Detta Dei XL, Via L.Spallanzani 7, 00161 Roma, Italy

Received  August 2016 Revised  July 2017 Published  August 2017

We introduce a family of kinetic vector fields on countable space-time grids and study related impulsive second order initial value Cauchy problems. We then construct special examples for which orbits and attractors display unusual analytic and geometric properties.

Citation: Umberto Mosco. Impulsive motion on synchronized spatial temporal grids. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6069-6098. doi: 10.3934/dcds.2017261
References:
[1]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Modern Birkhäuser Classics, Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. Google Scholar

[2]

M. T. Barlow and E. A. Perkins, Brownian motions on the Sierpinski gasket, Prob. Theo. Rel. Fields, 79 (1988), 543-623. doi: 10.1007/BF00318785. Google Scholar

[3]

M. CefaloM. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results and numerical approximation, Differential and Integral Equations, 26 (2013), 1027-1054. Google Scholar

[4]

M. G. Garroni and L. Menaldi, Second Order Elliptic Integro-Differential Problems, Research Notes in Math., 43, Chapman & Hall/CRC, Boca Raton, 221pp, 2002. doi: 10.1201/9781420035797. Google Scholar

[5]

D. Hilbert, Über the stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen, 38 (1891), 459-460. doi: 10.1007/BF01199431. Google Scholar

[6]

J. E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055. Google Scholar

[7]

J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar

[8]

H. von Koch, Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire, Arkiv für Matematik, Astronomi och Fysik, 1 (1904), 681-704. Google Scholar

[9]

S. Kusuoka, Diffusion Processes in Nested Fractals, Lecture Notes in Math. N. 1567, Springer V., 1993.Google Scholar

[10]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. Math. Anal., 42 (2010), 1539-1567. doi: 10.1137/090761173. Google Scholar

[11]

P. D. Lax, The differentiability of Pólya's function, Advances in Mathematics, 10 (1973), 456-464. doi: 10.1016/0001-8708(73)90125-4. Google Scholar

[12]

T. Lindström, Brownian motion on nested fractals Memoirs Amer. Math. Soc., 83 (1990), iv+128 pp. doi: 10.1090/memo/0420. Google Scholar

[13]

L. Menaldi, On the optimal stopping time problem for degenerate diffusions, SIAM J. Control Optim., 18 (1980), 722-739. doi: 10.1137/0318053. Google Scholar

[14]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Mathematics, 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7. Google Scholar

[15]

U. Mosco, An introduction to the approximate solution of variational inequalities, Constructive Aspects of Functional Analysis, 57 (2011), 497-682. doi: 10.1007/978-3-642-10984-3_5. Google Scholar

[16]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421. doi: 10.1006/jfan.1994.1093. Google Scholar

[17]

U. Mosco, Energy functionals on certain fractal structures, J. Convex Analysis, 9 (2002), 581-600. Google Scholar

[18]

U. Mosco, Analysis and numerics of some fractal boundary value problems, in "Analysis and Numerics of Partial Differential Equations–In Memory of Enrico Magenes", F. Brezzi, P. Colli Franzone, U. Gianazza, G. Gilardi eds., Springer INDAM Series, 4 (2013), 237–255. doi: 10.1007/978-88-470-2592-9_14. Google Scholar

[19]

U. Mosco, Time, space, similarity, New Trends in Differential Equations, Control Theory and Optimization, Eds. V. Barbu, C. Lefter, I. Vrabie, World Scientific, (2016), 261–276. Google Scholar

[20]

U. Mosco, Filling Attractors, manuscript, WPI, 2017.Google Scholar

[21]

U. Mosco, Finite-time self-organized-criticality on synchronized infinite grids, to appear.Google Scholar

[22]

U. Mosco and M. A. Vivaldi, Layered fractal fibers and potentials, Journal des Mathßmatiques Pures et Appliqußes, 103 (2015), 1198-1227. doi: 10.1016/j.matpur.2014.10.010. Google Scholar

[23]

G. Peano, Sur une courbe, qui remplit une aire plane, Mathematische Annalen, 36 (1890), 157-160. doi: 10.1007/BF01199438. Google Scholar

[24]

G. Polya, Über eine Peanosche kurve, Bull. Acad. Sci. Cracovie, Ser. A, (1913), 305-313. Google Scholar

show all references

References:
[1]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Modern Birkhäuser Classics, Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1. Google Scholar

[2]

M. T. Barlow and E. A. Perkins, Brownian motions on the Sierpinski gasket, Prob. Theo. Rel. Fields, 79 (1988), 543-623. doi: 10.1007/BF00318785. Google Scholar

[3]

M. CefaloM. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results and numerical approximation, Differential and Integral Equations, 26 (2013), 1027-1054. Google Scholar

[4]

M. G. Garroni and L. Menaldi, Second Order Elliptic Integro-Differential Problems, Research Notes in Math., 43, Chapman & Hall/CRC, Boca Raton, 221pp, 2002. doi: 10.1201/9781420035797. Google Scholar

[5]

D. Hilbert, Über the stetige Abbildung einer Linie auf ein Flächenstück, Mathematische Annalen, 38 (1891), 459-460. doi: 10.1007/BF01199431. Google Scholar

[6]

J. E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055. Google Scholar

[7]

J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar

[8]

H. von Koch, Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire, Arkiv für Matematik, Astronomi och Fysik, 1 (1904), 681-704. Google Scholar

[9]

S. Kusuoka, Diffusion Processes in Nested Fractals, Lecture Notes in Math. N. 1567, Springer V., 1993.Google Scholar

[10]

M. R. Lancia and P. Vernole, Irregular heat flow problems, SIAM J. Math. Anal., 42 (2010), 1539-1567. doi: 10.1137/090761173. Google Scholar

[11]

P. D. Lax, The differentiability of Pólya's function, Advances in Mathematics, 10 (1973), 456-464. doi: 10.1016/0001-8708(73)90125-4. Google Scholar

[12]

T. Lindström, Brownian motion on nested fractals Memoirs Amer. Math. Soc., 83 (1990), iv+128 pp. doi: 10.1090/memo/0420. Google Scholar

[13]

L. Menaldi, On the optimal stopping time problem for degenerate diffusions, SIAM J. Control Optim., 18 (1980), 722-739. doi: 10.1137/0318053. Google Scholar

[14]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Mathematics, 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7. Google Scholar

[15]

U. Mosco, An introduction to the approximate solution of variational inequalities, Constructive Aspects of Functional Analysis, 57 (2011), 497-682. doi: 10.1007/978-3-642-10984-3_5. Google Scholar

[16]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421. doi: 10.1006/jfan.1994.1093. Google Scholar

[17]

U. Mosco, Energy functionals on certain fractal structures, J. Convex Analysis, 9 (2002), 581-600. Google Scholar

[18]

U. Mosco, Analysis and numerics of some fractal boundary value problems, in "Analysis and Numerics of Partial Differential Equations–In Memory of Enrico Magenes", F. Brezzi, P. Colli Franzone, U. Gianazza, G. Gilardi eds., Springer INDAM Series, 4 (2013), 237–255. doi: 10.1007/978-88-470-2592-9_14. Google Scholar

[19]

U. Mosco, Time, space, similarity, New Trends in Differential Equations, Control Theory and Optimization, Eds. V. Barbu, C. Lefter, I. Vrabie, World Scientific, (2016), 261–276. Google Scholar

[20]

U. Mosco, Filling Attractors, manuscript, WPI, 2017.Google Scholar

[21]

U. Mosco, Finite-time self-organized-criticality on synchronized infinite grids, to appear.Google Scholar

[22]

U. Mosco and M. A. Vivaldi, Layered fractal fibers and potentials, Journal des Mathßmatiques Pures et Appliqußes, 103 (2015), 1198-1227. doi: 10.1016/j.matpur.2014.10.010. Google Scholar

[23]

G. Peano, Sur une courbe, qui remplit une aire plane, Mathematische Annalen, 36 (1890), 157-160. doi: 10.1007/BF01199438. Google Scholar

[24]

G. Polya, Über eine Peanosche kurve, Bull. Acad. Sci. Cracovie, Ser. A, (1913), 305-313. Google Scholar

Figure 1.  $\Gamma^{\beta, n}$ for different $\beta$
Figure 2.  Ring-like $\Gamma^{\beta, n}$
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