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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 and Continuous Dynamical Systems, 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.

[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.

[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. 

[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.

[5]

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

[6]

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

[7]

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

[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. 

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

[24]

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

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.

[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.

[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. 

[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.

[5]

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

[6]

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

[7]

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

[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. 

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

[24]

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

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