• Previous Article
    Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control
  • DCDS-S Home
  • This Issue
  • Next Article
    Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel
doi: 10.3934/dcdss.2020034

Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems

Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

* Corresponding author: franckemile2006@yahoo.ca

Received  April 2018 Revised  May 2018 Published  March 2019

Fund Project: This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa

Chaotic dynamical attractors are themselves very captivating in Science and Engineering, but systems with multi-dimensional and saturated chaotic attractors with many scrolls are even more fascinating for their multi-directional features. In this paper, the dynamics of a Caputo three-dimensional saturated system is successfully investigated by means of numerical techniques. The continuity property for the saturated function series involved in the model preludes suitable analytical conditions for existence and stability of the solution to the model. The Haar wavelet numerical method is applied to the saturated system and its convergence is shown thanks to error analysis. Therefore, the performance of numerical approximations clearly reveals that the Caputo model and its general initial conditions display some chaotic features with many directions. Such a chaos shows attractors with many scrolls and many directions. Then, the saturated Caputo system is indeed chaotic in the standard integer case (Caputo derivative order $ \alpha = 1 $) and this chaos remains in the fractional case ($ \alpha = 0.9 $). Moreover the dynamics of the system change depending on the parameter $ \alpha $, leading to an important observation that the saturated system is likely to be regulated or controlled via such a parameter.

Citation: Emile Franc Doungmo Goufo. Multi-directional and saturated chaotic attractors with many scrolls for fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020034
References:
[1]

P. ArenaS. BaglioL. Fortuna and G. Manganaro, Generation of n-double scrolls via cellular neural networks, International Journal of Circuit Theory and Applications, 24 (1996), 241-252. doi: 10.1002/(SICI)1097-007X(199605/06)24:3<241::AID-CTA912>3.0.CO;2-J. Google Scholar

[2] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, 2016. doi: 10.1016/B978-0-08-100644-3.00001-5. Google Scholar
[3]

E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, Journal of Computational and Applied Mathematics, 225 (2009), 87-95. doi: 10.1016/j.cam.2008.07.003. Google Scholar

[4]

E. G. Bazhlekova, Subordination principle for fractional evolution equations, Fractional Calculus and Applied Analysis, 3 (2000), 213-230. Google Scholar

[5]

D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Physical Review Letters, 98 (2007), 178-301. doi: 10.1103/PhysRevLett.98.178301. Google Scholar

[6]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent–Ⅱ, Geophysical Journal International, 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. Google Scholar

[7]

M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13. Google Scholar

[8]

G. Chen and J. Lü, Dynamics of the Lorenz system family: Analysis, control and synchronization, SciencePress, Beijing.Google Scholar

[9]

Y. ChenM. Yi and C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, Journal of Computational Science, 3 (2012), 367-373. doi: 10.1016/j.jocs.2012.04.008. Google Scholar

[10]

A. Coronel-EscamillaJ. Gómez-AguilarL. Torres and R. Escobar-Jiménez, A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014. Google Scholar

[11]

A. Coronel-EscamillaJ. Gómez-AguilarL. TorresR. Escobar-Jiménez and M. Valtierra-Rodríguez, Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order, Physica A: Statistical Mechanics and its Applications, 487 (2017), 1-21. doi: 10.1016/j.physa.2017.06.008. Google Scholar

[12]

A. Coronel-Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino and M. M. A. Qurashi, Bateman–feshbach tikochinsky and caldirola–kanai oscillators with new fractional differentiation, Entropy, 19 (2017), 55. doi: 10.3390/e19020055. Google Scholar

[13]

E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342. doi: 10.1016/j.cam.2017.08.026. Google Scholar

[14]

E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Mathematical Modelling and Analysis, 21 (2016), 188-198. doi: 10.3846/13926292.2016.1145607. Google Scholar

[15]

E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10pp. doi: 10.1063/1.4958921. Google Scholar

[16]

E. F. Doungmo Goufo, Stability and convergence analysis of a variable order replicator–mutator process in a moving medium, Journal of Theoretical Biology, 403 (2016), 178-187. doi: 10.1016/j.jtbi.2016.05.007. Google Scholar

[17]

E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos, Solitons & Fractals, 104 (2017), 443-451. doi: 10.1016/j.chaos.2017.08.038. Google Scholar

[18]

E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269.Google Scholar

[19]

G. Fubini, Opere scelte. Ⅱ, Cremonese, Roma, 1958. Google Scholar

[20]

J. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 13.Google Scholar

[21]

A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, 2006. Google Scholar

[22]

Ü. Lepik and H. Hein, Haar Wavelets: With Applications, Springer Science & Business Media, 2014. doi: 10.1007/978-3-319-04295-4. Google Scholar

[23]

J. LuG. ChenX. Yu and H. Leung, Design and analysis of multiscroll chaotic attractors from saturated function series, IEEE Transactions on Circuits and Systems I: Regular Papers, 51 (2004), 2476-2490. doi: 10.1109/TCSI.2004.838151. Google Scholar

[24]

J. LüF. HanX. Yu and G. Chen, Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method, Automatica, 40 (2004), 1677-1687. doi: 10.1016/j.automatica.2004.06.001. Google Scholar

[25]

K. S. Miller and B. Ross, An Introdution to the Fractional Calculus and Fractional Differential Equations, 1st edition, John Wiley & Sons, Inc, 1993, 1993.Google Scholar

[26] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, 1st edition, Academic Press, Inc, 1974. Google Scholar
[27]

I. Podlubny, The laplace transform method for linear differential equations of the fractional order, Website arXiv.org/pdf/func-an/9710005.pdf, 1997.Google Scholar

[28]

S. PoosehH. S. Rodrigues and D. F. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739-742. doi: 10.1063/1.3636838. Google Scholar

[29]

J. Prüss, Evolutionary Integral Equations and Applications, vol. 87, Birkhäuser, 2013.Google Scholar

[30]

J. A. Suykens and J. Vandewalle, Generation of n-double scrolls (n = 1, 2, 3, 4, ...), IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), 861-867. doi: 10.1109/81.251829. Google Scholar

[31]

W. K. TangG. ZhongG. Chen and K. Man, Generation of n-scroll attractors via sine function, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48 (2001), 1369-1372. doi: 10.1109/81.964432. Google Scholar

[32]

L. Tonelli, Sullintegrazione per parti, Rend. Acc. Naz. Lincei, 5 (1909), 246-253. Google Scholar

[33]

M. ur Rehman and R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 4163-4173. doi: 10.1016/j.cnsns.2011.01.014. Google Scholar

[34]

M. E. YALÇINJ. A. SuykensJ. Vandewalle and S. Özoğuz, Families of scroll grid attractors, International Journal of Bifurcation and Chaos, 12 (2002), 23-41. doi: 10.1142/S0218127402004164. Google Scholar

[35]

C. Zuñiga-Aguilar, J. Gómez-Aguilar, R. Escobar-Jiménez and H. Romero-Ugalde, Robust control for fractional variable-order chaotic systems with non-singular kernel, The European Physical Journal Plus, 133 (2018), 13.Google Scholar

[36]

C. Zúñiga-AguilarH. Romero-UgaldeJ. Gómez-AguilarR. Escobar-Jiménez and M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos, Solitons & Fractals, 103 (2017), 382-403. doi: 10.1016/j.chaos.2017.06.030. Google Scholar

show all references

References:
[1]

P. ArenaS. BaglioL. Fortuna and G. Manganaro, Generation of n-double scrolls via cellular neural networks, International Journal of Circuit Theory and Applications, 24 (1996), 241-252. doi: 10.1002/(SICI)1097-007X(199605/06)24:3<241::AID-CTA912>3.0.CO;2-J. Google Scholar

[2] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, 2016. doi: 10.1016/B978-0-08-100644-3.00001-5. Google Scholar
[3]

E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, Journal of Computational and Applied Mathematics, 225 (2009), 87-95. doi: 10.1016/j.cam.2008.07.003. Google Scholar

[4]

E. G. Bazhlekova, Subordination principle for fractional evolution equations, Fractional Calculus and Applied Analysis, 3 (2000), 213-230. Google Scholar

[5]

D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Physical Review Letters, 98 (2007), 178-301. doi: 10.1103/PhysRevLett.98.178301. Google Scholar

[6]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent–Ⅱ, Geophysical Journal International, 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. Google Scholar

[7]

M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13. Google Scholar

[8]

G. Chen and J. Lü, Dynamics of the Lorenz system family: Analysis, control and synchronization, SciencePress, Beijing.Google Scholar

[9]

Y. ChenM. Yi and C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, Journal of Computational Science, 3 (2012), 367-373. doi: 10.1016/j.jocs.2012.04.008. Google Scholar

[10]

A. Coronel-EscamillaJ. Gómez-AguilarL. Torres and R. Escobar-Jiménez, A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel, Physica A: Statistical Mechanics and its Applications, 491 (2018), 406-424. doi: 10.1016/j.physa.2017.09.014. Google Scholar

[11]

A. Coronel-EscamillaJ. Gómez-AguilarL. TorresR. Escobar-Jiménez and M. Valtierra-Rodríguez, Synchronization of chaotic systems involving fractional operators of Liouville–Caputo type with variable-order, Physica A: Statistical Mechanics and its Applications, 487 (2017), 1-21. doi: 10.1016/j.physa.2017.06.008. Google Scholar

[12]

A. Coronel-Escamilla, J. F. Gómez-Aguilar, D. Baleanu, T. Córdova-Fraga, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino and M. M. A. Qurashi, Bateman–feshbach tikochinsky and caldirola–kanai oscillators with new fractional differentiation, Entropy, 19 (2017), 55. doi: 10.3390/e19020055. Google Scholar

[13]

E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342. doi: 10.1016/j.cam.2017.08.026. Google Scholar

[14]

E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Mathematical Modelling and Analysis, 21 (2016), 188-198. doi: 10.3846/13926292.2016.1145607. Google Scholar

[15]

E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10pp. doi: 10.1063/1.4958921. Google Scholar

[16]

E. F. Doungmo Goufo, Stability and convergence analysis of a variable order replicator–mutator process in a moving medium, Journal of Theoretical Biology, 403 (2016), 178-187. doi: 10.1016/j.jtbi.2016.05.007. Google Scholar

[17]

E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos, Solitons & Fractals, 104 (2017), 443-451. doi: 10.1016/j.chaos.2017.08.038. Google Scholar

[18]

E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269.Google Scholar

[19]

G. Fubini, Opere scelte. Ⅱ, Cremonese, Roma, 1958. Google Scholar

[20]

J. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 13.Google Scholar

[21]

A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, 2006. Google Scholar

[22]

Ü. Lepik and H. Hein, Haar Wavelets: With Applications, Springer Science & Business Media, 2014. doi: 10.1007/978-3-319-04295-4. Google Scholar

[23]

J. LuG. ChenX. Yu and H. Leung, Design and analysis of multiscroll chaotic attractors from saturated function series, IEEE Transactions on Circuits and Systems I: Regular Papers, 51 (2004), 2476-2490. doi: 10.1109/TCSI.2004.838151. Google Scholar

[24]

J. LüF. HanX. Yu and G. Chen, Generating 3-D multi-scroll chaotic attractors: A hysteresis series switching method, Automatica, 40 (2004), 1677-1687. doi: 10.1016/j.automatica.2004.06.001. Google Scholar

[25]

K. S. Miller and B. Ross, An Introdution to the Fractional Calculus and Fractional Differential Equations, 1st edition, John Wiley & Sons, Inc, 1993, 1993.Google Scholar

[26] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, 1st edition, Academic Press, Inc, 1974. Google Scholar
[27]

I. Podlubny, The laplace transform method for linear differential equations of the fractional order, Website arXiv.org/pdf/func-an/9710005.pdf, 1997.Google Scholar

[28]

S. PoosehH. S. Rodrigues and D. F. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739-742. doi: 10.1063/1.3636838. Google Scholar

[29]

J. Prüss, Evolutionary Integral Equations and Applications, vol. 87, Birkhäuser, 2013.Google Scholar

[30]

J. A. Suykens and J. Vandewalle, Generation of n-double scrolls (n = 1, 2, 3, 4, ...), IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), 861-867. doi: 10.1109/81.251829. Google Scholar

[31]

W. K. TangG. ZhongG. Chen and K. Man, Generation of n-scroll attractors via sine function, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48 (2001), 1369-1372. doi: 10.1109/81.964432. Google Scholar

[32]

L. Tonelli, Sullintegrazione per parti, Rend. Acc. Naz. Lincei, 5 (1909), 246-253. Google Scholar

[33]

M. ur Rehman and R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 4163-4173. doi: 10.1016/j.cnsns.2011.01.014. Google Scholar

[34]

M. E. YALÇINJ. A. SuykensJ. Vandewalle and S. Özoğuz, Families of scroll grid attractors, International Journal of Bifurcation and Chaos, 12 (2002), 23-41. doi: 10.1142/S0218127402004164. Google Scholar

[35]

C. Zuñiga-Aguilar, J. Gómez-Aguilar, R. Escobar-Jiménez and H. Romero-Ugalde, Robust control for fractional variable-order chaotic systems with non-singular kernel, The European Physical Journal Plus, 133 (2018), 13.Google Scholar

[36]

C. Zúñiga-AguilarH. Romero-UgaldeJ. Gómez-AguilarR. Escobar-Jiménez and M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos, Solitons & Fractals, 103 (2017), 382-403. doi: 10.1016/j.chaos.2017.06.030. Google Scholar

Figure 1.  Representation of the phase portrait of saturated functions. In (a), the one-variable saturated function (4). In (b), the phase portrait of saturated function series (5) with $ k = 1, \ \ h = 4 $
Figure 2.  Plot representing multi-dimensional simulations of the saturated system (1) with $ \alpha = 1 $ (conventional case). In (a), projection of a one-dimensional saturated chaotic attractor with six scrolls. In (b), a projection of a two-dimensional saturated chaotic attractor with a grid of $ 6\times6 $ scrolls. In (c), a projection of a three-dimensional saturated chaotic attractor with a grid of $ 6\times6\times6 $ scrolls
Figure 3.  Plot representing multi-dimensional simulations of the saturated system (1) with $ \alpha = 0.9 $ (fractional case). The same dynamics as Fig. 2} are shown with a one-dimensional chaotic attractor with six scrolls in (a), two-dimensional saturated chaotic attractor with a grid of $ 6\times6 $ scrolls in (b) and a three-dimensional saturated chaotic attractor with a grid of $ 6\times6\times6 $ scrolls in (c)
[1]

Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705

[2]

Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219

[3]

Péter Bálint, Imre Péter Tóth. Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 37-59. doi: 10.3934/dcds.2006.15.37

[4]

Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861

[5]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57

[6]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[7]

Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038

[8]

Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652

[9]

Gerald Sommer, Di Zang. Parity symmetry in multi-dimensional signals. Communications on Pure & Applied Analysis, 2007, 6 (3) : 829-852. doi: 10.3934/cpaa.2007.6.829

[10]

Kang-Ling Liao, Chih-Wen Shih, Chi-Jer Yu. The snapback repellers for chaos in multi-dimensional maps. Journal of Computational Dynamics, 2018, 5 (1&2) : 81-92. doi: 10.3934/jcd.2018004

[11]

Takashi Suzuki, Shuji Yoshikawa. Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 209-217. doi: 10.3934/dcdss.2012.5.209

[12]

Tatsien Li, Wancheng Sheng. The general multi-dimensional Riemann problem for hyperbolic systems with real constant coefficients. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 737-744. doi: 10.3934/dcds.2002.8.737

[13]

Damien Allonsius, Franck Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019037

[14]

Xiaoling Sun, Xiaojin Zheng, Juan Sun. A Lagrangian dual and surrogate method for multi-dimensional quadratic knapsack problems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 47-60. doi: 10.3934/jimo.2009.5.47

[15]

Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011

[16]

Wen-Qing Xu. Boundary conditions for multi-dimensional hyperbolic relaxation problems. Conference Publications, 2003, 2003 (Special) : 916-925. doi: 10.3934/proc.2003.2003.916

[17]

Eugenii Shustin. Dynamics of oscillations in a multi-dimensional delay differential system. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 557-576. doi: 10.3934/dcds.2004.11.557

[18]

Jung-Chao Ban, Song-Sun Lin. Patterns generation and transition matrices in multi-dimensional lattice models. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 637-658. doi: 10.3934/dcds.2005.13.637

[19]

Shijin Deng, Weike Wang. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1107-1138. doi: 10.3934/dcds.2011.30.1107

[20]

Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (25)
  • HTML views (404)
  • Cited by (0)

Other articles
by authors

[Back to Top]