
-
Previous Article
Relative periodic solutions of the $ n $-vortex problem on the sphere
- JGM Home
- This Issue
-
Next Article
New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity
Self-organization on Riemannian manifolds
Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada |
We consider an aggregation model that consists of an active transport equation for the macroscopic population density, where the velocity has a nonlocal functional dependence on the density, modelled via an interaction potential. We set up the model on general Riemannian manifolds and provide a framework for constructing interaction potentials which lead to equilibria that are constant on their supports. We consider such potentials for two specific cases (the two-dimensional sphere and the two-dimensional hyperbolic space) and investigate analytically and numerically the long-time behaviour and equilibrium solutions of the aggregation model on these manifolds. Equilibria obtained numerically with other interaction potentials and an application of the model to aggregation on the rotation group $ SO(3) $ are also presented.
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1029-0. |
[2] |
B. Afsari,
Riemannian ${L}^{p}$ center of mass: Existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655-673.
doi: 10.1090/S0002-9939-2010-10541-5. |
[3] |
B. Afsari, R. Tron and R. Vidal,
On the convergence of gradient descent for finding the Riemannian center of mass, SIAM J. Control Optim., 51 (2013), 2230-2260.
doi: 10.1137/12086282X. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. |
[5] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.
doi: 10.1007/s00205-013-0644-6. |
[6] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.
doi: 10.1016/j.physd.2012.10.002. |
[7] |
A. J. Bernoff and C. M. Topaz,
A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.
doi: 10.1137/100804504. |
[8] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent,
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[9] |
A. L. Bertozzi and T. Laurent,
Finite-time blow-up of solutions of an aggregation equation in ${\bf{R}}^n$, Comm. Math. Phys., 274 (2007), 717-735.
doi: 10.1007/s00220-007-0288-1. |
[10] |
A. L. Bertozzi, T. Laurent and L. Flavien,
Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp.
doi: 10.1142/S0218202511400057. |
[11] |
A. L. Bertozzi, T. Laurent and J. Rosado,
$L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.
doi: 10.1002/cpa.20334. |
[12] |
M. Bodnar and J. J. L. Velazquez,
An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.
doi: 10.1016/j.jde.2005.07.025. |
[13] |
M. Burger and M. Di Francesco,
Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.
doi: 10.3934/nhm.2008.3.749. |
[14] |
S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Selforganization in Biological Systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003, Reprint of the 2001 original. |
[15] |
J. A. Cañizo, J. A. Carrillo and F. S. Patacchini,
Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217.
doi: 10.1007/s00205-015-0852-3. |
[16] |
J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997, 59-115. |
[17] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev,
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[18] |
J. A. Carrillo, Y. Huang and S. Martin,
Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578.
doi: 10.1017/S0956792514000126. |
[19] |
J. A. Carrillo, D. Slepčev and L. Wu,
Nonlocal-interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1209-1247.
doi: 10.3934/dcds.2016.36.1209. |
[20] |
J. A. Carrillo, R. J. McCann and C. Villani,
Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.
doi: 10.1007/s00205-005-0386-1. |
[21] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, vol. 553 of CISM Courses and Lect., Springer, Vienna, 2014, 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[22] |
R. Choksi, R. C. Fetecau and I. Topaloglu,
On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.
doi: 10.1016/j.anihpc.2014.09.004. |
[23] |
Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes,
State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[24] |
J. Cortés and M. Egerstedt, Coordinated control of multi-robot systems: A survey, SICE Journal of Control, Measurement, and System Integration, 10 (2017), 495-503. Google Scholar |
[25] |
I. D. Couzin, J. Krause, R. James, G. Ruxton and N. R. Franks,
Collective memory and spatial sorting in animal groups, J. Theor. Biol., 218 (2002), 1-11.
doi: 10.1006/jtbi.2002.3065. |
[26] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[27] |
A. Czirók, A.-L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Physical Review Letters, 82 (1999), 209-212. Google Scholar |
[28] |
R. Eftimie, G. de Vries and M. A. Lewis,
Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci., 104 (2007), 6974-6979.
doi: 10.1073/pnas.0611483104. |
[29] |
J. H. M. Evers, R. C. Fetecau and L. Ryzhik,
Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847-2871.
doi: 10.1088/0951-7715/28/8/2847. |
[30] |
K. Fellner and G. Raoul,
Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.
doi: 10.1142/S0218202510004921. |
[31] |
R. C. Fetecau and Y. Huang,
Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.
doi: 10.1016/j.physd.2012.11.004. |
[32] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov,
Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.
doi: 10.1088/0951-7715/24/10/002. |
[33] |
R. C. Fetecau and M. Kovacic,
Swarm equilibria in domains with boundaries, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1260-1308.
doi: 10.1137/17M1123900. |
[34] |
V. Gazi and K. M. Passino,
Stability analysis of swarms, IEEE Trans. Automat. Control, 48 (2003), 692-697.
doi: 10.1109/TAC.2003.809765. |
[35] |
J. Haile, Molecular Dynamics Simulation: Elementary Methods, John Wiley and Sons, Inc., New York, 1992. Google Scholar |
[36] |
D. D. Holm and V. Putkaradze,
Aggregation of finite-size particles with variable mobility, Phys Rev Lett., 95 (2005), 226106.
doi: 10.1103/PhysRevLett.95.226106. |
[37] |
J. L. A. Jadbabaie and A. S. Morse,
Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001.
doi: 10.1109/TAC.2003.812781. |
[38] |
M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Trans. Robot., 23 (2007), 693-703. Google Scholar |
[39] |
H. Karcher,
Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541.
doi: 10.1002/cpa.3160300502. |
[40] |
Y. Kimura,
Vortex motion on surfaces with constant curvature, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 245-259.
doi: 10.1098/rspa.1999.0311. |
[41] |
T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84 (2011), 015203(R). Google Scholar |
[42] |
W. Kühnel, Differential Geometry, vol. 16 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2002. |
[43] |
A. J. Leverentz, C. M. Topaz and A. J. Bernoff,
Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.
doi: 10.1137/090749037. |
[44] |
W. Li,
Collective motion of swarming agents evolving on a sphere manifold: A fundamental framework and characterization, Scientific Reports, 5 (2015), 13603.
doi: 10.1038/srep13603. |
[45] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[46] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[47] |
P. Petersen, Riemannian Geometry, vol. 171 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2006. |
[48] |
G. Savaré,
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris, 345 (2007), 151-154.
doi: 10.1016/j.crma.2007.06.018. |
[49] |
R. Sepulchre, Consensus on nonlinear spaces, Annual Reviews in Control, 35 (2011), 56-64. Google Scholar |
[50] |
R. Simione, D. Slepčev and I. Topaloglu,
Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986.
doi: 10.1007/s10955-015-1215-z. |
[51] |
R. Tron, B. Afsari and R. Vidal, Intrinsic consensus on $SO(3)$ with almost-global convergence, Proceedings of the 51st IEEE Conference on Decision and Control, 2013, 2052–2058.
doi: 10.1109/CDC.2012.6426677. |
[52] |
J. von Brecht and D. Uminsky,
On soccer balls and linearized inverse statistical mechanics, J. Nonlinear Sci., 22 (2012), 935-959.
doi: 10.1007/s00332-012-9132-7. |
[53] |
J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi,
Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012), 1140002, 31pp.
doi: 10.1142/S0218202511400021. |
[54] |
L. Wu and D. Slepčev,
Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281.
doi: 10.1080/03605302.2015.1015033. |
show all references
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 75 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1029-0. |
[2] |
B. Afsari,
Riemannian ${L}^{p}$ center of mass: Existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 139 (2011), 655-673.
doi: 10.1090/S0002-9939-2010-10541-5. |
[3] |
B. Afsari, R. Tron and R. Vidal,
On the convergence of gradient descent for finding the Riemannian center of mass, SIAM J. Control Optim., 51 (2013), 2230-2260.
doi: 10.1137/12086282X. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. |
[5] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Dimensionality of local minimizers of the interaction energy, Arch. Ration. Mech. Anal., 209 (2013), 1055-1088.
doi: 10.1007/s00205-013-0644-6. |
[6] |
D. Balagué, J. A. Carrillo, T. Laurent and G. Raoul,
Nonlocal interactions by repulsive-attractive potentials: Radial ins/stability, Phys. D, 260 (2013), 5-25.
doi: 10.1016/j.physd.2012.10.002. |
[7] |
A. J. Bernoff and C. M. Topaz,
A primer of swarm equilibria, SIAM J. Appl. Dyn. Syst., 10 (2011), 212-250.
doi: 10.1137/100804504. |
[8] |
A. L. Bertozzi, J. A. Carrillo and T. Laurent,
Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.
doi: 10.1088/0951-7715/22/3/009. |
[9] |
A. L. Bertozzi and T. Laurent,
Finite-time blow-up of solutions of an aggregation equation in ${\bf{R}}^n$, Comm. Math. Phys., 274 (2007), 717-735.
doi: 10.1007/s00220-007-0288-1. |
[10] |
A. L. Bertozzi, T. Laurent and L. Flavien,
Aggregation and spreading via the Newtonian potential: The dynamics of patch solutions, Math. Models Methods Appl. Sci., 22 (2012), 1140005, 39pp.
doi: 10.1142/S0218202511400057. |
[11] |
A. L. Bertozzi, T. Laurent and J. Rosado,
$L^p$ theory for the multidimensional aggregation equation, Comm. Pure Appl. Math., 64 (2011), 45-83.
doi: 10.1002/cpa.20334. |
[12] |
M. Bodnar and J. J. L. Velazquez,
An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations, 222 (2006), 341-380.
doi: 10.1016/j.jde.2005.07.025. |
[13] |
M. Burger and M. Di Francesco,
Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.
doi: 10.3934/nhm.2008.3.749. |
[14] |
S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonabeau, Selforganization in Biological Systems, Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 2003, Reprint of the 2001 original. |
[15] |
J. A. Cañizo, J. A. Carrillo and F. S. Patacchini,
Existence of compactly supported global minimisers for the interaction energy, Arch. Ration. Mech. Anal., 217 (2015), 1197-1217.
doi: 10.1007/s00205-015-0852-3. |
[16] |
J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, vol. 31 of Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997, 59-115. |
[17] |
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent and D. Slepčev,
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), 229-271.
doi: 10.1215/00127094-2010-211. |
[18] |
J. A. Carrillo, Y. Huang and S. Martin,
Explicit flock solutions for Quasi-Morse potentials, European J. Appl. Math., 25 (2014), 553-578.
doi: 10.1017/S0956792514000126. |
[19] |
J. A. Carrillo, D. Slepčev and L. Wu,
Nonlocal-interaction equations on uniformly prox-regular sets, Discrete Contin. Dyn. Syst. Ser. A, 36 (2016), 1209-1247.
doi: 10.3934/dcds.2016.36.1209. |
[20] |
J. A. Carrillo, R. J. McCann and C. Villani,
Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.
doi: 10.1007/s00205-005-0386-1. |
[21] |
J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds, vol. 553 of CISM Courses and Lect., Springer, Vienna, 2014, 1–46.
doi: 10.1007/978-3-7091-1785-9_1. |
[22] |
R. Choksi, R. C. Fetecau and I. Topaloglu,
On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.
doi: 10.1016/j.anihpc.2014.09.004. |
[23] |
Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes,
State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.
doi: 10.1016/j.physd.2007.05.007. |
[24] |
J. Cortés and M. Egerstedt, Coordinated control of multi-robot systems: A survey, SICE Journal of Control, Measurement, and System Integration, 10 (2017), 495-503. Google Scholar |
[25] |
I. D. Couzin, J. Krause, R. James, G. Ruxton and N. R. Franks,
Collective memory and spatial sorting in animal groups, J. Theor. Biol., 218 (2002), 1-11.
doi: 10.1006/jtbi.2002.3065. |
[26] |
F. Cucker and S. Smale,
Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862.
doi: 10.1109/TAC.2007.895842. |
[27] |
A. Czirók, A.-L. Barabási and T. Vicsek, Collective motion of self-propelled particles: Kinetic phase transition in one dimension, Physical Review Letters, 82 (1999), 209-212. Google Scholar |
[28] |
R. Eftimie, G. de Vries and M. A. Lewis,
Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci., 104 (2007), 6974-6979.
doi: 10.1073/pnas.0611483104. |
[29] |
J. H. M. Evers, R. C. Fetecau and L. Ryzhik,
Anisotropic interactions in a first-order aggregation model, Nonlinearity, 28 (2015), 2847-2871.
doi: 10.1088/0951-7715/28/8/2847. |
[30] |
K. Fellner and G. Raoul,
Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.
doi: 10.1142/S0218202510004921. |
[31] |
R. C. Fetecau and Y. Huang,
Equilibria of biological aggregations with nonlocal repulsive-attractive interactions, Phys. D, 260 (2013), 49-64.
doi: 10.1016/j.physd.2012.11.004. |
[32] |
R. C. Fetecau, Y. Huang and T. Kolokolnikov,
Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.
doi: 10.1088/0951-7715/24/10/002. |
[33] |
R. C. Fetecau and M. Kovacic,
Swarm equilibria in domains with boundaries, SIAM Journal on Applied Dynamical Systems, 16 (2017), 1260-1308.
doi: 10.1137/17M1123900. |
[34] |
V. Gazi and K. M. Passino,
Stability analysis of swarms, IEEE Trans. Automat. Control, 48 (2003), 692-697.
doi: 10.1109/TAC.2003.809765. |
[35] |
J. Haile, Molecular Dynamics Simulation: Elementary Methods, John Wiley and Sons, Inc., New York, 1992. Google Scholar |
[36] |
D. D. Holm and V. Putkaradze,
Aggregation of finite-size particles with variable mobility, Phys Rev Lett., 95 (2005), 226106.
doi: 10.1103/PhysRevLett.95.226106. |
[37] |
J. L. A. Jadbabaie and A. S. Morse,
Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, 48 (2003), 988-1001.
doi: 10.1109/TAC.2003.812781. |
[38] |
M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Trans. Robot., 23 (2007), 693-703. Google Scholar |
[39] |
H. Karcher,
Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 30 (1977), 509-541.
doi: 10.1002/cpa.3160300502. |
[40] |
Y. Kimura,
Vortex motion on surfaces with constant curvature, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 245-259.
doi: 10.1098/rspa.1999.0311. |
[41] |
T. Kolokolnikov, H. Sun, D. Uminsky and A. L. Bertozzi, A theory of complex patterns arising from 2D particle interactions, Phys. Rev. E, Rapid Communications, 84 (2011), 015203(R). Google Scholar |
[42] |
W. Kühnel, Differential Geometry, vol. 16 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2002. |
[43] |
A. J. Leverentz, C. M. Topaz and A. J. Bernoff,
Asymptotic dynamics of attractive-repulsive swarms, SIAM J. Appl. Dyn. Syst., 8 (2009), 880-908.
doi: 10.1137/090749037. |
[44] |
W. Li,
Collective motion of swarming agents evolving on a sphere manifold: A fundamental framework and characterization, Scientific Reports, 5 (2015), 13603.
doi: 10.1038/srep13603. |
[45] |
A. Mogilner and L. Edelstein-Keshet,
A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.
doi: 10.1007/s002850050158. |
[46] |
S. Motsch and E. Tadmor,
Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621.
doi: 10.1137/120901866. |
[47] |
P. Petersen, Riemannian Geometry, vol. 171 of Graduate Texts in Mathematics, 2nd edition, Springer, New York, 2006. |
[48] |
G. Savaré,
Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris, 345 (2007), 151-154.
doi: 10.1016/j.crma.2007.06.018. |
[49] |
R. Sepulchre, Consensus on nonlinear spaces, Annual Reviews in Control, 35 (2011), 56-64. Google Scholar |
[50] |
R. Simione, D. Slepčev and I. Topaloglu,
Existence of ground states of nonlocal-interaction energies, J. Stat. Phys., 159 (2015), 972-986.
doi: 10.1007/s10955-015-1215-z. |
[51] |
R. Tron, B. Afsari and R. Vidal, Intrinsic consensus on $SO(3)$ with almost-global convergence, Proceedings of the 51st IEEE Conference on Decision and Control, 2013, 2052–2058.
doi: 10.1109/CDC.2012.6426677. |
[52] |
J. von Brecht and D. Uminsky,
On soccer balls and linearized inverse statistical mechanics, J. Nonlinear Sci., 22 (2012), 935-959.
doi: 10.1007/s00332-012-9132-7. |
[53] |
J. von Brecht, D. Uminsky, T. Kolokolnikov and A. Bertozzi,
Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22 (2012), 1140002, 31pp.
doi: 10.1142/S0218202511400021. |
[54] |
L. Wu and D. Slepčev,
Nonlocal interaction equations in environments with heterogeneities and boundaries, Comm. Partial Differential Equations, 40 (2015), 1241-1281.
doi: 10.1080/03605302.2015.1015033. |









[1] |
Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 |
[2] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[3] |
Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206 |
[4] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[5] |
Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 |
[6] |
M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849 |
[7] |
Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513 |
[8] |
Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341 |
[9] |
Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20. |
[10] |
Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453 |
[11] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[12] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[13] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[14] |
Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 |
[15] |
Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 |
[16] |
Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190 |
[17] |
Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027 |
[18] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[19] |
Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207 |
[20] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]