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June  2014, 1(2): 307-338. doi: 10.3934/jcd.2014.1.307

Lattice structures for attractors I

William D. Kalies 1, , Konstantin Mischaikow 2, and  Robert C.A.M. Vandervorst 3,

1. 

Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, 33431 Boca Raton
2. 

Rutgers University, 110 Frelinghusen Road, Piscataway, NJ 08854
3. 

VU University, De Boelelaan 1081a, 1081 HV, Amsterdam, Netherlands

Received  July 2013 Revised  December 2013 Published  December 2014

We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation for the development of algorithms to manipulate these structures computationally.
Keywords: invariant set, Birkhoff's representation theorem., Attractor, attracting neighborhood, distributive lattice.
Mathematics Subject Classification: Primary: 37B25, 06D05; Secondary: 37B3.
Citation: William D. Kalies, Konstantin Mischaikow, Robert C.A.M. Vandervorst. Lattice structures for attractors I. Journal of Computational Dynamics, 2014, 1 (2) : 307-338. doi: 10.3934/jcd.2014.1.307
References:
[1]

E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.

[2]

Z. Arai, W. D. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 757-789. doi: 10.1137/080734935.

[3]

K. A. Baker and A. W. Hales, Distributive projective lattices, Canad. J. Math., 22 (1970), 472-475. doi: 10.4153/CJM-1970-054-0.

[4]

H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem, J. Comp. Nonlinear Dynamics, 1 (2006), 312-319. doi: 10.1115/1.2338651.

[5]

J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mischaikow, I. Obayashi and P. Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, CHAOS, 22 (2012), 047508. doi: 10.1063/1.4767672.

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, vol. 38 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I., 1978.

[7]

B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd edition, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9780511809088.

[8]

R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc., 298 (1986), 193-213. doi: 10.1090/S0002-9947-1986-0857439-7.

[9]

R. D. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations, 71 (1988), 270-287. doi: 10.1016/0022-0396(88)90028-9.

[10]

R. Freese and J. B. Nation, Projective lattices, Pacific J. Math., 75 (1978), 93-106, URL http://projecteuclid.org/euclid.pjm/1102810149. doi: 10.2140/pjm.1978.75.93.

[11]

G. Grätzer, Lattice Theory: Foundation, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0018-1.

[12]

W. Kalies, K. Mischaikow and R. C. Vandervorst, An algorithmic approach to chain recurrence, Found. Comput. Math., 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.

[13]

W. Kalies, K. Mischaikow and R. C. Vandervorst, Lattice structures for attractors II, Submitted for publication, arXiv:1409.5405.

[14]

E. Liz and P. Pilarczyk, Global dynamics in a stage-structured discrete-time population model with harvesting, J. Theoret. Biol., 297 (2012), 148-165. doi: 10.1016/j.jtbi.2011.12.012.

[15]

F. Miraglia, An Introduction to Partially Ordered Structures and Sheaves, vol. 1 of Contemporary Logic Series, Polimetrica Scientific Publisher, Milan, Italy, 2006.

[16]

J. W. Robbin and D. A. Salamon, Lyapunov maps, simplicial complexes and the Stone functor, Ergodic Theory Dynam. Systems, 12 (1992), 153-183. doi: 10.1017/S0143385700006647.

[17]

C. Robinson, Dynamical Systems, 2nd edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999, Stability, Symbolic dynamics, and Chaos.

[18]

S. Roman, Lattices and Ordered Sets, Springer, New York, 2008.

[19]

S. Vickers, Topology via Logic, vol. 5 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 1989.

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, vol. 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993.

[2]

Z. Arai, W. D. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 757-789. doi: 10.1137/080734935.

[3]

K. A. Baker and A. W. Hales, Distributive projective lattices, Canad. J. Math., 22 (1970), 472-475. doi: 10.4153/CJM-1970-054-0.

[4]

H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem, J. Comp. Nonlinear Dynamics, 1 (2006), 312-319. doi: 10.1115/1.2338651.

[5]

J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mischaikow, I. Obayashi and P. Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, CHAOS, 22 (2012), 047508. doi: 10.1063/1.4767672.

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, vol. 38 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I., 1978.

[7]

B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd edition, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9780511809088.

[8]

R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc., 298 (1986), 193-213. doi: 10.1090/S0002-9947-1986-0857439-7.

[9]

R. D. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations, 71 (1988), 270-287. doi: 10.1016/0022-0396(88)90028-9.

[10]

R. Freese and J. B. Nation, Projective lattices, Pacific J. Math., 75 (1978), 93-106, URL http://projecteuclid.org/euclid.pjm/1102810149. doi: 10.2140/pjm.1978.75.93.

[11]

G. Grätzer, Lattice Theory: Foundation, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0018-1.

[12]

W. Kalies, K. Mischaikow and R. C. Vandervorst, An algorithmic approach to chain recurrence, Found. Comput. Math., 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.

[13]

W. Kalies, K. Mischaikow and R. C. Vandervorst, Lattice structures for attractors II, Submitted for publication, arXiv:1409.5405.

[14]

E. Liz and P. Pilarczyk, Global dynamics in a stage-structured discrete-time population model with harvesting, J. Theoret. Biol., 297 (2012), 148-165. doi: 10.1016/j.jtbi.2011.12.012.

[15]

F. Miraglia, An Introduction to Partially Ordered Structures and Sheaves, vol. 1 of Contemporary Logic Series, Polimetrica Scientific Publisher, Milan, Italy, 2006.

[16]

J. W. Robbin and D. A. Salamon, Lyapunov maps, simplicial complexes and the Stone functor, Ergodic Theory Dynam. Systems, 12 (1992), 153-183. doi: 10.1017/S0143385700006647.

[17]

C. Robinson, Dynamical Systems, 2nd edition, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999, Stability, Symbolic dynamics, and Chaos.

[18]

S. Roman, Lattices and Ordered Sets, Springer, New York, 2008.

[19]

S. Vickers, Topology via Logic, vol. 5 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 1989.

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