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On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions
Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes
1. | Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany |
2. | Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049–001 Lisbon, Portugal |
$u_t = u_{xx} + f(x, u, u_x)\, , $ |
$0 < x <1$ |
$v_t = 0$ |
$\mathcal{A}$ |
$\mathcal{C}$ |
$x = 0$ |
$x = 1$ |
$\mathcal{A}$ |
References:
[1] |
S. Angenent,
The Morse-Smale property for a semi-linear parabolic equation, J. Diff. Eqns., 62 (1986), 427-442.
doi: 10.1016/0022-0396(86)90093-8. |
[2] |
S. Angenent,
The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.
doi: 10.1515/crll.1988.390.79. |
[3] |
V. I. Arnold,
A branched covering $CP^2 \to S^4$, hyperbolicity and projective topology, Sib. Math. J., 29 (1988), 717-726.
doi: 10.1007/BF00970265. |
[4] |
V. I. Arnold and M. I. Vishik,
Some solved and unsolved problems in the theory of differential equations and mathematical physics, Russ. Math. Surv., 44 (1989), 157-171.
|
[5] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam,
1992. |
[6] |
J. -M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müller. With an
appendix by François Laudenbach, Astérisque, 205, Soc. Math. de France, 1992. |
[7] |
R. Bott,
Morse theory indomitable, Public. Math. I.H. É.S., 68 (1988), 99-114.
|
[8] |
P. Brunovský and B. Fiedler,
Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89.
|
[9] |
P. Brunovský and B. Fiedler,
Connecting orbits in scalar reaction diffusion equations Ⅱ: The complete solution, J. Diff. Eqns., 81 (1989), 106-135.
doi: 10.1016/0022-0396(89)90180-0. |
[10] |
N. Chafee and E. Infante,
A bifurcation problem for a nonlinear parabolic equation, J. Applicable Analysis, 4 (1974), 17-37.
doi: 10.1080/00036817408839081. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloq.
AMS, Providence, 2002. |
[12] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative
Evolution Equations, Wiley, Chichester, 1994. |
[13] |
B. Fiedler,
Global attractors of one-dimensional parabolic equations: Sixteen examples, Tatra Mountains Math. Publ., 4 (1994), 67-92.
|
[14] |
B. Fiedler (ed.), Handbook of Dynamical Systems, 2, Elsevier, Amsterdam, 2002. |
[15] |
B. Fiedler and C. Rocha,
Heteroclinic orbits of semilinear parabolic equations, J. Diff. Eqns., 125 (1996), 239-281.
doi: 10.1006/jdeq.1996.0031. |
[16] |
B. Fiedler and C. Rocha,
Realization of meander permutations by boundary value problems, J. Diff. Eqns., 156 (1999), 282-308.
doi: 10.1006/jdeq.1998.3532. |
[17] |
B. Fiedler and C. Rocha,
Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc., 352 (2000), 257-284.
doi: 10.1090/S0002-9947-99-02209-6. |
[18] |
B. Fiedler and C. Rocha,
Connectivity and design of planar global attractors of Sturm type, Ⅱ: Connection graphs, J. Diff. Eqns., 244 (2008), 1255-1286.
doi: 10.1016/j.jde.2007.09.015. |
[19] |
B. Fiedler and C. Rocha,
Connectivity and design of planar global attractors of Sturm type, Ⅰ: Bipolar orientations and Hamiltonian paths, J. Reine Angew. Math., 635 (2009), 71-96.
doi: 10.1515/CRELLE.2009.076. |
[20] |
B. Fiedler and C. Rocha,
Connectivity and design of planar global attractors of Sturm type, Ⅲ: Small and Platonic examples, J. Dyn. Diff. Eqns., 22 (2010), 121-162.
doi: 10.1007/s10884-009-9149-2. |
[21] |
B. Fiedler and C. Rocha,
Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes, Discr. Cont. Dyn. Sys., 34 (2014), 5099-5122.
doi: 10.3934/dcds.2014.34.5099. |
[22] |
B. Fiedler and C. Rocha,
Schoenflies spheres as boundaries of bounded unstable manifolds in gradient Sturm systems, J. Dyn. Diff. Eqns., 27 (2015), 597-626.
doi: 10.1007/s10884-013-9311-8. |
[23] |
B. Fiedler and C. Rocha, Sturm 3-balls and global attractors 1: Thom-Smale complexes and
meanders, arXiv: 1611.02003, 2016; São Paulo J. Math. Sc. (2017).
doi: 10.1007/s40863-017-0082-8. |
[24] |
B. Fiedler and C. Rocha, Sturm 3-balls and global attractors 2: Design of Thom-Smale
complexes, arXiv: 1704.00344, 2017; to appear in J. Dyn. Diff. Eqns. |
[25] |
B. Fiedler and C. Rocha, Boundary orders of equilibria in Sturm global attractors, In preparation, 2018. |
[26] |
B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In Trends
in Nonlinear Analysis, M. Kirkilionis et al. (eds.), Springer-Verlag, Berlin, 2003, 23–152. |
[27] |
B. Fiedler, C. Rocha and M. Wolfrum,
A permutation characterization of Sturm global attractors of Hamiltonian type, J. Diff. Eqns., 252 (2012), 588-623.
doi: 10.1016/j.jde.2011.08.013. |
[28] |
B. Fiedler, C. Grotta-Ragazzo and C. Rocha,
An explicit Lyapunov function for reflection symmetric parabolic differential equations on the circle, Russ. Math. Surveys., 69 (2014), 419-433.
|
[29] |
J. M. Franks,
Morse-Smale flows and homotopy theory, Topology, 18 (1979), 199-215.
doi: 10.1016/0040-9383(79)90003-X. |
[30] |
R. Fritsch and R. A. Piccinini, Cellular Structures in Topology, Cambridge University Press,
1990.
doi: 10.1017/CBO9780511983948. |
[31] |
G. Fusco and W. Oliva,
Jacobi matrices and transversality, Proc. Royal Soc. Edinburgh A, 109 (1988), 231-243.
doi: 10.1017/S0308210500027748. |
[32] |
G. Fusco and C. Rocha,
A permutation related to the dynamics of a scalar parabolic PDE, J. Diff. Eqns., 91 (1991), 111-137.
doi: 10.1016/0022-0396(91)90134-U. |
[33] |
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall, Boca Raton, 2004.
doi: 10.1201/9780203998069. |
[34] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surv., 25. AMS, Providence,
1988. |
[35] |
J. K. Hale, L. T. Magalh˜aes and W. M. Oliva, Dynamics in Infinite Dimensions, SpringerVerlag, New York, 2002.
doi: 10.1007/b100032. |
[36] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 804,
Springer-Verlag, New York, 1981. |
[37] |
D. Henry,
Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Eqns., 59 (1985), 165-205.
doi: 10.1016/0022-0396(85)90153-6. |
[38] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lect. Notes Math. 2018,
Springer-Verlag, Berlin, 2011.
doi: 10.1007/978-3-642-18460-4. |
[39] |
A. Karnauhova, Meanders, de Gruyter, Berlin, 2017.
doi: 10.1515/9783110533026. |
[40] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511569418. |
[41] |
Ph. Lappicy, B. Fiedler, A Lyapunov function for fully nonlinear parabolic equations in one
spatial variable, arXiv: 1802.09754 [math. DS], submitted 2018. |
[42] |
J. Mallet-Paret,
Morse decompositions for delay-differential equations, J. Diff. Eqns., 72 (1988), 270-315.
doi: 10.1016/0022-0396(88)90157-X. |
[43] |
H. Matano,
Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227.
doi: 10.1215/kjm/1250522572. |
[44] |
H. Matano,
Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. Fac. Sci. Univ. Tokyo Sec. IA, 29 (1982), 401-441.
|
[45] |
H. Matano and K.-I. Nakamura,
The global attractor of semilinear parabolic equations on ${S^1}$, Discr. Cont. Dyn. Sys., 3 (1997), 1-24.
|
[46] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, SpringerVerlag, New York, 1982. |
[47] |
J. Palis and S. Smale, Structural stability theorems, Global Analysis. Proc. Simp. in Pure
Math. AMS, Providence, (1970), 223–231. |
[48] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,
Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[49] |
G. Raugel, Global attractors in partial differential equations, In [14], (2002), 885–982. |
[50] |
C. Rocha,
Properties of the attractor of a scalar parabolic PDE, J. Dyn. Diff. Eqns., 3 (1991), 575-591.
doi: 10.1007/BF01049100. |
[51] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York,
2002.
doi: 10.1007/978-1-4757-5037-9. |
[52] |
C. Sturm,
Mémoire sur une classe d'équations à différences partielles, Collected Works of Charles François Sturm, (2009), 505-576.
doi: 10.1007/978-3-7643-7990-2_33. |
[53] | |
[54] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[55] |
T. I. Zelenyak,
Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, (Russian) Differencial’nye Uravnenija, 4 (1968), 34-45.
|
show all references
References:
[1] |
S. Angenent,
The Morse-Smale property for a semi-linear parabolic equation, J. Diff. Eqns., 62 (1986), 427-442.
doi: 10.1016/0022-0396(86)90093-8. |
[2] |
S. Angenent,
The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390 (1988), 79-96.
doi: 10.1515/crll.1988.390.79. |
[3] |
V. I. Arnold,
A branched covering $CP^2 \to S^4$, hyperbolicity and projective topology, Sib. Math. J., 29 (1988), 717-726.
doi: 10.1007/BF00970265. |
[4] |
V. I. Arnold and M. I. Vishik,
Some solved and unsolved problems in the theory of differential equations and mathematical physics, Russ. Math. Surv., 44 (1989), 157-171.
|
[5] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam,
1992. |
[6] |
J. -M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müller. With an
appendix by François Laudenbach, Astérisque, 205, Soc. Math. de France, 1992. |
[7] |
R. Bott,
Morse theory indomitable, Public. Math. I.H. É.S., 68 (1988), 99-114.
|
[8] |
P. Brunovský and B. Fiedler,
Connecting orbits in scalar reaction diffusion equations, Dynamics Reported, 1 (1988), 57-89.
|
[9] |
P. Brunovský and B. Fiedler,
Connecting orbits in scalar reaction diffusion equations Ⅱ: The complete solution, J. Diff. Eqns., 81 (1989), 106-135.
doi: 10.1016/0022-0396(89)90180-0. |
[10] |
N. Chafee and E. Infante,
A bifurcation problem for a nonlinear parabolic equation, J. Applicable Analysis, 4 (1974), 17-37.
doi: 10.1080/00036817408839081. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloq.
AMS, Providence, 2002. |
[12] |
A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative
Evolution Equations, Wiley, Chichester, 1994. |
[13] |
B. Fiedler,
Global attractors of one-dimensional parabolic equations: Sixteen examples, Tatra Mountains Math. Publ., 4 (1994), 67-92.
|
[14] |
B. Fiedler (ed.), Handbook of Dynamical Systems, 2, Elsevier, Amsterdam, 2002. |
[15] |
B. Fiedler and C. Rocha,
Heteroclinic orbits of semilinear parabolic equations, J. Diff. Eqns., 125 (1996), 239-281.
doi: 10.1006/jdeq.1996.0031. |
[16] |
B. Fiedler and C. Rocha,
Realization of meander permutations by boundary value problems, J. Diff. Eqns., 156 (1999), 282-308.
doi: 10.1006/jdeq.1998.3532. |
[17] |
B. Fiedler and C. Rocha,
Orbit equivalence of global attractors of semilinear parabolic differential equations, Trans. Amer. Math. Soc., 352 (2000), 257-284.
doi: 10.1090/S0002-9947-99-02209-6. |
[18] |
B. Fiedler and C. Rocha,
Connectivity and design of planar global attractors of Sturm type, Ⅱ: Connection graphs, J. Diff. Eqns., 244 (2008), 1255-1286.
doi: 10.1016/j.jde.2007.09.015. |
[19] |
B. Fiedler and C. Rocha,
Connectivity and design of planar global attractors of Sturm type, Ⅰ: Bipolar orientations and Hamiltonian paths, J. Reine Angew. Math., 635 (2009), 71-96.
doi: 10.1515/CRELLE.2009.076. |
[20] |
B. Fiedler and C. Rocha,
Connectivity and design of planar global attractors of Sturm type, Ⅲ: Small and Platonic examples, J. Dyn. Diff. Eqns., 22 (2010), 121-162.
doi: 10.1007/s10884-009-9149-2. |
[21] |
B. Fiedler and C. Rocha,
Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes, Discr. Cont. Dyn. Sys., 34 (2014), 5099-5122.
doi: 10.3934/dcds.2014.34.5099. |
[22] |
B. Fiedler and C. Rocha,
Schoenflies spheres as boundaries of bounded unstable manifolds in gradient Sturm systems, J. Dyn. Diff. Eqns., 27 (2015), 597-626.
doi: 10.1007/s10884-013-9311-8. |
[23] |
B. Fiedler and C. Rocha, Sturm 3-balls and global attractors 1: Thom-Smale complexes and
meanders, arXiv: 1611.02003, 2016; São Paulo J. Math. Sc. (2017).
doi: 10.1007/s40863-017-0082-8. |
[24] |
B. Fiedler and C. Rocha, Sturm 3-balls and global attractors 2: Design of Thom-Smale
complexes, arXiv: 1704.00344, 2017; to appear in J. Dyn. Diff. Eqns. |
[25] |
B. Fiedler and C. Rocha, Boundary orders of equilibria in Sturm global attractors, In preparation, 2018. |
[26] |
B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In Trends
in Nonlinear Analysis, M. Kirkilionis et al. (eds.), Springer-Verlag, Berlin, 2003, 23–152. |
[27] |
B. Fiedler, C. Rocha and M. Wolfrum,
A permutation characterization of Sturm global attractors of Hamiltonian type, J. Diff. Eqns., 252 (2012), 588-623.
doi: 10.1016/j.jde.2011.08.013. |
[28] |
B. Fiedler, C. Grotta-Ragazzo and C. Rocha,
An explicit Lyapunov function for reflection symmetric parabolic differential equations on the circle, Russ. Math. Surveys., 69 (2014), 419-433.
|
[29] |
J. M. Franks,
Morse-Smale flows and homotopy theory, Topology, 18 (1979), 199-215.
doi: 10.1016/0040-9383(79)90003-X. |
[30] |
R. Fritsch and R. A. Piccinini, Cellular Structures in Topology, Cambridge University Press,
1990.
doi: 10.1017/CBO9780511983948. |
[31] |
G. Fusco and W. Oliva,
Jacobi matrices and transversality, Proc. Royal Soc. Edinburgh A, 109 (1988), 231-243.
doi: 10.1017/S0308210500027748. |
[32] |
G. Fusco and C. Rocha,
A permutation related to the dynamics of a scalar parabolic PDE, J. Diff. Eqns., 91 (1991), 111-137.
doi: 10.1016/0022-0396(91)90134-U. |
[33] |
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall, Boca Raton, 2004.
doi: 10.1201/9780203998069. |
[34] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surv., 25. AMS, Providence,
1988. |
[35] |
J. K. Hale, L. T. Magalh˜aes and W. M. Oliva, Dynamics in Infinite Dimensions, SpringerVerlag, New York, 2002.
doi: 10.1007/b100032. |
[36] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 804,
Springer-Verlag, New York, 1981. |
[37] |
D. Henry,
Some infinite dimensional Morse-Smale systems defined by parabolic differential equations, J. Diff. Eqns., 59 (1985), 165-205.
doi: 10.1016/0022-0396(85)90153-6. |
[38] |
B. Hu, Blow-up Theories for Semilinear Parabolic Equations, Lect. Notes Math. 2018,
Springer-Verlag, Berlin, 2011.
doi: 10.1007/978-3-642-18460-4. |
[39] |
A. Karnauhova, Meanders, de Gruyter, Berlin, 2017.
doi: 10.1515/9783110533026. |
[40] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, 1991.
doi: 10.1017/CBO9780511569418. |
[41] |
Ph. Lappicy, B. Fiedler, A Lyapunov function for fully nonlinear parabolic equations in one
spatial variable, arXiv: 1802.09754 [math. DS], submitted 2018. |
[42] |
J. Mallet-Paret,
Morse decompositions for delay-differential equations, J. Diff. Eqns., 72 (1988), 270-315.
doi: 10.1016/0022-0396(88)90157-X. |
[43] |
H. Matano,
Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18 (1978), 221-227.
doi: 10.1215/kjm/1250522572. |
[44] |
H. Matano,
Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation, J. Fac. Sci. Univ. Tokyo Sec. IA, 29 (1982), 401-441.
|
[45] |
H. Matano and K.-I. Nakamura,
The global attractor of semilinear parabolic equations on ${S^1}$, Discr. Cont. Dyn. Sys., 3 (1997), 1-24.
|
[46] |
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, SpringerVerlag, New York, 1982. |
[47] |
J. Palis and S. Smale, Structural stability theorems, Global Analysis. Proc. Simp. in Pure
Math. AMS, Providence, (1970), 223–231. |
[48] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,
Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[49] |
G. Raugel, Global attractors in partial differential equations, In [14], (2002), 885–982. |
[50] |
C. Rocha,
Properties of the attractor of a scalar parabolic PDE, J. Dyn. Diff. Eqns., 3 (1991), 575-591.
doi: 10.1007/BF01049100. |
[51] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York,
2002.
doi: 10.1007/978-1-4757-5037-9. |
[52] |
C. Sturm,
Mémoire sur une classe d'équations à différences partielles, Collected Works of Charles François Sturm, (2009), 505-576.
doi: 10.1007/978-3-7643-7990-2_33. |
[53] | |
[54] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[55] |
T. I. Zelenyak,
Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, (Russian) Differencial’nye Uravnenija, 4 (1968), 34-45.
|








































γ | κ | ρ | κρ, double dual |
δ in Σδj | −δ for j = 0, 1, 2 | −δ for j = 1 | −δ for j = 0, 2 |
bipolarity | reverse | keep | reverse |
3-cell orientation | reverse | reverse | keep |
poles N, S | N ↔ S | keep | N ↔ S |
meridians WE, EW | WE ↔ EW | WE ↔ EW | keep |
hemispheres W, E | W ↔ E | keep | W ↔ E |
faces NE, NW, SE, SW | NE ↔ SE NW ↔ SW |
NE ↔ NW SE ↔ SW |
NE ↔ SW NW ↔ SE |
w±ι | |
w±1-ι | |
hι | κhικ | ρh1−ι | κρh1−ικ |
σ | κσκ | σ−1 | κσ−1κ |
γ | κ | ρ | κρ, double dual |
δ in Σδj | −δ for j = 0, 1, 2 | −δ for j = 1 | −δ for j = 0, 2 |
bipolarity | reverse | keep | reverse |
3-cell orientation | reverse | reverse | keep |
poles N, S | N ↔ S | keep | N ↔ S |
meridians WE, EW | WE ↔ EW | WE ↔ EW | keep |
hemispheres W, E | W ↔ E | keep | W ↔ E |
faces NE, NW, SE, SW | NE ↔ SE NW ↔ SW |
NE ↔ NW SE ↔ SW |
NE ↔ SW NW ↔ SE |
w±ι | |
w±1-ι | |
hι | κhικ | ρh1−ι | κρh1−ικ |
σ | κσκ | σ−1 | κσ−1κ |
N | 7 | 9 | 11 | 11 | 13 | 13 |
(m, n) | (1, 1) | (1, 2) | (1, 3) | (2, 2) | (1, 4) | (2, 3) |
case | 5.2|5.2 | 7.3|7.3 | 9.4 − 1|9.4 − 1 | 9.4 − 2|9.4 − 2 | 11.5 − 1|11.5 − 1 | 11.5 − 2|11.5 − 2 |
N | 7 | 9 | 11 | 11 | 13 | 13 |
(m, n) | (1, 1) | (1, 2) | (1, 3) | (2, 2) | (1, 4) | (2, 3) |
case | 5.2|5.2 | 7.3|7.3 | 9.4 − 1|9.4 − 1 | 9.4 − 2|9.4 − 2 | 11.5 − 1|11.5 − 1 | 11.5 − 2|11.5 − 2 |
N | 7 | 9 | 11 | 11 | 13 | 13 |
(m, n) | (1, 1) | (2, 1) | (3, 1) | (2, 2) | (4, 1) | (3, 2) |
case | 5.2|5.2 | 5.2|7.22 | 5.2|9.23 | 7.22|7.22 | 5.2|11.24 | 7.22|9.23 |
N | 7 | 9 | 11 | 11 | 13 | 13 |
(m, n) | (1, 1) | (2, 1) | (3, 1) | (2, 2) | (4, 1) | (3, 2) |
case | 5.2|5.2 | 5.2|7.22 | 5.2|9.23 | 7.22|7.22 | 5.2|11.24 | 7.22|9.23 |
M\N | 11 | 13 | 13 | 13 |
0 | (5.2|9.32) | (5.2|11.322) | (5.2|11.42) | (7.22|9.32) |
1 | (7.3|9.32−1) | (7.3|11.322−1) | (7.3|11.43−1) | (9.32−1|9.32−1) |
2 | – | (9.4|11.32−1) | (9.4|11.42−1) | – |
M\N | 11 | 13 | 13 | 13 |
0 | (5.2|9.32) | (5.2|11.322) | (5.2|11.42) | (7.22|9.32) |
1 | (7.3|9.32−1) | (7.3|11.322−1) | (7.3|11.43−1) | (9.32−1|9.32−1) |
2 | – | (9.4|11.32−1) | (9.4|11.42−1) | – |
M\N | 11 | 13 | 13 | 13 |
1 ref | (7.3|9.32 − 1) | (7.3|11.322 − 1) | (7.3|11.43 − 1) | (9.32 − 1|9.32 − 1) |
1 attr | (7.3|9.32 − 2) | (7.3|11.322 − 2) (7.3|11.322 − 3) |
(7.3|11.43 − 2) | (9.32 − 1|9.32 − 2) |
2 ref | – | (9.4|11.32 − 1) | (9.4|11.42 − 1) | – |
2 attr | – | (9.4|11.42 − 2) | (9.4|11.32 − 2) (9.4|11.42 − 3) |
– |
M\N | 11 | 13 | 13 | 13 |
1 ref | (7.3|9.32 − 1) | (7.3|11.322 − 1) | (7.3|11.43 − 1) | (9.32 − 1|9.32 − 1) |
1 attr | (7.3|9.32 − 2) | (7.3|11.322 − 2) (7.3|11.322 − 3) |
(7.3|11.43 − 2) | (9.32 − 1|9.32 − 2) |
2 ref | – | (9.4|11.32 − 1) | (9.4|11.42 − 1) | – |
2 attr | – | (9.4|11.42 − 2) | (9.4|11.32 − 2) (9.4|11.42 − 3) |
– |
# | case | sec | Sturm permutation σ | iso | pitch | remarks |
1 | 5.2|5.2 | 6.2 | 1 6 3 4 5 2 7 | κ, ρ | √ | Chafee-Infante |
2 | 5.2|7.22 | 6.3 | 1 8 3 4 7 6 5 2 9 | ρ | √ | 2, 1-gon, susp |
3 | 7.3|7.3 | 6.2 | 1 6 7 8 3 4 5 2 9 | κρ | √ | 1, 2-gon, pitch |
4 | 5.2|9.23 | 6.3 | 1 10 3 4 9 8 7 6 5 2 11 | ρ | √ | 3, 1-gon, susp |
5 | 5.2|9.32 | 6.5 | 1 10 3 4 9 6 7 8 5 2 11 | ρ | √ | 2, 1 multi-striped |
6 | 7.22|7.22 | 6.3 | 1 10 5 4 3 6 9 8 7 2 11 | κ, ρ | √ | 2, 2-gon, susp |
7 | 7.3|9.32 − 1 | 6.5 | 1 8 9 10 3 4 7 6 5 2 11 | – | √ | 2, 1 multi-striped |
8 | 7.3|9.32 − 2 | 6.6 | 1 6 7 10 3 4 9 8 5 2 11 | – | – | from 7, non-pitch |
9 | 9.4 − 1|9.4 − 1 | 6.2 | 1 6 7 8 9 10 3 4 5 2 11 | κρ | √ | 1, 3-gon, pitch |
10 | 9.4 − 2|9.4 − 2 | 6.2 | 1 8 9 10 5 6 7 2 3 4 11 | κ, ρ | √ | 2, 2-gon, pitch |
# | case | sec | Sturm permutation σ | iso | pitch | remarks |
1 | 5.2|5.2 | 6.2 | 1 6 3 4 5 2 7 | κ, ρ | √ | Chafee-Infante |
2 | 5.2|7.22 | 6.3 | 1 8 3 4 7 6 5 2 9 | ρ | √ | 2, 1-gon, susp |
3 | 7.3|7.3 | 6.2 | 1 6 7 8 3 4 5 2 9 | κρ | √ | 1, 2-gon, pitch |
4 | 5.2|9.23 | 6.3 | 1 10 3 4 9 8 7 6 5 2 11 | ρ | √ | 3, 1-gon, susp |
5 | 5.2|9.32 | 6.5 | 1 10 3 4 9 6 7 8 5 2 11 | ρ | √ | 2, 1 multi-striped |
6 | 7.22|7.22 | 6.3 | 1 10 5 4 3 6 9 8 7 2 11 | κ, ρ | √ | 2, 2-gon, susp |
7 | 7.3|9.32 − 1 | 6.5 | 1 8 9 10 3 4 7 6 5 2 11 | – | √ | 2, 1 multi-striped |
8 | 7.3|9.32 − 2 | 6.6 | 1 6 7 10 3 4 9 8 5 2 11 | – | – | from 7, non-pitch |
9 | 9.4 − 1|9.4 − 1 | 6.2 | 1 6 7 8 9 10 3 4 5 2 11 | κρ | √ | 1, 3-gon, pitch |
10 | 9.4 − 2|9.4 − 2 | 6.2 | 1 8 9 10 5 6 7 2 3 4 11 | κ, ρ | √ | 2, 2-gon, pitch |
# | case | sec | Sturm permutation σ | iso | pitch | remarks |
11 | 5.2|11.24 | 6.3 | 1 12 3 4 11 10 9 8 7 6 5 2 13 | ρ | √ | 4, 1-gon, susp |
12 | 5.2|11.322 | 6.5 | 1 12 3 4 11 8 9 10 7 6 5 2 13 | – | √ | 3, 1 multi-striped |
13 | 5.2|11.322+ | 6.4 | 1 12 3 4 11 6 7 10 9 8 5 2 13 | ρ | √ | triangle core |
14 | 5.2|11.322− | 6.4 | 1 12 3 4 11 8 7 6 9 10 5 2 13 | ρ | √ | triangle core |
15 | 5.2|11.42 | 6.5 | 1 12 3 4 11 6 7 8 9 10 5 2 13 | ρ | √ | 2, 1 multi-striped |
16 | 7.22|9.23 | 6.3 | 1 12 5 4 3 6 11 10 9 8 7 2 13 | ρ | √ | 3, 2-gon, susp |
17 | 7.22|9.32 | 6.5 | 1 12 5 4 3 6 11 8 9 10 7 2 13 | ρ | √ | 2, 2 multi-striped |
18 | 7.3|11.322−1 | 6.5 | 1 10 11 12 3 4 9 8 7 6 5 2 13 | – | √ | 3, 1 multi-striped |
19 | 7.3|11.322−2 | 6.6 | 1 8 9 12 3 4 11 10 7 6 5 2 13 | – | – | from 18 |
20 | 7.3|11.322−3 | 6.6 | 1 6 7 12 3 4 11 10 9 8 5 2 13 | – | – | from 18 |
21 | 7.3|11.43−1 | 6.5 | 1 10 11 12 3 4 9 6 7 8 5 2 13 | – | √ | 2, 1 multi-striped |
22 | 7.3|11.43−2 | 6.6 | 1 6 7 12 3 4 11 8 9 10 5 2 13 | – | √ | from 21 |
23 | 9.32−1|9.32−1 | 6.5 | 1 10 11 12 5 4 3 6 9 8 7 2 13 | κρ | √ | 2, 2 multi-striped |
24 | 9.32−1|9.32−2 | 6.6 | 1 8 9 12 5 4 3 6 11 10 7 2 13 | – | – | from 23 |
25 | 9.4|11.32−1 | 6.5 | 1 10 11 12 5 6 9 8 7 2 3 4 13 | ρ | √ | 2, 1 multi-striped |
26 | 9.4|11.32−2 | 6.6 | 1 6 7 10 11 12 3 4 9 8 5 2 13 | – | – | from 27 |
27 | 9.4|11.42−1 | 6.5 | 1 8 9 10 11 12 3 4 7 6 5 2 13 | – | √ | 2, 1 multi-striped |
28 | 9.4|11.42−2 | 6.6 | 1 8 9 12 5 6 11 10 7 2 3 4 13 | – | – | from 25 |
29 | 9.4|11.42−3 | 6.6 | 1 6 7 8 9 12 3 4 11 10 5 2 13 | – | – | from 27 |
30 | 11.5 − 1|11.5 − 1 | 6.2 | 1 6 7 8 9 10 11 12 3 4 5 2 13 | κρ | √ | 1, 4-gon, pitch |
31 | 11.5−2|11.5−2 | 6.2 | 1 8 9 10 11 12 5 6 7 2 3 4 13 | κρ | √ | 2, 3-gon, pitch |
# | case | sec | Sturm permutation σ | iso | pitch | remarks |
11 | 5.2|11.24 | 6.3 | 1 12 3 4 11 10 9 8 7 6 5 2 13 | ρ | √ | 4, 1-gon, susp |
12 | 5.2|11.322 | 6.5 | 1 12 3 4 11 8 9 10 7 6 5 2 13 | – | √ | 3, 1 multi-striped |
13 | 5.2|11.322+ | 6.4 | 1 12 3 4 11 6 7 10 9 8 5 2 13 | ρ | √ | triangle core |
14 | 5.2|11.322− | 6.4 | 1 12 3 4 11 8 7 6 9 10 5 2 13 | ρ | √ | triangle core |
15 | 5.2|11.42 | 6.5 | 1 12 3 4 11 6 7 8 9 10 5 2 13 | ρ | √ | 2, 1 multi-striped |
16 | 7.22|9.23 | 6.3 | 1 12 5 4 3 6 11 10 9 8 7 2 13 | ρ | √ | 3, 2-gon, susp |
17 | 7.22|9.32 | 6.5 | 1 12 5 4 3 6 11 8 9 10 7 2 13 | ρ | √ | 2, 2 multi-striped |
18 | 7.3|11.322−1 | 6.5 | 1 10 11 12 3 4 9 8 7 6 5 2 13 | – | √ | 3, 1 multi-striped |
19 | 7.3|11.322−2 | 6.6 | 1 8 9 12 3 4 11 10 7 6 5 2 13 | – | – | from 18 |
20 | 7.3|11.322−3 | 6.6 | 1 6 7 12 3 4 11 10 9 8 5 2 13 | – | – | from 18 |
21 | 7.3|11.43−1 | 6.5 | 1 10 11 12 3 4 9 6 7 8 5 2 13 | – | √ | 2, 1 multi-striped |
22 | 7.3|11.43−2 | 6.6 | 1 6 7 12 3 4 11 8 9 10 5 2 13 | – | √ | from 21 |
23 | 9.32−1|9.32−1 | 6.5 | 1 10 11 12 5 4 3 6 9 8 7 2 13 | κρ | √ | 2, 2 multi-striped |
24 | 9.32−1|9.32−2 | 6.6 | 1 8 9 12 5 4 3 6 11 10 7 2 13 | – | – | from 23 |
25 | 9.4|11.32−1 | 6.5 | 1 10 11 12 5 6 9 8 7 2 3 4 13 | ρ | √ | 2, 1 multi-striped |
26 | 9.4|11.32−2 | 6.6 | 1 6 7 10 11 12 3 4 9 8 5 2 13 | – | – | from 27 |
27 | 9.4|11.42−1 | 6.5 | 1 8 9 10 11 12 3 4 7 6 5 2 13 | – | √ | 2, 1 multi-striped |
28 | 9.4|11.42−2 | 6.6 | 1 8 9 12 5 6 11 10 7 2 3 4 13 | – | – | from 25 |
29 | 9.4|11.42−3 | 6.6 | 1 6 7 8 9 12 3 4 11 10 5 2 13 | – | – | from 27 |
30 | 11.5 − 1|11.5 − 1 | 6.2 | 1 6 7 8 9 10 11 12 3 4 5 2 13 | κρ | √ | 1, 4-gon, pitch |
31 | 11.5−2|11.5−2 | 6.2 | 1 8 9 10 11 12 5 6 7 2 3 4 13 | κρ | √ | 2, 3-gon, pitch |
N | n | d | c0 | c1 | c2 | ϑ | dual | |
15 | 3 | 3 | 4 | 6 | 4 | 1 | ||
27 | 3 | 4 | 6 | 12 | 8 | 2 | ||
63 | 3 | 5 | 12 | 30 | 20 | 3 | ||
27 | 4 | 3 | 8 | 12 | 6 | 3 | ||
63 | 5 | 3 | 20 | 30 | 12 | 5 |
N | n | d | c0 | c1 | c2 | ϑ | dual | |
15 | 3 | 3 | 4 | 6 | 4 | 1 | ||
27 | 3 | 4 | 6 | 12 | 8 | 2 | ||
63 | 3 | 5 | 12 | 30 | 20 | 3 | ||
27 | 4 | 3 | 8 | 12 | 6 | 3 | ||
63 | 5 | 3 | 20 | 30 | 12 | 5 |
# | δ | η | Sturm permutation σ | iso | pitch |
1 | 1 | 1 14 5 6 13 10 9 2 3 8 11 12 7 4 15 | – | – | |
1 | 2 | 1 8 9 12 5 4 13 14 3 6 11 10 7 2 15 | κρ | – |
# | δ | η | Sturm permutation σ | iso | pitch |
1 | 1 | 1 14 5 6 13 10 9 2 3 8 11 12 7 4 15 | – | – | |
1 | 2 | 1 8 9 12 5 4 13 14 3 6 11 10 7 2 15 | κρ | – |
# | δ | η | Sturm permutation σ | iso | pitch |
1 | 1 | 1 1 26 5 6 25 14 15 24 23 20 19 16 13 12 11 2 3 10 17 18 9 8 21 22 7 4 27 |
– | – | |
2 | 1 | 1 1 26 5 6 25 22 21 18 17 2 3 16 15 8 9 14 19 20 13 12 23 24 11 10 7 4 27 |
– | – | |
3 | |
1 | 1 1 26 5 6 25 22 21 12 11 2 3 10 13 20 19 14 9 8 15 18 23 24 17 16 7 4 27 |
– | – |
4 | 1 | 1 1 26 5 6 25 18 17 12 11 2 3 10 13 16 19 24 23 20 15 14 9 8 21 22 7 4 27 |
– | – | |
5 | 1 | 2 1 16 17 26 7 6 5 8 25 22 21 18 15 14 13 2 3 12 19 20 11 10 23 24 9 4 27 |
ρ | – |
# | δ | η | Sturm permutation σ | iso | pitch |
1 | 1 | 1 1 26 5 6 25 14 15 24 23 20 19 16 13 12 11 2 3 10 17 18 9 8 21 22 7 4 27 |
– | – | |
2 | 1 | 1 1 26 5 6 25 22 21 18 17 2 3 16 15 8 9 14 19 20 13 12 23 24 11 10 7 4 27 |
– | – | |
3 | |
1 | 1 1 26 5 6 25 22 21 12 11 2 3 10 13 20 19 14 9 8 15 18 23 24 17 16 7 4 27 |
– | – |
4 | 1 | 1 1 26 5 6 25 18 17 12 11 2 3 10 13 16 19 24 23 20 15 14 9 8 21 22 7 4 27 |
– | – | |
5 | 1 | 2 1 16 17 26 7 6 5 8 25 22 21 18 15 14 13 2 3 12 19 20 11 10 23 24 9 4 27 |
ρ | – |
# | δ | η | Sturm permutation σ | fig. | iso | pitch | |
1 | 1 | 1 | 1 26 7 8 25 20 19 2 3 16 15 4 5 10 11 14 17 18 21 22 13 12 23 24 9 6 27 |
7.14 | – | – | |
2 | 1 | 1 | 1 26 7 8 25 20 19 2 3 12 13 18 21 22 17 14 11 4 5 10 15 16 23 24 9 6 27 |
7.14 | – | – | |
3 | 1 | 1 | 1 26 7 8 25 14 15 22 21 16 13 2 3 12 17 18 11 4 5 10 19 20 23 24 9 6 27 |
7.14 | – | – | |
4 | 2 | 1 | 1 18 19 26 5 6 25 20 17 14 13 2 3 8 9 12 15 16 21 22 11 10 23 24 7 4 27 |
7.14 | – | – | |
5 | 2 | 1 | 1 18 19 26 5 6 25 20 17 10 11 16 21 22 15 12 9 2 3 8 13 14 23 24 7 4 27 |
7.14 | ρ | – | |
6 | 2 | 2 | 1 12 13 18 19 24 7 6 25 26 5 8 23 20 17 14 11 2 3 10 15 16 21 22 9 4 27 |
7.12 | – | – | |
7 | |
3 | 3 | 1 18 19 24 13 6 7 12 25 26 11 8 5 14 23 20 17 2 3 16 21 22 15 4 9 10 27 |
7.10 | κ, ρ, κρ | – |
# | δ | η | Sturm permutation σ | fig. | iso | pitch | |
1 | 1 | 1 | 1 26 7 8 25 20 19 2 3 16 15 4 5 10 11 14 17 18 21 22 13 12 23 24 9 6 27 |
7.14 | – | – | |
2 | 1 | 1 | 1 26 7 8 25 20 19 2 3 12 13 18 21 22 17 14 11 4 5 10 15 16 23 24 9 6 27 |
7.14 | – | – | |
3 | 1 | 1 | 1 26 7 8 25 14 15 22 21 16 13 2 3 12 17 18 11 4 5 10 19 20 23 24 9 6 27 |
7.14 | – | – | |
4 | 2 | 1 | 1 18 19 26 5 6 25 20 17 14 13 2 3 8 9 12 15 16 21 22 11 10 23 24 7 4 27 |
7.14 | – | – | |
5 | 2 | 1 | 1 18 19 26 5 6 25 20 17 10 11 16 21 22 15 12 9 2 3 8 13 14 23 24 7 4 27 |
7.14 | ρ | – | |
6 | 2 | 2 | 1 12 13 18 19 24 7 6 25 26 5 8 23 20 17 14 11 2 3 10 15 16 21 22 9 4 27 |
7.12 | – | – | |
7 | |
3 | 3 | 1 18 19 24 13 6 7 12 25 26 11 8 5 14 23 20 17 2 3 16 21 22 15 4 9 10 27 |
7.10 | κ, ρ, κρ | – |
w−0 | w−1 | w+0 | w+1 | w−0 |
A | B | C | D | E |
E | C | A | ||
B | C | D | E | B |
C | ||||
C | A | - | - | - |
w−0 | w−1 | w+0 | w+1 | w−0 |
A | B | C | D | E |
E | C | A | ||
B | C | D | E | B |
C | ||||
C | A | - | - | - |
Case | δ | η | Sturm permutation σ | iso | pitch | |
1 | 1 | 2 | 1 20 21 62 7 6 5 8 61 58 57 40 39 22 19 18 23 38 41 56 55 46 45 42 37 36 35 24 17 16 15 2 3 14 25 34 33 26 13 12 27 32 43 44 31 30 47 54 53 48 29 28 11 10 49 52 59 60 51 50 9 4 63 |
– | – | |
2 | 2 | 2 | 1 26 27 38 39 52 53 60 9 8 61 62 7 10 59 54 51 40 37 28 25 2 3 14 15 24 29 30 31 36 41 42 43 50 55 56 49 44 35 32 23 16 17 22 33 34 45 46 V21 18 13 4 5 12 19 20 47 48 57 58 11 6 63 |
– | – |
Case | δ | η | Sturm permutation σ | iso | pitch | |
1 | 1 | 2 | 1 20 21 62 7 6 5 8 61 58 57 40 39 22 19 18 23 38 41 56 55 46 45 42 37 36 35 24 17 16 15 2 3 14 25 34 33 26 13 12 27 32 43 44 31 30 47 54 53 48 29 28 11 10 49 52 59 60 51 50 9 4 63 |
– | – | |
2 | 2 | 2 | 1 26 27 38 39 52 53 60 9 8 61 62 7 10 59 54 51 40 37 28 25 2 3 14 15 24 29 30 31 36 41 42 43 50 55 56 49 44 35 32 23 16 17 22 33 34 45 46 V21 18 13 4 5 12 19 20 47 48 57 58 11 6 63 |
– | – |
Case | δ | η | Sturm permutation σ | iso | pitch |
1 | 1 | 1 | 1 28 29 80 3 6 69 58 17 18 39 40 57 70 77 50 47 32 25 24 33 46 51 76 71 56 41 38 19 16 59 68 7 8 67 60 15 20 37 42 55 72 73 54 43 36 21 14 61 66 9 10 65 62 13 22 35 44 53 74 75 52 45 34 23 26 31 48 49 78 5 4 79 30 27 12 63 64 11 2 81 |
– | – |
2 | 1 | 1 | 1 80 5 72 71 6 33 34 51 52 79 76 55 48 37 30 9 10 29 38 47 56 65 20 19 66 75 2 3 74 67 18 21 64 57 46 39 28 11 12 27 40 45 58 63 22 17 68 69 16 23 62 59 44 41 26 13 8 31 36 49 54 77 78 53 50 35 32 7 14 25 42 43 60 61 24 15 70 73 4 81 |
– | – |
Case | δ | η | Sturm permutation σ | iso | pitch |
1 | 1 | 1 | 1 28 29 80 3 6 69 58 17 18 39 40 57 70 77 50 47 32 25 24 33 46 51 76 71 56 41 38 19 16 59 68 7 8 67 60 15 20 37 42 55 72 73 54 43 36 21 14 61 66 9 10 65 62 13 22 35 44 53 74 75 52 45 34 23 26 31 48 49 78 5 4 79 30 27 12 63 64 11 2 81 |
– | – |
2 | 1 | 1 | 1 80 5 72 71 6 33 34 51 52 79 76 55 48 37 30 9 10 29 38 47 56 65 20 19 66 75 2 3 74 67 18 21 64 57 46 39 28 11 12 27 40 45 58 63 22 17 68 69 16 23 62 59 44 41 26 13 8 31 36 49 54 77 78 53 50 35 32 7 14 25 42 43 60 61 24 15 70 73 4 81 |
– | – |
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