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Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents
The graph of the logistic map is a tower
1. | Department of Mathematics, Howard University, Washington, DC 20059, USA |
2. | Institute for Physical Science and Technology and the Departments of Mathematics and Physics, University of Maryland, College Park, MD 20742, USA |
The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node $ A $ to node $ B $ if, using arbitrary small perturbations, a trajectory starting from any point of $ A $ can be steered to any point of $ B $. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node $ A $ to node $ B $, the unstable manifold of some periodic orbit in $ A $ contains points that eventually map onto $ B $. For special parameter values, this tower has infinitely many nodes.
References:
[1] |
K. Alligood and J. Yorke,
Accessible saddles on fractal basin boundaries, Ergodic Theory and Dynamical Systems, 12 (1992), 377-400.
doi: 10.1017/S0143385700006842. |
[2] |
G. Birkhoff, Sur quelques courbes fermées remarquables, Bulletin de la Société mathématique de France, 60 (1932), 1-26. |
[3] |
A. Blokh, The "spectral" decomposition for one-dimensional maps, in Dynamics Reported, Springer, 1995, 1-59. |
[4] |
A. Blokh and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, in Annales Scientifiques de l'Ecole Normale Supérieure, 24 (1991), 545-573.
doi: 10.24033/asens.1636. |
[5] |
R. Bowen,
$\omega$-limit sets for axiom A diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[6] |
J. Buescu, Exotic Attractors: From Liapunov Stability to Riddled Basins, 153, Birkhäuser, 2012.
doi: 10.1007/978-3-0348-7421-2. |
[7] |
X. Chen and P. Polácik,
Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, Journal of Dynamics and Differential Equations, 7 (1995), 73-107.
doi: 10.1007/BF02218815. |
[8] |
C. Conley, On a generalization of the morse index, in Ordinary Differential Equations, Elsevier, 1972, 27-33. |
[9] |
C. Conley, Isolated Invariant Sets and the Morse Index, 38, American Mathematical Soc., 1978. |
[10] |
R. De Leo and J. Yorke, Infinite towers in the graph of a dynamical system, Nonlinear Dynamics (to appear) |
[11] |
W. de Melo and S. van Strien,
A structure theorem in one dimensional dynamics, Annals of Mathematics, 129 (1989), 519-546.
doi: 10.2307/1971516. |
[12] |
W. de Melo and S. van Strien, One-Dimensional Dynamics, 25, Springer Science & Business Media, 1993. Available from: URL http://www2.imperial.ac.uk/ svanstri/Files/demelo-strien.pdf.
doi: 10.1007/978-3-642-78043-1. |
[13] |
M. Feigenbaum,
Quantitative universality for a class of nonlinear transformations, Journal of Statistical Physics, 19 (1978), 25-52.
doi: 10.1007/BF01020332. |
[14] |
J. Graczyk and G. Swiatek, Generic hyperbolicity in the logistic family, Annals of Mathematics, 1-52.
doi: 10.2307/2951831. |
[15] |
C. Grebogi, E. Ott and J. Yorke, Chaotic attractors in crisis, Physical Review Letters, 48 (1982), 1507.
doi: 10.1103/PhysRevLett.48.1507. |
[16] |
C. Grebogi, E. Ott and J. Yorke,
Basin boundary metamorphoses: Changes in accessible boundary orbits, Nuclear Physics B-Proceedings Supplements, 2 (1987), 281-300.
doi: 10.1016/0167-2789(87)90078-9. |
[17] |
J. Guckenheimer,
Sensitive dependence to initial conditions for one dimensional maps, Communications in Mathematical Physics, 70 (1979), 133-160.
doi: 10.1007/BF01982351. |
[18] |
J. Hale, L. Magalhães and W. Oliva, Dynamics in Infinite Dimensions, 47, Springer Science & Business Media, 2006. |
[19] |
M. Hirsch, H. Smith and X. Zhao,
Chain transitivity, attractivity, and strong repellors for semidynamical systems, Journal of Dynamics and Differential Equations, 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
[20] |
P. Holmes and D. Whitley,
Bifurcations of one-and two-dimensional maps, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 311 (1984), 43-102.
doi: 10.1098/rsta.1984.0020. |
[21] |
M. Hurley,
Chain recurrence and attraction in non-compact spaces, Ergodic Theory and Dynamical Systems, 11 (1991), 709-729.
doi: 10.1017/S014338570000643X. |
[22] |
M. Jakobson,
Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88.
doi: 10.1007/BF01941800. |
[23] |
L. Jonker and D. Rand,
Bifurcations in one dimension, Inventiones Mathematicae, 62 (1980), 347-365.
doi: 10.1007/BF01394248. |
[24] |
M. Lyubich,
Dynamics of quadratic polynomials, I-II, Acta Mathematica, 178 (1997), 185-297.
doi: 10.1007/BF02392694. |
[25] |
M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Mathematics, 1-78.
doi: 10.2307/3597183. |
[26] |
J. Mallet-Paret,
Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315.
doi: 10.1016/0022-0396(88)90157-X. |
[27] |
J. Milnor, On the concept of attractor, in The Theory of Chaotic Attractors, Springer, 1985,243-264.
doi: 10.1007/BF01212280. |
[28] |
D. Norton,
The Conley decomposition theorem for maps: a metric approach, Rikkyo Daigaku Sugaku Zasshi, 44 (1995), 151-173.
|
[29] |
D. Norton,
The fundamental theorem of dynamical systems, Commentationes Mathematicae Universitatis Carolinae, 36 (1995), 585-597.
|
[30] |
M. Patrão,
Morse decomposition of semiflows on topological spaces, Journal of Dynamics and Differential Equations, 19 (2007), 181-198.
doi: 10.1007/s10884-006-9033-2. |
[31] |
K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, 1987.
doi: 10.1007/978-3-642-72833-4. |
[32] |
A. Sharkovsky, S. Kolyada, A. Sivak and V. Fedorenko, Dynamics of One-Dimensional Maps, 407, Springer Science & Business Media, 1997.
doi: 10.1007/978-94-015-8897-3. |
[33] |
D. Singer,
Stable orbits and bifurcation of maps of the interval, SIAM Journal on Applied Mathematics, 35 (1978), 260-267.
doi: 10.1137/0135020. |
[34] |
S. Smale, On gradient dynamical systems, Annals of Mathematics, 199-206.
doi: 10.2307/1970311. |
[35] |
S. Smale,
Differentiable dynamical systems, Bulletin of the American mathematical Society, 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[36] |
S. Smale and R. Williams,
The qualitative analysis of a difference equation of population growth, Journal of Mathematical Biology, 3 (1976), 1-4.
doi: 10.1007/BF00307853. |
[37] |
S. van Strien, On the bifurcations creating horseshoes, in Dynamical Systems and Turbulence, Warwick 1980, Springer, 1981,316-351. |
[38] |
F. Wilson and J. Yorke,
Lyapunov functions and isolating blocks, Journal of Differential Equations, 13 (1973), 106-123.
doi: 10.1016/0022-0396(73)90034-X. |
show all references
References:
[1] |
K. Alligood and J. Yorke,
Accessible saddles on fractal basin boundaries, Ergodic Theory and Dynamical Systems, 12 (1992), 377-400.
doi: 10.1017/S0143385700006842. |
[2] |
G. Birkhoff, Sur quelques courbes fermées remarquables, Bulletin de la Société mathématique de France, 60 (1932), 1-26. |
[3] |
A. Blokh, The "spectral" decomposition for one-dimensional maps, in Dynamics Reported, Springer, 1995, 1-59. |
[4] |
A. Blokh and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, in Annales Scientifiques de l'Ecole Normale Supérieure, 24 (1991), 545-573.
doi: 10.24033/asens.1636. |
[5] |
R. Bowen,
$\omega$-limit sets for axiom A diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339.
doi: 10.1016/0022-0396(75)90065-0. |
[6] |
J. Buescu, Exotic Attractors: From Liapunov Stability to Riddled Basins, 153, Birkhäuser, 2012.
doi: 10.1007/978-3-0348-7421-2. |
[7] |
X. Chen and P. Polácik,
Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations, Journal of Dynamics and Differential Equations, 7 (1995), 73-107.
doi: 10.1007/BF02218815. |
[8] |
C. Conley, On a generalization of the morse index, in Ordinary Differential Equations, Elsevier, 1972, 27-33. |
[9] |
C. Conley, Isolated Invariant Sets and the Morse Index, 38, American Mathematical Soc., 1978. |
[10] |
R. De Leo and J. Yorke, Infinite towers in the graph of a dynamical system, Nonlinear Dynamics (to appear) |
[11] |
W. de Melo and S. van Strien,
A structure theorem in one dimensional dynamics, Annals of Mathematics, 129 (1989), 519-546.
doi: 10.2307/1971516. |
[12] |
W. de Melo and S. van Strien, One-Dimensional Dynamics, 25, Springer Science & Business Media, 1993. Available from: URL http://www2.imperial.ac.uk/ svanstri/Files/demelo-strien.pdf.
doi: 10.1007/978-3-642-78043-1. |
[13] |
M. Feigenbaum,
Quantitative universality for a class of nonlinear transformations, Journal of Statistical Physics, 19 (1978), 25-52.
doi: 10.1007/BF01020332. |
[14] |
J. Graczyk and G. Swiatek, Generic hyperbolicity in the logistic family, Annals of Mathematics, 1-52.
doi: 10.2307/2951831. |
[15] |
C. Grebogi, E. Ott and J. Yorke, Chaotic attractors in crisis, Physical Review Letters, 48 (1982), 1507.
doi: 10.1103/PhysRevLett.48.1507. |
[16] |
C. Grebogi, E. Ott and J. Yorke,
Basin boundary metamorphoses: Changes in accessible boundary orbits, Nuclear Physics B-Proceedings Supplements, 2 (1987), 281-300.
doi: 10.1016/0167-2789(87)90078-9. |
[17] |
J. Guckenheimer,
Sensitive dependence to initial conditions for one dimensional maps, Communications in Mathematical Physics, 70 (1979), 133-160.
doi: 10.1007/BF01982351. |
[18] |
J. Hale, L. Magalhães and W. Oliva, Dynamics in Infinite Dimensions, 47, Springer Science & Business Media, 2006. |
[19] |
M. Hirsch, H. Smith and X. Zhao,
Chain transitivity, attractivity, and strong repellors for semidynamical systems, Journal of Dynamics and Differential Equations, 13 (2001), 107-131.
doi: 10.1023/A:1009044515567. |
[20] |
P. Holmes and D. Whitley,
Bifurcations of one-and two-dimensional maps, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 311 (1984), 43-102.
doi: 10.1098/rsta.1984.0020. |
[21] |
M. Hurley,
Chain recurrence and attraction in non-compact spaces, Ergodic Theory and Dynamical Systems, 11 (1991), 709-729.
doi: 10.1017/S014338570000643X. |
[22] |
M. Jakobson,
Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Communications in Mathematical Physics, 81 (1981), 39-88.
doi: 10.1007/BF01941800. |
[23] |
L. Jonker and D. Rand,
Bifurcations in one dimension, Inventiones Mathematicae, 62 (1980), 347-365.
doi: 10.1007/BF01394248. |
[24] |
M. Lyubich,
Dynamics of quadratic polynomials, I-II, Acta Mathematica, 178 (1997), 185-297.
doi: 10.1007/BF02392694. |
[25] |
M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Mathematics, 1-78.
doi: 10.2307/3597183. |
[26] |
J. Mallet-Paret,
Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315.
doi: 10.1016/0022-0396(88)90157-X. |
[27] |
J. Milnor, On the concept of attractor, in The Theory of Chaotic Attractors, Springer, 1985,243-264.
doi: 10.1007/BF01212280. |
[28] |
D. Norton,
The Conley decomposition theorem for maps: a metric approach, Rikkyo Daigaku Sugaku Zasshi, 44 (1995), 151-173.
|
[29] |
D. Norton,
The fundamental theorem of dynamical systems, Commentationes Mathematicae Universitatis Carolinae, 36 (1995), 585-597.
|
[30] |
M. Patrão,
Morse decomposition of semiflows on topological spaces, Journal of Dynamics and Differential Equations, 19 (2007), 181-198.
doi: 10.1007/s10884-006-9033-2. |
[31] |
K. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, 1987.
doi: 10.1007/978-3-642-72833-4. |
[32] |
A. Sharkovsky, S. Kolyada, A. Sivak and V. Fedorenko, Dynamics of One-Dimensional Maps, 407, Springer Science & Business Media, 1997.
doi: 10.1007/978-94-015-8897-3. |
[33] |
D. Singer,
Stable orbits and bifurcation of maps of the interval, SIAM Journal on Applied Mathematics, 35 (1978), 260-267.
doi: 10.1137/0135020. |
[34] |
S. Smale, On gradient dynamical systems, Annals of Mathematics, 199-206.
doi: 10.2307/1970311. |
[35] |
S. Smale,
Differentiable dynamical systems, Bulletin of the American mathematical Society, 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
[36] |
S. Smale and R. Williams,
The qualitative analysis of a difference equation of population growth, Journal of Mathematical Biology, 3 (1976), 1-4.
doi: 10.1007/BF00307853. |
[37] |
S. van Strien, On the bifurcations creating horseshoes, in Dynamical Systems and Turbulence, Warwick 1980, Springer, 1981,316-351. |
[38] |
F. Wilson and J. Yorke,
Lyapunov functions and isolating blocks, Journal of Differential Equations, 13 (1973), 106-123.
doi: 10.1016/0022-0396(73)90034-X. |









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