T. Y. Li and J. A. Yorke. Period three implies chaos. Amer. Math. Monthly, 82(10):985--992, 1975. Home/Search Document Not in Database Summary Related Articles Check |
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Shortcuts to Kaos: Complex Behaviour From Simple Dynamics - Ushaw, Bell (2000) (Correct)
....initial conditions to the actual system. Hence the model and the system will diverge over time. Expansion and contraction of phase space. A good populist account of the history and theory of the study of chaotic systems can be found in [2] Further seminal works on the subject are found in [3] [4], 5] and [6] The related field of fractals was most famously presented in [7] Chaos Theory In Game AI As game environments become more complex and more extensive, the behaviour which the player expects to see within that environment will also become more complex. An area of research which may ....
T.Y.Li and J.Yorke, "Period three implies chaos", American Mathematical Monthly, vol 82, pp 985-92, 1975
On a Linear Chaotic Quantum Harmonic Oscillator - Duan, Fu, Pei-De Liu, Manning (1997) (Correct)
....set for the operator B, and therefore show that the above linear quantum harmonic oscillator is chaotic in the sense of Li Yorke. 2 Chaotic Set of Operator B We now show that the above weighted shift operator B is chaotic in the sense of Li Yorke. We first recall the chaos definition of Li Yorke [7]. Definition Let M be a metric space with metric ae and f : M M be a continuous map. The discrete dynamical system (M; f) is called chaotic in the sense of Li Yorke if there exists an uncountable subset S of nonwandering non periodic points such that whenever x; y 2 S; x 6= y , the following ....
....such that whenever x; y 2 S; x 6= y , the following conditions hold, i) lim sup n 1 ae(f n (x) f n (y) 0 (ii) lim inf n 1 ae(f n (x) f n (y) 0 ; The subset S above is called a chaotic set for f . 2 Remark The original characterization of chaos in Li Yorke s theorem [7] is via three conditions. The third one is : iii) lim sup n 1 ae(f n (x) f n (p) 0; 8x 2 S; 8p 2 P (f) This condition means that no point in S is asymptotically periodic. From conditions (i) and (ii) in the Definition, S contains at most one asymptotically periodic point [11] So ....
T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82(1975), 985-992.
Snap-Back Repellers And Scrambled Sets In General.. - Boyarsky, Gora, Lioubimov (Correct)
....2 # S with s 1 #= s 2 : lim sup k## #T k (s 1 ) T k (s 2 )# 0, Typeset by A M S T E X 1 (iii) an uncountable subset S 0 # S such that for every s 1 , s 2 # S 0 : lim inf k## #T k (s 1 ) T k (s 2 )# = 0. Conditions (i) iii) were established earlier by Li and Yorke [2] for one dimensional transformations having a point of period 3. In this paper we introduce the concepts of a snap back repeller and various scrambled sets for a continuous transformation from a general Hausdor# topological space to itself and show that chaotic behavior of a transformation with a ....
T.-Y. Li and J. A. Yorke, Period Three Implies Chaos, Amer. Math. Monthly 82 (1975), 985---992.
Infinite-Dimensional Linear Dynamical Systems with Chaoticity - Fu (1998) (Correct)
....on infinite dimensional linear dynamical systems with chaoticity. In Section 2 we consider the backward shift map in the space of infinite sequences of elements from a general Fr echet space, and show that this backward shift map itself is chaotic in both senses of Wiggins, and of Li Yorke [19] in terms of orbit separation of nonwandering non periodic points. In Section 3 we show that the translation map in the space of real continuous functions is chaotic in both senses of Wiggins, and of Li Yorke. Note that we have only proved the backward shift itself is chaotic in the space of ....
....(linear) differential 2 equations, than the space of complex entire functions as used by Godefroy and Shapiro [13] Again, our work is in both senses of Wiggins and of Li Yorke. There are a few definitions of chaos . Although the definition in Wiggins [25] seems a popular one, other definitions [19], 18] which capture or describe various different dynamical behavior of a system, are proposed and used in nonlinear science. Li Yorke s definition of chaos does characterize the complexity of dynamical systems. For example, for onedimensional dynamical systems (the iteration of continuous maps ....
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T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82(1975), 985-992 13
Weakly Chaotic Functions With Zero Topological Entropy And.. - Jim Enez Opez (Correct)
....when ffi 0, a ffi scrambled set. Received September 25, 1991. 1980 Mathematics Subject Classification (1985 Revision) Primary 58F13, 58F08; Secondary 28A18, 54H20. 196 V. JIM ENEZ L OPEZ This definition is equivalent (see Kuchta and Sm ital [9] to the original Li and Yorke s version [10]. In particular any weakly chaotic function possesses a Cantor type ffi scrambled set (see Jankov a and Sm ital [7] Sm ital [17] The relations between weak and strong chaos were analyzed in a number of papers. So it is well known (see for example Graw [6] or Jankov a and Sm ital [7] that any ....
Li T. and Yorke J., Period three implies chaos, Amer. Math. Monthly 82 (1975), 985--992.
Fractal and Chaotic Dynamics in Nervous Systems - King (1991) (3 citations) (Correct)
....Period Windows : Intermittency and Crises There are a series of windows in the chaotic region where chaotic behavior is abruptly interrupted by new periodic regimes of periods 3, 5, etc. bvi ) These windows contain for example 3. 2 n bifurcation sequences similar to that of (2) As a result of Li Yorke (1975), the existence of a period 3 attractor guarantees the existence of periods of all orders and uncountably many aperiodic orbits (chaos) By Sarkovski, the periods follow the causal sequence : 3 5 7 . 2 n .3 2 n .5 . 2 4 2 3 2 2 2 1 n = 1, 2, 3, At the left hand end of ....
....Various electronic orbital structures such as delocalized electrons, spin orbit coupling, triplet state electrons and dipole structures which could enable Bose condensation of coherent photons have been proposed in biomolecular systems which profoundly complicate this fractal structure. Frlich (1968,1975) has suggested that the cyto skeleton and membrane may admit coherent states. The complexity of three dimensional structure of a protein, resulting from the hierarchy of strong and weak bonding water structures, and the many conformational changes possible make for a high order of variation in ....
Li T.Y., Yorke J.A., (1975), Period three implies chaos, Am. Math. Mon. 82, 985.
One-Dimensional Topological Dynamics - Llibre (Correct)
....maps in one dimension. Here we deal with this last topic, see Llibre [Ll] for the topological dynamics in dimension two or larger. Various notions of chaotic behavior started to form and interval maps proved to be sufficiently complex to display most kind of chaos. The paper of Li and Yorke [LY] Period three implies chaos helped very much to make those ideas popular. The simplicity of tools necessary to study these systems attracted mathematicians (and physicists) from many areas. A finite graph (simply a graph) G is a Hausdorff space which has a finite subspace V (points of V are ....
....1 ; 13) f2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 21; 22; 23; 28; 29g if G = 1 2 ; 14) f2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 21; 22; 23; 28; 29g if G = T . Statement (1) of the above theorem follows from Sharkovskii s Theorem, see also Li and Yorke [LY]. Statement (2) was proved by Mumbr u, it also can be obtained from [ALM1] and [B1] Statements (3) 4) and (5) are due to Alsed a and Moreno [AM] Statement (6) is due to Block [Bl] The remainder statements are proved by Leseduarte and Llibre [LL2] If we consider the trefoil and one of its ....
T. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985--992.
Word of Mouth Learning in the Battle of the Sexes - Dawid (Correct)
....towards the mixed strategy equilibrium (x; y) 0:75; 0:25) The flip bifurcation at fl can be seen quite clearly and we observe that further period doubling bifurcations occur for fl fl . For fl = 3:65 a cycle of period three can be detected which implies by the Li Yorke theorem (Li and Yorke, 1975) that our learning process shows topological chaos for these paramter values (see Li and Yorke, 1975 or Lorenz, 1993 for a definition of topological chaos) Generally speaking we realize that oscillating and cyclical behavior arises in our learning model if both populations start from completely ....
....be seen quite clearly and we observe that further period doubling bifurcations occur for fl fl . For fl = 3:65 a cycle of period three can be detected which implies by the Li Yorke theorem (Li and Yorke, 1975) that our learning process shows topological chaos for these paramter values (see Li and Yorke, 1975 or Lorenz, 1993 for a definition of topological chaos) Generally speaking we realize that oscillating and cyclical behavior arises in our learning model if both populations start from completely symmetric initial distrubtions and if the players adopt higher paying strategies with a high ....
Li, T. and Yorke J. (1975), "Period Three implies Chaos," American Mathematical Monthly, 82, 985-992.
Runge-Kutta Solutions of a Hyperbolic Conservation Law.. - Aves, Griffiths, Higham (1998) (Correct)
....to study the fixed points of the system (6) 2.1 Initial BVP For the ibvp problem, 6) has a fixed point when U j Gamma1 = U j Gamma Deltax a g(U j ) j = 1; 2; N; 10) with U 0 = u 0 . Maps of this type have received considerable attention in the literature; see, for example, [16]. Generically, there are parameter ranges for which periodic cycles of any period may be generated. The following results give insight into the behaviour of the map (10) around a zero of g. Theorem 1 Suppose g 2 C 1 with g(fi) 0 and g 0 (fi) 0 for some fi 2 R. Let I R be an open, ....
T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), pp. 985--992.
On Li-Yorke Pairs - Blanchard, Glasner, Kolyada, Maass (2001) Self-citation (Li Yorke) (Correct)
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Li, T. Y. and Yorke, J. A., Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992.
On Li-Yorke Pairs - Blanchard, Glasner, Kolyada, Maass Self-citation (Li Yorke) (Correct)
....when transitive, and the property is stable under factor maps, arbitrary products and inverse limits. Finally it is proven that minimal systems without Li Yorke pairs are disjoint from scattering systems. 0. Introduction The term chaos in connection with a map was introduced by Li and Yorke [LY], although without a formal definition. Today there are various definitions of what it means for a map to be chaotic, some of them working reasonably only in particular phase spaces; most of the existing ones were reviewed in [KS] Although one could say that as many authors, as many definitions ....
....of Topological Dynamics; formerly it had been studied mainly in the setting of interval maps. Let (X, T ) be a topological dynamical system, with X a compact metric space with metric #, and T a surjective continuous map from X to itself. The definition of Li Yorke chaos is based on ideas in [LY]. A pair of points x, y # X is said to be a Li Yorke pair (with modulus #) if one has simultaneously lim sup n## #(T n x, T n y) # 0 and lim inf n## #(T n x, T n y) 0. A set S # X is called scrambled if any pair of distinct points x, y # S is a Li Yorke pair. ....
Li, T. Y. and Yorke, J. A., Period three implies chaos, Amer. Math. Monthly 82 (1975), 985--992.
Numerical integration of ODEs with Preisach - Nonlinearity Pokrovskii And (Correct)
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T. Y. Li and J. A. Yorke. Period three implies chaos. Amer. Math. Monthly, 82(10):985--992, 1975.
Asymptotic Pairs in Positive-Entropy Systems - Blanchard, Host, Ruette (2001) (Correct)
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Li, T.Y., Yorke, J.A.: Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985-992
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T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985--992.
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T.-Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), pp. 985--992.
Rigorous analysis of complicated behaviour in a.. - Rasskazov, Huyet, .. (2001) (Correct)
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T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), pp. 985-992.
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T. Li and J. Yorke, "Period three implies chaos", American Mathematical Monthly, Vol. 82(10), pp. 985-992, December 1975.
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Li T-Y, Yorke JA, 1975, Period three implies chaos, Am Math Monthly 82, 985-992.
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Li, Tien-Yien & Yorke, J.A., 1975, Period Three Implies Chaos, American Mathematical Monthly, 2.
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Li, Tien-Yien & Yorke, J.A., 1975, Period Three Implies Chaos, American Mathematical Monthly, 2.
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T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985--992.
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Li, Tien-Yien & Yorke, J.A., 1975, Period Three Implies Chaos, American Mathematical Monthly, 2.
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T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82, (1975) pp. 985-992.
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