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Articles de revues sur le sujet « Doubling property »

Auteur : Grafiati

Publié le 9 mars 2023

Mis à jour le 31 juillet 2025

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1

Su, Bo. "Doubling property of elliptic equations." Communications on Pure & Applied Analysis 7, no. 1 (2008): 143–47. http://dx.doi.org/10.3934/cpaa.2008.7.143.

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Wang, Liqiu, and Mingtian Xu. "Property of period-doubling bifurcations." Chaos, Solitons & Fractals 24, no. 2 (2005): 527–32. http://dx.doi.org/10.1016/j.chaos.2004.09.045.

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3

Shkredov, I. D. "On sets with small doubling property." Mathematical Notes 84, no. 5-6 (2008): 859–78. http://dx.doi.org/10.1134/s000143460811028x.

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4

Aimar, Hugo, Marilina Carena, and Bibiana Iaffei. "Gradual doubling property of Hutchinson orbits." Czechoslovak Mathematical Journal 65, no. 1 (2015): 191–205. http://dx.doi.org/10.1007/s10587-015-0168-3.

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5

Le Donne, Enrico. "Doubling Property for BiLipschitz Homogeneous Geodesic Surfaces." Journal of Geometric Analysis 21, no. 4 (2010): 783–806. http://dx.doi.org/10.1007/s12220-010-9167-7.

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6

Jin, Renling. "Freiman's inverse problem with small doubling property." Advances in Mathematics 216, no. 2 (2007): 711–52. http://dx.doi.org/10.1016/j.aim.2007.06.002.

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7

Heer, Loreno. "Some Invariant Properties of Quasi-Möbius Maps." Analysis and Geometry in Metric Spaces 5, no. 1 (2017): 69–77. http://dx.doi.org/10.1515/agms-2017-0004.

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Abstract We investigate properties which remain invariant under the action of quasi-Möbius maps of quasimetric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)Möbius maps. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)Möbius maps as well.

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8

Zhu, Jiuyi. "Doubling Property and Vanishing Order of Steklov Eigenfunctions." Communications in Partial Differential Equations 40, no. 8 (2015): 1498–520. http://dx.doi.org/10.1080/03605302.2015.1025980.

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Baudoin, Fabrice, and Nicola Garofalo. "Perelman’s Entropy and Doubling Property on Riemannian Manifolds." Journal of Geometric Analysis 21, no. 4 (2010): 1119–31. http://dx.doi.org/10.1007/s12220-010-9180-x.

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10

Caffarelli, Luis A., and Michael G. Crandall. "Relations between geometric convexity, doubling measures and property $\Gamma $." Proceedings of the American Mathematical Society 142, no. 7 (2014): 2395–406. http://dx.doi.org/10.1090/s0002-9939-2014-11940-x.

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11

Zongo, Frédéric D. Y., Salifou Korbeogo, and Arouna Ouédraogo. "On Some Fractional Magnetic Problems." Pan-American Journal of Mathematics 2 (December 19, 2023): 17. http://dx.doi.org/10.28919/cpr-pajm/2-17.

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In this article, we study existence of weak solutions and the unique continuation property of the nonlocal fractional magnetic equation. First, we use a variational technique to prove existence of weak solutions for the fractional Schrödinger equation with magnetic field. Moreover, we show the doubling property main argument to unique continuation property via Carleman estimates.

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12

Xu, Mingtian. "Property of period-doubling bifurcation cascades of discrete dynamical systems." Chaos, Solitons & Fractals 33, no. 2 (2007): 455–62. http://dx.doi.org/10.1016/j.chaos.2006.01.022.

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Adamowicz, Tomasz, Antoni Kijowski, and Elefterios Soultanis. "Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces." Analysis and Geometry in Metric Spaces 10, no. 1 (2022): 344–72. http://dx.doi.org/10.1515/agms-2022-0143.

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Abstract We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

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14

Evans, Benny. "The Long Annulus Theorem." Canadian Mathematical Bulletin 29, no. 3 (1986): 257–67. http://dx.doi.org/10.4153/cmb-1986-040-4.

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AbstractGiven a properly embedded incompressible surface F in a Haken manifold M, there is an integer n depending only on M and F with the following property: If there is a singular annulus in M that meets F in more then n nontrivial loops that are not freely homotopic on F then M contains an essential torus or annulus, or M is a bundle with fiber F, or M is a doubled twisted I-bundle with doubling surface F.

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15

Suzuki, S. "Bing doubling and the colored Jones polynomial." International Journal of Mathematics 25, no. 08 (2014): 1450074. http://dx.doi.org/10.1142/s0129167x14500748.

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Bing doubling is a satellite operation on links which replaces a knot component with a 2-component link in a certain way. In this paper we give a formula for the reduced colored Jones polynomial of a Bing double in terms of that of the companion. Using this formula we derive a divisibility property of the unified Witten–Reshetikhin–Turaev invariant of integral homology spheres obtained by ±1-surgery along Bing doubles of knots. This result is applied to the Witten–Reshetikhin–Turaev invariant and the Ohtsuki series of these integral homology spheres.

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16

Elsner, James B., Emily Ryan, and Georgianna Strode. "Structural Property Losses from Tornadoes in Florida." Weather, Climate, and Society 10, no. 2 (2018): 253–58. http://dx.doi.org/10.1175/wcas-d-17-0055.1.

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Abstract Property losses from tornadoes in Florida are estimated by combining a 1-km spatial grid of structural values from the Department of Revenue’s 2014 cadastral database with historical tornado events since 1950. There are 91 180 grid cells in the state with at least some structural value. Total and residential structural values total $942 billion and $619 billion, respectively. Over the period 1950 through 2015 there were 3233 individual tornado reports in the state with a peak frequency during June. The property value exposed to tornadoes is estimated using a geometric model for the pa

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17

Wang Zheng-Ping, Teng Bing, Du Chen-Lin, et al. "Frequency doubling property of the low symmetric nonlinear optical crystal BIBO." Acta Physica Sinica 52, no. 9 (2003): 2176. http://dx.doi.org/10.7498/aps.52.2176.

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18

Mauldin, R. Daniel, and Mariusz Urbański. "The doubling property of conformal measures of infinite iterated function systems." Journal of Number Theory 102, no. 1 (2003): 23–40. http://dx.doi.org/10.1016/s0022-314x(03)00065-9.

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19

Rubin, Edward J. "Obligatory dative clitic-doubling of type III experiencers in Bulgnais." Proceedings of the Linguistic Society of America 3, no. 1 (2018): 18. http://dx.doi.org/10.3765/plsa.v3i1.4299.

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The correlation between the position of the Dative experiencer of a type III psych-verb relative to the verb itself and the obligatory vs. optional nature of an associated Dative clitic has seldom been noted in the literature, and it has never previously been explained. This paper presents relevant new data from Bulgnais (Bologna, Italy), and it proposes that these verbs, in the languages that require the Dative clitic with the preverbal Dative experiencer, have an additional strong lexical property beyond inherent Case licensing. Like Case licensing, this property requires feature checking, w

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20

LIU, HAIRONG, FANG LIU, and HUI WU. "THE UNIQUE CONTINUATION PROPERTY OF -HARMONIC FUNCTIONS ON THE HEISENBERG GROUP." Bulletin of the Australian Mathematical Society 99, no. 2 (2018): 219–30. http://dx.doi.org/10.1017/s0004972718001016.

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We introduce an Almgren frequency function of the sub-$p$-Laplace equation on the Heisenberg group to establish a doubling estimate under the assumption that the frequency function is locally bounded. From this, we obtain some partial results on unique continuation for the sub-$p$-Laplace equation.

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21

Mohd Roslan, Ummu Atiqah. "Sensitive Dependence on Initial Conditions for an Example of Markov Map: Skewed Doubling Map." MATEMATIKA 34, no. 1 (2018): 13–21. http://dx.doi.org/10.11113/matematika.v34.n1.1002.

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Markov map is one example of interval maps where it is a piecewise ex-panding map and obeys the Markov property. One well-known example of Markov map is the doubling map, a map which has two subintervals with equal partitions. In this paper, we are interested to investigate another type of Markov map, the so-called skewed doubling map. This map is a more generalized map than the doubling map. Thus, the aims of this paper are to nd the xed points as well as the periodic points for the skewed doubling map and to investigate the sensitive dependence on initial conditions of this map. The method c

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22

Ranjbar-Motlagh, Alireza. "Poincaré inequality for abstract spaces." Bulletin of the Australian Mathematical Society 71, no. 2 (2005): 193–204. http://dx.doi.org/10.1017/s000497270003817x.

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The Poincaré inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincaré inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

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23

Llorente, Jos� G. "On the Gehring-Hayman property, the Privalov-Riesz theorems, and doubling measures." Michigan Mathematical Journal 52, no. 3 (2004): 553–71. http://dx.doi.org/10.1307/mmj/1100623413.

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24

Li, Hui, Chun Wei, and Sheng Wen. "Doubling property of self-affine measures on carpets of Bedford and McMullen." Indiana University Mathematics Journal 65, no. 3 (2016): 833–65. http://dx.doi.org/10.1512/iumj.2016.65.5826.

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25

Stanchescu, Yonutz. "On the structure of sets with small doubling property on the plane (I)." Acta Arithmetica 83, no. 2 (1998): 127–41. http://dx.doi.org/10.4064/aa-83-2-127-141.

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26

Zheng, Jianyu, Wenhui Sun, Wenting Wang, et al. "Frequency-Doubling OEO Using the Polarization Property of LiNbO₃ Modulator." IEEE Photonics Technology Letters 27, no. 17 (2015): 1864–67. http://dx.doi.org/10.1109/lpt.2015.2443974.

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27

Wang, Li, and N. D. Kazarinoff. "A metric property of period doubling for nonisosceles trapezoidal maps on an interval." Advances in Applied Mathematics 8, no. 2 (1987): 208–21. http://dx.doi.org/10.1016/0196-8858(87)90014-5.

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28

Marshall, James. "Fractional Integrals of Imaginary Order Supported on Convex Curves, and the Doubling Property." Rocky Mountain Journal of Mathematics 28, no. 1 (1998): 287–302. http://dx.doi.org/10.1216/rmjm/1181071834.

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29

Baudoin, Fabrice, Michel Bonnefont, and Nicola Garofalo. "A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality." Mathematische Annalen 358, no. 3-4 (2013): 833–60. http://dx.doi.org/10.1007/s00208-013-0961-y.

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30

Cohen, Nathann. "A Doubling Construction for 3-Uniform Friendship Hypergraphs with the Universal Pairs Property." Journal of Combinatorial Designs 24, no. 7 (2015): 336–39. http://dx.doi.org/10.1002/jcd.21509.

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31

Hall, WB, GM Mckeon, JO Carter, et al. "Climate Change in Queensland's Grazing Lands: Ii. An Assessment of the Impact on Animal Production From Native Pastures." Rangeland Journal 20, no. 2 (1998): 177. http://dx.doi.org/10.1071/rj9980177.

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The 160 million ha of grazing land in Queensland support approximately 10 million beef equivalents (9.8 million cattle and 10.7 million sheep) with treed and cleared native pastures as the major forage source. The complexity of these biophysical systems and their interaction with pasture and stock management, economic and social forces limits our ability to easily calculate the impact of climate change scenarios. We report the application of a systems approach in simulating the flow of plant dry matter and utilisation of forage by animals. Our review of available models highlighted the lack of

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32

Yao, Jiangyan, and Wei Han. "The Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity." Discrete Dynamics in Nature and Society 2019 (July 7, 2019): 1–10. http://dx.doi.org/10.1155/2019/3713909.

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In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a (2p-1)-linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that Mb≠∅. The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter b↘0.

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33

Li, Dongyan. "Singularity and Decay Estimates for a Degenerate Parabolic Equation." Complexity 2021 (March 10, 2021): 1–6. http://dx.doi.org/10.1155/2021/5517008.

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In this paper, a degenerate parabolic equation u t − div x θ ∇ u = x a u p with p > 1 and θ < 2 , a ∈ ℝ , is considered. Based on rescaling arguments combined with a doubling property, the space-time singularity and decay estimates are established. Moreover, a universal and a priori bound of global nonnegative solutions for the corresponding initial boundary value problem is derived.

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34

Tao, Xiangxing, and Qinqin Chen. "The Boundedness of Maximal Functions in Orlicz–Campanato Spaces of Homogeneous Type." gmj 15, no. 2 (2008): 377–88. http://dx.doi.org/10.1515/gmj.2008.377.

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Abstract Let (𝑋,𝑑,μ) be a normal space of homogeneous type, 𝑋+ be the upper half-space equipped with a Carleson measure β, and let Φ be an N-function and φ a suitable function satisfying the doubling property. We prove that the generalized Hardy–Littlewood maximal operator 𝑀 is bounded from the Orlicz–Campanato space 𝐿Φ,φ (𝑋,μ) to 𝐿Φ,φ (𝑋+,β).

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Coulhon, Thierry, Baptiste Devyver, and Adam Sikora. "Gaussian heat kernel estimates: From functions to forms." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 761 (2020): 25–79. http://dx.doi.org/10.1515/crelle-2018-0021.

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AbstractOn a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

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Mirotin, A. R. "Hausdorff Operators on Real Hardy Spaces H1 Over Homogeneous Spaces with Local Doubling Property." Analysis Mathematica 47, no. 2 (2021): 385–403. http://dx.doi.org/10.1007/s10476-021-0087-5.

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Lin, Haibo, Eiichi Nakai та Dachun Yang. "Boundedness of Lusin-area andgλ*functions on localized Morrey-Campanato spaces over doubling metric measure spaces". Journal of Function Spaces and Applications 9, № 3 (2011): 245–82. http://dx.doi.org/10.1155/2011/187597.

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Letχbe a doubling metric measure space andρan admissible function onχ. In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spacesερα,p(χ)and Morrey-Campanato-BLO spacesε̃ρα,p(χ)whenα∈(-∞,0)andp∈[1,∞). Ifχhas the volume regularity Property(P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, fromερa,p(χ)toε̃ρa,p(χ)without invoking any regularity of considered kernels. The same is true for thegλ*function and, unlike the Lusin-are

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38

Han, Yongsheng, Detlef Müller, and Dachun Yang. "A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces." Abstract and Applied Analysis 2008 (2008): 1–250. http://dx.doi.org/10.1155/2008/893409.

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We work on RD-spaces𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterizati

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39

Eriksson-Bique, Sylvester, and Pietro Poggi-Corradini. "Density of continuous functions in Sobolev spaces with applications to capacity." Transactions of the American Mathematical Society, Series B 11, no. 27 (2024): 901–44. http://dx.doi.org/10.1090/btran/188.

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We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if ( X , d , μ ) (X,d,\mu ) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N 1 , p ( X ) N^{1,p}(X) . Here the measure μ \mu is Borel and is finite and positive on all metric balls. In particular, we don’t assume properness of X X , doubling of μ \mu or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn

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Zhu, Xi Feng, and Quan Fu Gao. "Periodic-Doubling and Hopf Bifurcations of a Two-Degree-of-Freedom System with Elastic Constraints and Clearances." Advanced Materials Research 1006-1007 (August 2014): 285–89. http://dx.doi.org/10.4028/www.scientific.net/amr.1006-1007.285.

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Based on the study of a dual component system with elastic constraints, the stability and local bifurcations of the soft-impacts system, such as piecewise property and singularity, was analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and period-doubling bifurcation, were investigated exactly. In the large constraint stiffness case, the period-doubling and Hopf bifurcation exist in the two-degree-of-freedom system with elastic constraints and clearances. The clearances of the system, stiffness and dam

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Al-Wardat, Rami. "Object marking in Jordanian Arabic." Even Yearbook 16 (2024): 14–31. https://doi.org/10.57133/evenyrbk.24wa.

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This paper aims to investigate object marking in Jordanian Arabic (henceforth, JA). Object marking is more distant from the stem than subject marking. The latter is arguably inflectional: it exhibits a sensitivity to tense characteristic of inflectional morphology. Object marking, by contrast, is tense invariant, a property typical of clitics. Together, their peripherality to subject marking and their tense-invariance make a case for object marking in JA involving cliticisation. The argument is based on predictions from binding facts and theories of agreement versus clitic doubling, with evide

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Rowland, Eric S. "Local Nested Structure in Rule 30." Complex Systems 16, no. 3 (2006): 239–58. http://dx.doi.org/10.25088/complexsystems.16.3.239.

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At row 2n in the cellular automaton rule 30, a region of the initial condition reappears on the right side, which causes the automaton to "begin again" locally. As a result, local nested structure is produced. This phenomenon is ultimately due to the property that rule 30 is reversible in time under the condition that the right half of each row is white. The main result of the paper establishes the presence of local nested structure in k-color rules with this bijectivity property, and we explore a class of integer sequences characterizing the nested structure. We also prove an observation of W

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Chang, Der-Chen, Shu-Cheng Chang, Yingbo Han, and Jingzhi Tie. "A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth." Canadian Journal of Mathematics 71, no. 6 (2019): 1367–94. http://dx.doi.org/10.4153/cjm-2018-024-3.

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AbstractIn this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

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Zhang, Songyan. "The existence and boundedness of square functions on generalized Orlicz–Campanato spaces." gmj 17, no. 2 (2010): 405–21. http://dx.doi.org/10.1515/gmj.2010.019.

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Abstract Let T(ƒ) denote the Littlewood–Paley square operators, including the g-function g(ƒ), Luzin area function S(ƒ) and Stein's function , on the generalized Orlicz–Campanato spaces , where Φ is a N-function satisfying the Δ2 condition and φ a positive function satisfying the doubling property. It is proved that if T(ƒ)(x 0) < ∞ for a single point , then T(ƒ)(x) exists almost everywhere in , and T(ƒ) is bounded on the spaces .

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Björn, Anders, and Daniel Hansevi. "Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces." Analysis and Geometry in Metric Spaces 7, no. 1 (2019): 179–96. http://dx.doi.org/10.1515/agms-2019-0009.

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Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

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Hua, Bobo, Jürgen Jost, and Shiping Liu. "Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature." Journal für die reine und angewandte Mathematik (Crelles Journal) 2015, no. 700 (2015): 1–36. http://dx.doi.org/10.1515/crelle-2013-0015.

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AbstractWe apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature. We obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point. Moreover, we show the volume doubling property and the Poincaré inequality on such graphs. The quadratic volume growth of these graphs implies the parabolicity. Finally, we prove the polynomial growth harmonic function theorem analogous to the case of Riemannian manifolds.

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Effah-Poku, S., W. Obeng-Denteh, and I. K. Dontwi. "A Study of Chaos in Dynamical Systems." Journal of Mathematics 2018 (2018): 1–5. http://dx.doi.org/10.1155/2018/1808953.

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The behavior of systems such as periodicity, fixed points, and most importantly chaos has evolved as an integral part of mathematics, especially in dynamical system. This research presents a study on chaos as a property of nonlinear science. Systems with at least two of the following properties are considered to be chaotic in a certain sense: bifurcation and period doubling, period three, transitivity and dense orbit, sensitive dependence to initial conditions, and expansivity. These are termed as the routes to chaos.

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48

An, Cun-bin, Jiangyan Yao, and Wei Han. "The Existence of the Sign-Changing Solutions for the Kirchhoff-Schrödinger-Poisson System in Bounded Domains." Advances in Mathematical Physics 2020 (May 19, 2020): 1–10. http://dx.doi.org/10.1155/2020/8254898.

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In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution u0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.

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Grigor'yan, Alexander, and Jiaxin Hu. "Heat Kernels and Green Functions on Metric Measure Spaces." Canadian Journal of Mathematics 66, no. 3 (2014): 641–99. http://dx.doi.org/10.4153/cjm-2012-061-5.

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AbstractWe prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

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Chen, Danfeng, and Cong Wang. "Prediction of Period-Doubling Bifurcation Based on Dynamic Recognition and Its Application to Power Systems." International Journal of Bifurcation and Chaos 26, no. 09 (2016): 1650157. http://dx.doi.org/10.1142/s0218127416501571.

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In this paper, a bifurcation prediction approach is proposed based on dynamic recognition and further applied to predict the period-doubling bifurcation (PDB) of power systems. Firstly, modeling of the internal dynamics of nonlinear systems is obtained through deterministic learning (DL), and the modeling results are applied for constructing the dynamic training pattern database. Specifically, training patterns are chosen according to the hierarchical structured knowledge representation based on the qualitative property of dynamical systems, which is capable of arranging the dynamical models i

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