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Volumn 4, Issue 4, 2016, Pages

Visual analysis of nonlinear dynamical systems: Chaos, fractals, self-similarity and the limits of prediction

Author keywords

Attractor; Bifurcation; Chaos; Dynamical systems; Fractal; Logistic map; Nonlinear dynamics; Prediction; Python; Visualization

Indexed keywords


EID: 85111351559     PISSN: None     EISSN: 20798954     Source Type: Journal    
DOI: 10.3390/systems4040037     Document Type: Article
Times cited : (268)

References (71)
  • 2
    • 35848954313 scopus 로고    scopus 로고
    • A Simple Guide to Chaos and Complexity
    • [CrossRef] [PubMed]
    • Rickles, D.; Hawe, P.; Shiell, A. A Simple Guide to Chaos and Complexity. J. Epidemiol. Commun. Health 2007, 61, 933–937. [CrossRef] [PubMed]
    • (2007) J. Epidemiol. Commun. Health , vol.61 , pp. 933-937
    • Rickles, D.1    Hawe, P.2    Shiell, A.3
  • 3
    • 84866950574 scopus 로고    scopus 로고
    • Manifold Learning Approach for Chaos in the Dripping Faucet
    • [CrossRef] [PubMed]
    • Suetani, H.; Soejima, K.; Matsuoka, R.; Parlitz, U.; Hata, H. Manifold Learning Approach for Chaos in the Dripping Faucet. Phys. Rev. E 2012, 86, 036209. [CrossRef] [PubMed]
    • (2012) Phys. Rev. E , vol.86 , pp. 036209
    • Suetani, H.1    Soejima, K.2    Matsuoka, R.3    Parlitz, U.4    Hata, H.5
  • 5
    • 84883809867 scopus 로고    scopus 로고
    • Poincaré Plot Indexes of Heart Rate Variability: Relationships with Other Nonlinear Variables
    • [CrossRef] [PubMed]
    • Hoshi, R.A.; Pastre, C.M.; Vanderlei, L.C.M.; Godoy, M.F. Poincaré Plot Indexes of Heart Rate Variability: Relationships with Other Nonlinear Variables. Auton. Neurosci. 2013, 177, 271–274. [CrossRef] [PubMed]
    • (2013) Auton. Neurosci , vol.177 , pp. 271-274
    • Hoshi, R.A.1    Pastre, C.M.2    Vanderlei, L.C.M.3    Godoy, M.F.4
  • 6
    • 85095303942 scopus 로고    scopus 로고
    • Initiation of Ventricular Fibrillation by a Single Ectopic Beat in Three Dimensional Numerical Models of Ischemic Heart Disease: Abrupt Transition to Chaos
    • [CrossRef]
    • Babbs, C.F. Initiation of Ventricular Fibrillation by a Single Ectopic Beat in Three Dimensional Numerical Models of Ischemic Heart Disease: Abrupt Transition to Chaos. J. Clin. Exp. Cardiol. 2014, 5, 2–11. [CrossRef]
    • (2014) J. Clin. Exp. Cardiol , vol.5 , pp. 2-11
    • Babbs, C.F.1
  • 7
    • 67651100776 scopus 로고    scopus 로고
    • Introduction to Controversial Topics in Nonlinear Science: Is the Normal Heart Rate Chaotic?
    • [CrossRef] [PubMed]
    • Glass, L. Introduction to Controversial Topics in Nonlinear Science: Is the Normal Heart Rate Chaotic? Chaos 2009, 19, 028501. [CrossRef] [PubMed]
    • (2009) Chaos , vol.19 , pp. 028501
    • Glass, L.1
  • 11
    • 85059167728 scopus 로고    scopus 로고
    • From Individuals to Groups: The Aggregation of Votes and Chaotic Dynamics
    • Kiel, L.D., Elliott, E., Eds.; University of Michigan Press: Ann Arbor, MI, USA
    • Richards, D. From Individuals to Groups: The Aggregation of Votes and Chaotic Dynamics. In Chaos Theory in the Social Sciences; Kiel, L.D., Elliott, E., Eds.; University of Michigan Press: Ann Arbor, MI, USA, 1996; pp. 89–116.
    • (1996) Chaos Theory in the Social Sciences , pp. 89-116
    • Richards, D.1
  • 13
    • 0000356817 scopus 로고    scopus 로고
    • Self-Organized Criticality and Urban Development
    • [CrossRef]
    • Batty, M.; Xie, Y. Self-Organized Criticality and Urban Development. Discret. Dyn. Nat. Soc. 1999, 3, 109–124. [CrossRef]
    • (1999) Discret. Dyn. Nat. Soc , vol.3 , pp. 109-124
    • Batty, M.1    Xie, Y.2
  • 15
    • 0036335994 scopus 로고    scopus 로고
    • Fractal Dimension and Fractal Growth of Urbanized Areas
    • [CrossRef]
    • Shen, G. Fractal Dimension and Fractal Growth of Urbanized Areas. Int. J. Geogr. Inf. Sci. 2002, 16, 419–437. [CrossRef]
    • (2002) Int. J. Geogr. Inf. Sci , vol.16 , pp. 419-437
    • Shen, G.1
  • 16
    • 34548534433 scopus 로고    scopus 로고
    • Scaling Laws and Indications of Self-Organized Criticality in Urban Systems
    • [CrossRef]
    • Chen, Y.; Zhou, Y. Scaling Laws and Indications of Self-Organized Criticality in Urban Systems. Chaos Solitons Fractals 2008, 35, 85–98. [CrossRef]
    • (2008) Chaos Solitons Fractals , vol.35 , pp. 85-98
    • Chen, Y.1    Zhou, Y.2
  • 17
    • 38649117916 scopus 로고    scopus 로고
    • Nonlinear Dynamics and Chaos in a Fractional-Order Financial System
    • [CrossRef]
    • Chen, W.C. Nonlinear Dynamics and Chaos in a Fractional-Order Financial System. Chaos Solitons Fractals 2008, 36, 1305–314. [CrossRef]
    • (2008) Chaos Solitons Fractals , vol.36 , pp. 1305-1314
    • Chen, W.C.1
  • 18
    • 67349155616 scopus 로고    scopus 로고
    • Chaos in Economics and Finance
    • [CrossRef]
    • Guégan, D. Chaos in Economics and Finance. Annu. Rev. Control 2009, 33, 89–93. [CrossRef]
    • (2009) Annu. Rev. Control , vol.33 , pp. 89-93
    • Guégan, D.1
  • 20
    • 85059139734 scopus 로고    scopus 로고
    • Chaos Theory and Rationality in Economics
    • Kiel, L.D., Elliott, E., Eds.; University of Michigan Press: Ann Arbor, MI, USA
    • Rosser, J.B. Chaos Theory and Rationality in Economics. In Chaos Theory in the Social Sciences; Kiel, L.D., Elliott, E., Eds.; University of Michigan Press: Ann Arbor, MI, USA, 1996; pp. 199–213.
    • (1996) Chaos Theory in the Social Sciences , pp. 199-213
    • Rosser, J.B.1
  • 21
    • 33845696713 scopus 로고    scopus 로고
    • Economics on the Edge of Chaos: Some Pitfalls of Linearizing Complex Systems
    • [CrossRef]
    • Oxley, L.; George, D.A.R. Economics on the Edge of Chaos: Some Pitfalls of Linearizing Complex Systems. Environ. Model. Softw. 2007, 22, 580–589. [CrossRef]
    • (2007) Environ. Model. Softw , vol.22 , pp. 580-589
    • Oxley, L.1    George, D.A.R.2
  • 22
    • 70349980917 scopus 로고    scopus 로고
    • Can Chaos Theory Explain Complexity In Urban Fabric? Applications in Traditional Muslim Settlements
    • [CrossRef]
    • Hamouche, M.B. Can Chaos Theory Explain Complexity In Urban Fabric? Applications in Traditional Muslim Settlements. Nexus Netw. J. 2009, 11, 217–242. [CrossRef]
    • (2009) Nexus Netw. J , vol.11 , pp. 217-242
    • Hamouche, M.B.1
  • 23
    • 84881038165 scopus 로고    scopus 로고
    • The Fractal Analysis of Architecture: Calibrating the Box-Counting Method Using Scaling Coefficient and Grid Disposition Variables
    • [CrossRef]
    • Ostwald, M.J. The Fractal Analysis of Architecture: Calibrating the Box-Counting Method Using Scaling Coefficient and Grid Disposition Variables. Environ. Plan B 2013, 40, 644–663. [CrossRef]
    • (2013) Environ. Plan B , vol.40 , pp. 644-663
    • Ostwald, M.J.1
  • 24
    • 0025936716 scopus 로고
    • Planning and Chaos Theory
    • [CrossRef]
    • Cartwright, T.J. Planning and Chaos Theory. J. Am. Plan. Assoc. 1991, 57, 44–56. [CrossRef]
    • (1991) J. Am. Plan. Assoc , vol.57 , pp. 44-56
    • Cartwright, T.J.1
  • 26
    • 84925271334 scopus 로고    scopus 로고
    • The Origins of Complexity Theory in Cities and Planning
    • Portugali, J., Meyer, H., Stolk, E., Tan, E., Eds.; Springer: Berlin, Germany
    • Batty, M.; Marshall, S. The Origins of Complexity Theory in Cities and Planning. In Complexity Theories of Cities Have Come of Age; Portugali, J., Meyer, H., Stolk, E., Tan, E., Eds.; Springer: Berlin, Germany, 2012; pp. 21–45.
    • (2012) Complexity Theories of Cities Have Come of Age , pp. 21-45
    • Batty, M.1    Marshall, S.2
  • 28
    • 84996190375 scopus 로고    scopus 로고
    • Dynamical System Visualization and Analysis via Performance Maps
    • [CrossRef]
    • Alpigini, J.J. Dynamical System Visualization and Analysis via Performance Maps. Inf. Vis. 2004, 3, 271–287. [CrossRef]
    • (2004) Inf. Vis , vol.3 , pp. 271-287
    • Alpigini, J.J.1
  • 31
    • 0000241853 scopus 로고
    • Deterministic Nonperiodic Flow
    • [CrossRef]
    • Lorenz, E.N. Deterministic Nonperiodic Flow. J. Atmos. Sci. 1963, 20, 130–141. [CrossRef]
    • (1963) J. Atmos. Sci , vol.20 , pp. 130-141
    • Lorenz, E.N.1
  • 32
    • 0017185443 scopus 로고
    • Simple Mathematical Models with Very Complicated Dynamics
    • [CrossRef] [PubMed]
    • May, R.M. Simple Mathematical Models with Very Complicated Dynamics. Nature 1976, 261, 459–467. [CrossRef] [PubMed]
    • (1976) Nature , vol.261 , pp. 459-467
    • May, R.M.1
  • 34
    • 22944444400 scopus 로고    scopus 로고
    • Time Series Analysis
    • 2nd ed.; Hand, D., Berthold, M., Eds.; Springer: Berlin, Germany
    • Bradley, E. Time Series Analysis. In Intelligent Data Analysis: An Introduction, 2nd ed.; Hand, D., Berthold, M., Eds.; Springer: Berlin, Germany, 2003.
    • (2003) Intelligent Data Analysis: An Introduction
    • Bradley, E.1
  • 35
    • 84927732320 scopus 로고    scopus 로고
    • Nonlinear Time-Series Analysis Revisited
    • [CrossRef] [PubMed]
    • Bradley, E.; Kantz, H. Nonlinear Time-Series Analysis Revisited. Chaos 2015, 25, 097610. [CrossRef] [PubMed]
    • (2015) Chaos , vol.25 , pp. 097610
    • Bradley, E.1    Kantz, H.2
  • 36
    • 33646427717 scopus 로고    scopus 로고
    • Metaphors in Complexity Theory and Planning
    • [CrossRef]
    • Chettiparamb, A. Metaphors in Complexity Theory and Planning. Plan Theory 2006, 5, 71–91. [CrossRef]
    • (2006) Plan Theory , vol.5 , pp. 71-91
    • Chettiparamb, A.1
  • 38
    • 0034738985 scopus 로고    scopus 로고
    • The Lorenz Attractor Exists
    • [CrossRef] [PubMed]
    • Stewart, I. The Lorenz Attractor Exists. Nature 2000, 406, 948–949. [CrossRef] [PubMed]
    • (2000) Nature , vol.406 , pp. 948-949
    • Stewart, I.1
  • 39
    • 85020361339 scopus 로고    scopus 로고
    • (accessed on 1 September 2016)
    • Danforth, C.M. Chaos in an Atmosphere Hanging on a Wall. Available online: http://mpe2013.org/2013/03/17/chaos-in-an-atmosphere-hanging-on-a-wall/(accessed on 1 September 2016).
    • Chaos in an Atmosphere Hanging on a Wall
    • Danforth, C.M.1
  • 41
    • 0016394337 scopus 로고
    • Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos
    • [CrossRef] [PubMed]
    • May, R.M. Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos. Science 1974, 186, 645–647. [CrossRef] [PubMed]
    • (1974) Science , vol.186 , pp. 645-647
    • May, R.M.1
  • 42
    • 79959759804 scopus 로고    scopus 로고
    • Optimal Harvesting Policy for Stochastic Logistic Population Model
    • [CrossRef]
    • Li, W.; Wang, K.; Su, H. Optimal Harvesting Policy for Stochastic Logistic Population Model. Appl. Math. Comput. 2011, 218, 157–162. [CrossRef]
    • (2011) Appl. Math. Comput , vol.218 , pp. 157-162
    • Li, W.1    Wang, K.2    Su, H.3
  • 43
    • 63349088494 scopus 로고    scopus 로고
    • Chaotic Growth with the Logistic Model of P.F. Verhulst
    • Ausloos, M., Dirickx, M., Eds.; Springer: Berlin, Germany
    • Pastijn, H. Chaotic Growth with the Logistic Model of P.F. Verhulst. In The Logistic Map and the Route to Chaos; Ausloos, M., Dirickx, M., Eds.; Springer: Berlin, Germany, 2006; pp. 3–11.
    • (2006) The Logistic Map and the Route to Chaos , pp. 3-11
    • Pastijn, H.1
  • 45
    • 34250451946 scopus 로고
    • On the Nature of Turbulence
    • [CrossRef]
    • Ruelle, D.; Takens, F. On the Nature of Turbulence. Commun. Math. Phys. 1971, 20, 167–192. [CrossRef]
    • (1971) Commun. Math. Phys , vol.20 , pp. 167-192
    • Ruelle, D.1    Takens, F.2
  • 46
    • 67650854000 scopus 로고    scopus 로고
    • Bifurcations and Strange Attractors
    • Shilnikov, L. Bifurcations and Strange Attractors. Proc. Int. Congr. Math. 2002, 3, 349–372.
    • (2002) Proc. Int. Congr. Math , vol.3 , pp. 349-372
    • Shilnikov, L.1
  • 47
    • 37049182192 scopus 로고
    • Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics
    • [CrossRef] [PubMed]
    • Grebogi, C.; Ott, E.; Yorke, J.A. Chaos, Strange Attractors, and Fractal Basin Boundaries in Nonlinear Dynamics. Science 1987, 238, 632–638. [CrossRef] [PubMed]
    • (1987) Science , vol.238 , pp. 632-638
    • Grebogi, C.1    Ott, E.2    Yorke, J.A.3
  • 49
    • 84891517157 scopus 로고    scopus 로고
    • Discrete Fractional Logistic Map and Its Chaos
    • [CrossRef]
    • Wu, G.C.; Baleanu, D. Discrete Fractional Logistic Map and Its Chaos. Nonlinear Dyn. 2014, 75, 283–287. [CrossRef]
    • (2014) Nonlinear Dyn , vol.75 , pp. 283-287
    • Wu, G.C.1    Baleanu, D.2
  • 50
    • 0000100336 scopus 로고
    • Period Three Implies Chaos
    • [CrossRef]
    • Li, T.Y.; Yorke, J.A. Period Three Implies Chaos. Am. Math. Mon. 1975, 82, 985–992. [CrossRef]
    • (1975) Am. Math. Mon , vol.82 , pp. 985-992
    • Li, T.Y.1    Yorke, J.A.2
  • 51
    • 0001816458 scopus 로고
    • Quantitative Universality for a Class of Nonlinear Transformations
    • [CrossRef]
    • Feigenbaum, M.J. Quantitative Universality for a Class of Nonlinear Transformations. J. Stat. Phys. 1978, 19, 25–52. [CrossRef]
    • (1978) J. Stat. Phys , vol.19 , pp. 25-52
    • Feigenbaum, M.J.1
  • 52
    • 48749145356 scopus 로고
    • Universal Behavior in Nonlinear Systems
    • [CrossRef]
    • Feigenbaum, M.J. Universal Behavior in Nonlinear Systems. Phys. Nonlinear Phenom. 1983, 7, 16–39. [CrossRef]
    • (1983) Phys. Nonlinear Phenom , vol.7 , pp. 16-39
    • Feigenbaum, M.J.1
  • 53
    • 0002591468 scopus 로고
    • A Two-Dimensional Mapping with a Strange Attractor
    • [CrossRef]
    • Hénon, M. A Two-Dimensional Mapping with a Strange Attractor. Commun. Math. Phys. 1976, 50, 69–77. [CrossRef]
    • (1976) Commun. Math. Phys , vol.50 , pp. 69-77
    • Hénon, M.1
  • 54
    • 48749145669 scopus 로고
    • The Dimension of Chaotic Attractors
    • [CrossRef]
    • Farmer, J.D.; Ott, E.; Yorke, J.A. The Dimension of Chaotic Attractors. Phys. Nonlinear Phenom. 1983, 7, 153–180. [CrossRef]
    • (1983) Phys. Nonlinear Phenom , vol.7 , pp. 153-180
    • Farmer, J.D.1    Ott, E.2    Yorke, J.A.3
  • 55
    • 33646981873 scopus 로고
    • Characterization of Strange Attractors
    • [CrossRef]
    • Grassberger, P.; Procaccia, I. Characterization of Strange Attractors. Phys. Rev. Lett. 1983, 50, 346–349. [CrossRef]
    • (1983) Phys. Rev. Lett , vol.50 , pp. 346-349
    • Grassberger, P.1    Procaccia, I.2
  • 56
    • 5444244687 scopus 로고
    • How Long Is the Coast of Britain?
    • [CrossRef] [PubMed]
    • Mandelbrot, B.B. How Long Is the Coast of Britain? Science 1967, 156, 636–638. [CrossRef] [PubMed]
    • (1967) Science , vol.156 , pp. 636-638
    • Mandelbrot, B.B.1
  • 59
    • 0342445412 scopus 로고    scopus 로고
    • Diamond Study Group. Fractal Correlation Properties of RR Interval Dynamics and Mortality in Patients with Depressed Left Ventricular Function after an Acute Myocardial Infarction
    • [CrossRef] [PubMed]
    • Huikuri, H.V.; Mäkikallio, T.H.; Peng, C.K.; Goldberger, A.L.; Hintze, U.; Møller, M. Diamond Study Group. Fractal Correlation Properties of RR Interval Dynamics and Mortality in Patients with Depressed Left Ventricular Function after an Acute Myocardial Infarction. Circulation 2000, 101, 47–53. [CrossRef] [PubMed]
    • (2000) Circulation , vol.101 , pp. 47-53
    • Huikuri, H.V.1    Mäkikallio, T.H.2    Peng, C.K.3    Goldberger, A.L.4    Hintze, U.5    Møller, M.6
  • 60
    • 0000779360 scopus 로고
    • Detecting Strange Attractors in Turbulence
    • Rand, D., Young, L.S., Eds.; Springer: Berlin, Germany
    • Takens, F. Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence; Rand, D., Young, L.S., Eds.; Springer: Berlin, Germany, 1981; pp. 366–381.
    • (1981) Dynamical Systems and Turbulence , pp. 366-381
    • Takens, F.1
  • 61
    • 84975613293 scopus 로고
    • Estimating Fractal Dimension
    • [CrossRef]
    • Theiler, J. Estimating Fractal Dimension. J. Opt. Soc. Am. A 1990, 7, 1055–1073. [CrossRef]
    • (1990) J. Opt. Soc. Am. A , vol.7 , pp. 1055-1073
    • Theiler, J.1
  • 62
    • 85070538805 scopus 로고    scopus 로고
    • A Study of Period Doubling in Logistic Map for Shift Parameter
    • [CrossRef]
    • Kekre, H.B.; Sarode, T.; Halarnkar, P.N. A Study of Period Doubling in Logistic Map for Shift Parameter. Int. J. Eng. Trends Technol. 2014, 13, 281–286. [CrossRef]
    • (2014) Int. J. Eng. Trends Technol , vol.13 , pp. 281-286
    • Kekre, H.B.1    Sarode, T.2    Halarnkar, P.N.3
  • 63
    • 0022928633 scopus 로고
    • Computation of the Fractal Dimension of Topographic Surfaces Using the Triangular Prism Surface Area Method
    • [CrossRef]
    • Clarke, K.C. Computation of the Fractal Dimension of Topographic Surfaces Using the Triangular Prism Surface Area Method. Comput. Geosci. 1986, 12, 713–722. [CrossRef]
    • (1986) Comput. Geosci , vol.12 , pp. 713-722
    • Clarke, K.C.1
  • 64
    • 84929179953 scopus 로고    scopus 로고
    • The Many Facets of Chaos
    • [CrossRef]
    • Sander, E.; Yorke, J.A. The Many Facets of Chaos. Int. J. Bifurc. Chaos 2015, 25, 1530011. [CrossRef]
    • (2015) Int. J. Bifurc. Chaos , vol.25 , pp. 1530011
    • Sander, E.1    Yorke, J.A.2
  • 65
    • 84938629862 scopus 로고    scopus 로고
    • Classifying and Quantifying Basins of Attraction
    • [CrossRef] [PubMed]
    • Sprott, J.C.; Xiong, A. Classifying and Quantifying Basins of Attraction. Chaos 2015, 25, 083101. [CrossRef] [PubMed]
    • (2015) Chaos , vol.25 , pp. 083101
    • Sprott, J.C.1    Xiong, A.2
  • 66
    • 85059131692 scopus 로고    scopus 로고
    • Measuring Chaos Using the Lyapunov Exponent
    • Kiel, L.D., Elliott, E., Eds.; University of Michigan Press: Ann Arbor, MI, USA
    • Brown, T.A. Measuring Chaos Using the Lyapunov Exponent. In Chaos Theory in the Social Sciences; Kiel, L.D., Elliott, E., Eds.; University of Michigan Press: Ann Arbor, MI, USA, 1996; pp. 53–66.
    • (1996) Chaos Theory in the Social Sciences , pp. 53-66
    • Brown, T.A.1
  • 70
    • 84938315026 scopus 로고    scopus 로고
    • Defining Chaos
    • [CrossRef] [PubMed]
    • Hunt, B.R.; Ott, E. Defining Chaos. Chaos 2015, 25, 097618. [CrossRef] [PubMed]
    • (2015) Chaos , vol.25 , pp. 097618
    • Hunt, B.R.1    Ott, E.2
  • 71
    • 84878810963 scopus 로고    scopus 로고
    • The Problem of Spurious Lyapunov Exponents in Time Series Analysis and Its Solution by Covariant Lyapunov Vectors
    • [CrossRef]
    • Kantz, H.; Radons, G.; Yang, H. The Problem of Spurious Lyapunov Exponents in Time Series Analysis and Its Solution by Covariant Lyapunov Vectors. J. Phys. Math. Theor. 2013, 46, 254009. [CrossRef]
    • (2013) J. Phys. Math. Theor , vol.46 , pp. 254009
    • Kantz, H.1    Radons, G.2    Yang, H.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.