Skip to main content
SpringerLink
Log in

Chaotic phenomena

  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Literature

  • Allwirght, D. J. 1978. “Hypergraphic Functions and Bifurcations in Recurrence Relations.”SIAM J. appl. Math. 34, 687–691.

    Article  MathSciNet  Google Scholar 

  • Babin, A. V. and M. I. Vishik. 1983. “Attractors of Partial Differential Evolution Equations and Estimates of their Dimension.”Russian Math. Surveys 38, 151–213.

    Article  MATH  MathSciNet  Google Scholar 

  • Baillieul, J., R. W. Brockett and R. B. Washburn. 1980. “Chaotic Motions in Nonlinear Feedback Systems.”IEEE Trans. CAS-27, 990–997.

    MATH  MathSciNet  Google Scholar 

  • Barna, B. 1975. “Uber die Iterationen reeler Funktionen III.”Publ. Math. Debrecen 22, 269–278.

    MATH  MathSciNet  Google Scholar 

  • Beddington, J. R., C. A. Free and J. M. Lawton. 1975. “Dynamic Complexity in Predator-Prey Models Framed in Difference Equations.”Nature 255, 58–60.

    Article  Google Scholar 

  • Block, L. 1978. “Homoclinic Points of Mappings of the Interval.”Proc. Am. math. Soc. 72, 576–580.

    Article  MATH  MathSciNet  Google Scholar 

  • —— 1981. “Stability of Periodic Orbits in the Theorem of Sharkovskii.”Proc. Am. math. Soc. 81, 333–336.

    Article  MATH  MathSciNet  Google Scholar 

  • —— and D. Hart. 1982. “The Bifurcation of Homoclinic Orbits of Maps of the Interval.”Ergod. Theor. Dynam. Syst. 2, 131–138.

    MATH  MathSciNet  Google Scholar 

  • Block, L. and W. A. Coppel. 1984. “Stratification of Continuous Maps of an Interval.” Research Report No. 32-1984 Department of Mathematics, IAS Australian National University.

  • Blokh, A. M. 1982. “On Sensitive Mappings of the Interval.”Uspekhi Mat. Nauk. 37, 189–190.

    MATH  MathSciNet  Google Scholar 

  • Brandstäter, A., J. Swift, H. L. Swinney, A. Wolf, J. D. Farmer, E. Jen and P. J. Crutchfield. 1983. “Low-dimensional Chaos in a Hydrodynamic System.”Phys. Rev. Lett. 51(16), 1442–1445.

    Article  MathSciNet  Google Scholar 

  • Brunovsky, P. 1983. “Notes on Chaos in the Cell Population Partial Differential Equation.”Nonlin. Analysis TMA-7, 167–176.

    Article  MATH  MathSciNet  Google Scholar 

  • —— and J. Komornik. 1984. “Egodicity and Exactness of the Shift onC[0, ∞) and the Semiflow of a First-order Partial Differential Equation.”J. math. Analysis Applic. 104, 235–245.

    Article  MATH  MathSciNet  Google Scholar 

  • Bunow, B. and G. H. Weiss. 1979. “How Chaotic is Chaos? Chaotic and Other Noisy Dynamics in Frequency Domain.”Mathl Biosci. 47, 221–237.

    Article  MATH  MathSciNet  Google Scholar 

  • Carpenter, G. A. 1979. “Bursting Phenomena in Excitable Membranes.”SIAM J. appl. Math. 36, 334–372.

    Article  MATH  MathSciNet  Google Scholar 

  • Cartwright, M. 1967. “Almost Periodic Flows and Solutions of Differential Equations.”Proc. Lond. math. Soc. 17, 355–380.

    MATH  MathSciNet  Google Scholar 

  • Cartwright, M. L. and J. E. Littlewood. 1945. “On Nonlinear Differential Equations of the Second Order.”J. Lond. math. Soc. 20, 180–189.

    MATH  MathSciNet  Google Scholar 

  • Collet, P. and J.-P. Eckmann. 1980. “Iterated Maps on the Interval as Dynamical Systems.” Basel: Birkhauser.

    MATH  Google Scholar 

  • Coppel, W. A. 1983a. “Maps of an Interval.” IMA Preprint Series No. 26, University of Minnesota.

  • —— 1983b. “An Interesting Cantor Set.”Am. math. Mon. 90, 456–460.

    Article  MATH  MathSciNet  Google Scholar 

  • Coppel, W. A. 1984. “Continuous Maps of an Interval.” Lecture notes, Mathematics Department, IAS, Australian National University.

  • —— 1955. “The Solution of Equations by Iteration.”Proc. Camb. phil. Soc. 51, 41–43.

    MATH  MathSciNet  Google Scholar 

  • Cronin, J. 1973. “The Danziger-Elmergreen Theory of Periodic Catatonic Schizophrenia.”Bull. math. Biol. 35, 689–706.

    MATH  MathSciNet  Google Scholar 

  • Cvitanovic, P. and J. Myrheim. 1984. “Complex Universality.” Nordita Preprint 84/5.

  • Daido, H. 1982. “Period-doubling Bifurcations and Associated Universal Properties Including Parameter Dependence.”Prog. theor. Phys. 67, 1698–1723.

    Article  MATH  MathSciNet  Google Scholar 

  • Delbourgo, R., W. Hart and B. G. Kenny. 1985. “Dependence of Univeral Constants upon Multiplication Period in Nonlinear Maps.”Phys. Rev. A31, 514–516.

    Article  Google Scholar 

  • Devaney, R. L. 1984. “Homoclinic Bifurcations and the Area Conserving Henon Mapping.”J. diff. Equat. 51, 254–266.

    Article  MATH  MathSciNet  Google Scholar 

  • Douady, A. 1982. “Systèmes Dynamiques Holomorphes.” Seminare Bourbaki 1982/93 No. 599.

  • Du, B.-S. 1983. “Are Chaotic Functions Really Chaotic?”Bull. Aust. math. Soc. 28, 53–66.

    MATH  Google Scholar 

  • Dutton, J. and R. Wells. 1984. “Topological Issues in Hydrodynamic Predictability.” InPredictability of Fluid Motions, G. Holloway and B. West (Eds), pp. 11–43. New York: American Institute of Physics.

    Google Scholar 

  • Eckmann, J. P. 1981. “Roads to Turbulence in Dissipative Systems.”Rev. mod. Phys. 53, 643–654.

    Article  MathSciNet  MATH  Google Scholar 

  • Farmer, J. D., E. Ott and J. E. Yorke. “The Dimension of Chaotic Attractors.”Physica 7D, 153–180.

  • Feigenbaum, M. 1979. “Quantitative Universality for a Class of Nonlinear Transformations.”J. statist. Phys. 19 (1978), 25–52;21 (1979), 669–706.

    MathSciNet  Google Scholar 

  • FitzHugh, R. 1961. “Impulses and Physiological States in Theoretical Models of Nerve Membrane.”Biophysical J. 1, 445–466.

    Google Scholar 

  • Foias, C. and G. Prodi. 1967. “Sur le Comportement Global des Solutions Nonstationaires des Equations de Navier-Stokes en Dimension 2.”Rc. Semin. mat. Univ. Padova 39, 1–34.

    MATH  MathSciNet  Google Scholar 

  • ——, O. Manley, R. Temam and Y. Treve. 1983. “Asymptotic Analysis of the Navier-Stokes Equations.”Physica 9D, 157–188.

    MathSciNet  Google Scholar 

  • Gambaudo, J. M. and C. Tresser. 1983. “Some Difficulties Generated by Small Sinks in the Numerical Study of Dynamical Systems: Two Examples.”Phys. Lett. 94A, 412–414.

    MathSciNet  Google Scholar 

  • Glendinning, P. and C. Sparrow. 1984. “Local and Global Behaviour Near Homoclinic Orbits.”J. statist. Phys. 35, 645–697.

    Article  MATH  MathSciNet  Google Scholar 

  • Grassberger, P. 1983. “Generalized Dimensions of Strange Attractors.”Phys. Lett. 97A, 227–230.

    MathSciNet  Google Scholar 

  • —— and I. Procaccia. 1983a. “Measuring the Strangeness of Strange Attractors.”Physica 9D, 189–208.

    MathSciNet  Google Scholar 

  • ——. 1983b. “Characterization of Strange Attractors.”Phys. Rev. Lett. 50, 346–349.

    Article  MathSciNet  Google Scholar 

  • ——. 1983c. “Estimation of the Kolmogorov Entropy from a Chaotic Signal.”Phys. Rev. A28, 2591–2593.

    Article  Google Scholar 

  • ——. 1984. “Dimensions and Entropies of Strange Attractors from a Fluctuating Dynamics Approach.”Physica 13D, 34–54.

    MathSciNet  Google Scholar 

  • Greenside, H. S., A. Wolf, J. Swift and T. Pignataro. 1982. “Impracticality of a Box-counting Algorithm for Calculating the Dimension of Strange Attractors.”Phys. Rev. A25, 3453–3456.

    Article  MathSciNet  Google Scholar 

  • Griewank, A. and M. R. Osborne. 1983. “Analysis of Newton's Method at Irregular Singularities.”SIAM J. numer. Anal. 20, 747–773.

    Article  MATH  MathSciNet  Google Scholar 

  • Guckenheimer, J. 1976. “A Strange, Strange Attractor.” InThe Hopf Bifurcation and its Applications, J. E. Marsden and M. McCracken (Eds).Applied Mathematical Sciences, No. 1. 19. New York: Springer.

    Google Scholar 

  • —— 1979. “Sensitive Dependence on Initial Conditions for One-dimensional Maps.”Commun. Math. Phys. 70, 133–160.

    Article  MATH  MathSciNet  Google Scholar 

  • Guckenheimer, J. 1982. “Noise in Chaotic Systems.”Nature 298, 358–361.

    Article  Google Scholar 

  • —— 1984. “Dimension Estimates for Attractors.”Contemp. Math. 28, 357–367.

    MATH  MathSciNet  Google Scholar 

  • —— and G. Buzyna. 1983. “Dimension Measurements for Geostrophic Turbulence.”Phys. Rev. Lett. 51, 1438–1441.

    Article  MathSciNet  Google Scholar 

  • —— and P. Holmes. 1983. “Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields.”Applied Mathematical Sciences, Vol. 43, New York: Springer.

    MATH  Google Scholar 

  • ——, J. Moser and S. E. Newhouse. 1980. “Dynamical Systems.” C.I.M.E. Lectures, Bressanone, Italy, June 1978. Basel: Birkhauser.

    MATH  Google Scholar 

  • ——, G. Oster and A. Ipaktchi. 1977. “Dynamics of Density Dependent Population Models.”J. math. Biol. 4, 101–147.

    MATH  MathSciNet  Google Scholar 

  • Guevara, M. R. and L. Glass. 1982. “Phase Locking, Period Doubling Bifurcations and Chaos in a Model of a Periodically Driven Oscillator: A Theory for the Entrainment of Biological Oscillations and the Generation of Cardiac Dysrhythmias.”J. math. Biol. 14, 1–24.

    Article  MATH  MathSciNet  Google Scholar 

  • —— and A. Schrier. 1981. “Phase Locking, Period Doubling Bifurcations, and Irregular Dynamics in Periodically Stimulated Cardiac Cells.”Science, Wash. 214, 1350–1353.

    Google Scholar 

  • Gumowski, I. and C. Mira. 1980. “Recurrences and Discrete Dynamical Systems.”Lecture Notes in Mathematics, Vol. 809, Berlin: Springer.

    Google Scholar 

  • Haken, H. 1975. “Analogy Between Higher Instabilities in Fluids and Lasers.”Phys. Lett. 53A, 77.

    Google Scholar 

  • —— 1983.Synergetics: An Introduction, 3rd edn. Berlin: Springer.

    MATH  Google Scholar 

  • Hassel, M. P., J. P. Lawton and R. M. May. 1976. “Patterns of Dynamical Behaviour in Single-Species Populations.”J. Anim. Ecol. 45, 471–486.

    Article  Google Scholar 

  • Hénon, M. 1976. “A Two Dimensional Mapping with a Strange Attractor.”Commun. Math. Phys. 50, 69–77.

    Article  MATH  Google Scholar 

  • Hirsch, M. W. 1984. “The Dynamical Systems Approach to Differential Equations.”Bull. Am. math. Soc. (NS) 11, 1–64.

    MATH  MathSciNet  Google Scholar 

  • —— and S. Smale. 1974. “Differential Equations, Dynamical Systems, and Linear Algebra.” New York: Academic Press.

    MATH  Google Scholar 

  • Holden, A. V., W. Winlow and P. G. Heydon. 1982.Biol. Cybernet. 43, 163–173.

    Article  Google Scholar 

  • —— and S. M. Ramadan. 1981. “The Response of a Molluscan Neurone to a Cyclic Input: Entrainment and Phase Locking.”Biol. Cybertnet. 41, 157–163.

    Article  Google Scholar 

  • Hopf, E. 1937.Ergoden theorie. Berlin: Springer.

    Google Scholar 

  • Huppert, H. E. and D. R. Moore. 1976. “Nonlinear Double-diffusive Convection.”J. Fluid Mech. 78, 821–854.

    Article  MATH  Google Scholar 

  • Julia, G. 1918. “Memoire sur l'Iteration des Functions Rationelles.”J. Math (Ser. 7)4, 47–245.

    Google Scholar 

  • Kaplan, J. L. and J. A. Yorke. 1979. “Chaotic Behaviour of Multidimensional Difference Equations.”Lecture Notes in Mathematics, Vol. 730. Berlin: Springer.

    Google Scholar 

  • Kazmarek, L. K. and A. Babloyantz. 1977. “Spatiotemporal Patterns in Epileptic Seizures.”Biol. Cybernet. 26, 199–208.

    Article  Google Scholar 

  • Keener, J. P. 1981. “Chaotic Cardiac Dynamics.” InMathematical Aspects of Physiology (AMS Lectures in Applied Mathematics, Vol. 19.), F. C. Hoppensteadt (Ed.), pp. 299–325. Providence, RI: American Mathematical Society.

    Google Scholar 

  • King, R., J. D. Barchas and B. A. Huberman. 1984. “Chaotic Behaviour in Dopamine Neurodynamics.” Preprint, Stanford University School of Medicine.

  • Kloeden, P. E. 1976. “Chaotic Difference Equations are Dense.”Bull. Aust. math. Soc. 15, 371–380.

    Article  MATH  MathSciNet  Google Scholar 

  • —— 1979. “On Sharkovsky's Cycle Coexistence Ordering.”Bull. Aust. math. Soc. 20, 171–178.

    MATH  MathSciNet  Google Scholar 

  • —— 1981a. “Chaotic Difference Equations inR n.”J. Aust. math. Soc. XXXIA, 217–225.

    MathSciNet  Google Scholar 

  • Kloeden, P. E. 1981b. “Cycles and Chaos in Higher Dimensional Difference Equations.”Proc. IXth International Conference on Nonlinear Oscillations, Kiev.

  • Kloeden, P. E., M. A. B. Deakin and A. Tirkel. 1976. “A Precise Definition of Chaos.”Nature 264, 295.

    Article  Google Scholar 

  • Kolyada, S. F. and A. G. Sivak. 1982. “Universal Constants for One Parameter Family of Mappings.” InOscillations and Stability of Solutions of Functional Differential Equations, A. N. Sharkovsky (Ed.), pp. 53–60. Kiev: Institute of Mathematics, Ukranian Academy of Sciences.

    Google Scholar 

  • Lanford, O. 1982. “A Computer Assisted Proof of the Feigenbaum Conjectures.”Bull. Am. math. Soc. (NS) 6, 427–434.

    MATH  MathSciNet  Google Scholar 

  • Lasota, A. 1981. “Stable and Chaotic Solutions of a First-order Partial Differential Equation.”Nonlin. Analysis TMA-5, 1181–1193.

    Article  MATH  MathSciNet  Google Scholar 

  • Levi, M. 1981. “Qualitative Analysis of the Periodically Forced Relaxation Oscillations.”Mem. Am. math. Soc. 244.

  • Levinson, N. 1949. “A Second Order Differential Equation with Singular Solutions.”Ann. Math. 50, 127–153.

    Article  MATH  MathSciNet  Google Scholar 

  • Li, T.-Y. and J. A. Yorke. 1975. “Period Three Implies Chaos.”Am. math. Mon. 82, 985–992.

    Article  MATH  MathSciNet  Google Scholar 

  • Lorenz, E. N. 1963. “Deterministic Nonperiodic Flow.”J. atmos. Sci. 20, 130–141.

    Article  Google Scholar 

  • Mackey, M. C. and U. An der Heiden. 1983. “Dynamics of Recurrent Inhibition.” Preprint, McGill University, Montreal.

    Google Scholar 

  • —— and L. Glass. 1977. “Oscillation and Chaos in Physiological Control Systems.”Science 197, 287–289.

    Google Scholar 

  • Mallet-Paret, J. 1976. “Negatively Invariant Sets of Compact Maps and an Extension of a Theorem of Cartwright.”J. diff. Equat. 22, 331–348.

    Article  MATH  MathSciNet  Google Scholar 

  • Mandelbrot, B. B. 1977.Fractals: Form, Chance and Dimension. San Francisco: Freeman.

    MATH  Google Scholar 

  • Marotto, F. R. 1978. “Snap-back Repellers Imply Chaos inR n.”J. math. Analysis Applic. 63, 199–223.

    Article  MATH  MathSciNet  Google Scholar 

  • Matsumoto, T., L. O. Chua and M. Komuro. 1984. “A Double-scroll Chaotic Attractor From a Third-order Reciprocal Circuit.” Memorandum UCB/ERL/M84/73, University of California at Berkeley.

  • May, R. M. 1976. “Simple Mathematical Models with Very Complicated Dynamics.”Nature 261, 459–467.

    Article  Google Scholar 

  • —— 1983. “Nonlinear Problems in Ecology and Resource Management.” InChaotic Behaviour of Deterministic Systems, Les Houches, Session XXXVI, 1981, G. Iooss, R. Helleman and R. Stora (Eds). Amsterdam: North-Holland.

    Google Scholar 

  • May, R. M. 1984. “Regulation of Populations with Nonoverlapping Generations by Microparasites: A Purely Chaotic System.” Preprint.

  • —— and G. F. Oster. 1976. “Bifurcation and Dynamic Complexity in Simple Ecological Models.”Am. Nat. 110, 573–599.

    Article  Google Scholar 

  • McGuire, J. B. and C. J. Thompson. 1982. “Asymptotic Properties of Sequences of Iterates of Nonlinear Transformations.”J. statist. Phys. 27, 183–200.

    Article  MATH  MathSciNet  Google Scholar 

  • Mees, A. I. 1985. “Chaos in Feedback Systems.” InChaos: An Introduction, Vol. 1 ofNonlinear Science; Theory and Applications, A. V. Holden (Ed.). Manchester: Manchester University Press.

    Google Scholar 

  • —— and P. E. Rapp. 1978. “Periodic Metabolic Systems.”J. math. Biol. 5, 99–114.

    MATH  MathSciNet  Google Scholar 

  • —— and C. T. Sparrow. 1981. “Chaos.”IEE Proc. 128D, 201–205.

    MathSciNet  Google Scholar 

  • Mees, A. I. and P. Chapman. 1985. “Homoclinic and Heteroclinic Bifurcations in the Double-scroll Attractor Circuit.”IEEE Trans. CAS, under consideration.

  • Mizutani, M., T. Niwa and T. Ohno. 1983. “Chaos and Bifurcation in Limiting Central Difference Scheme.”J. Math. Kyoto Univ. 23, 29–54.

    MathSciNet  Google Scholar 

  • Myrberg, P. J. 1962. “Sur l'lteration des Polynomes Reels Quadratiques.”J. Math. Pures Appl. 41, 339–351.

    MATH  MathSciNet  Google Scholar 

  • Nagumo, J., S. Arimoto and S. Yoshizawa. 1962. “An Active Pulse Transmission Line Simulating Nerve Axon.”Proc. Inst. Radio Engrs 50, 2061–2070.

    Google Scholar 

  • Nicolis, C. and G. Nicolis. 1984. “Is There a Climatic Attractor?”Nature 311, 529–532.

    Article  Google Scholar 

  • Nitecki, Z. 1971.Differentiable Dynamics. Cambridge, MA: MIT Press.

    MATH  Google Scholar 

  • Olsen, L. F. and H. Degn. 1978. “Oscillatory Kinetics of the Peroxidase-Oxidase Reaction in an Open System. Experimental and Theoretical Studies.”Biochem. biophys. Acta 523, 321–334.

    Google Scholar 

  • Oono, Y. 1978. “Period ≠ 2n Implies Chaos.”Prog. theor. Phys. 59, 1028–1030.

    Article  Google Scholar 

  • Peitgen, H. O., D. Saupe and F. van Hoesler. 1984. “Cayley's Problem and Julia Sets.”The Mathematical Intelligencer 6, 11–20.

    Article  MATH  MathSciNet  Google Scholar 

  • Pianigiani, G. 1979. “Absolutely Continuous Invariant Measures for the Processx n+1=Axn(1−xn).”Boll. Un. mat. ital. 16, 374–378.

    MATH  MathSciNet  Google Scholar 

  • Preston, C. 1983. “Iterates of Maps on an Interval.”Lecture Notes in Mathematics, Vol. 999. Berlin: Springer.

    MATH  Google Scholar 

  • Rapp, P. E. 1985. “Chaotic Neural Dynamics: Turbulent Behaviour in a Biophysical Control System.” InChaos: An Introduction, A. V. Holden (Ed.). Manchester: Manchester University Press.

    Google Scholar 

  • ——, A. I. Mees and C. T. Sparrow. 1980. “Frequency Dependent Biochemical Regulation is More Accurate than Amplitude Dependent Control.”J. theor. Biol. 90, 531–544.

    Article  Google Scholar 

  • Rapp, P. E., I. D. Zimmerman, A. M. Albano, G. C. deGuzman and N. N. Greenbaum. 1985. “Dynamics of Spontaneous Neural Activity in the Simian Meter Cortex: The Dimension of Chaotic Neurones.” Submitted toPhys. Lett.

  • Rogers, T. D. 1981. “Chaos in Systems in Population Biology.”Prog. theor. Biol. 6, 91–145.

    Google Scholar 

  • Rössler, O. E. 1977. “Continuous Chaos.” InSynergetics: A Workshop, H. Haken (Ed.). Berlin: Springer.

    Google Scholar 

  • —— 1978. “Chaos and Strange Attractors in Chemical Kinetics.” InLoin de l'Equilibre, A. Pacault and C. Vidal (Eds). Berlin: Springer.

    Google Scholar 

  • —— 1982. “The Chaotic Hierarchy.”Z. Naturforsch. 38A, 788–801.

    Google Scholar 

  • Roux, J.-C., A. Rossi, S. Bachelart and C. Vidal. 1980. “Representation of a Strange Attractor from an Experimental Study of Chemical Turbulence.”Phys. Lett. 77A, 391–393.

    MathSciNet  Google Scholar 

  • Ruelle, D. 1981. “Differentiable Dynamical Systems and the Problem of Turbulence.”Bull. Am. math. Soc. (NS) 5, 29–72.

    MATH  MathSciNet  Google Scholar 

  • —— and F. Takens. 1971. “On the Nature of Turbulence.”Commun. Math. Phys. 20, 167–192.

    Article  MATH  MathSciNet  Google Scholar 

  • Saari, D. and J. Urenko. 1984. “Newton's Method, Circle Maps and Chaotic Motion.”Am. math. Mon. 91, 3–18.

    Article  MATH  MathSciNet  Google Scholar 

  • Salam, F. M. A., J. E. Marsden and P. Varaiya. 1983. “Chaos and Arnold Diffusion in Dynamical Systems.”IEEE Trans. CAS-30, 697–708.

    MATH  MathSciNet  Google Scholar 

  • Sejnowski, T. J. and G. E. Hinton. 1984. “Parallel Stochastic Search in Early Vision.” InVision, Brain and Cooperative Computation, M. Arbib and A. R. Hanson (Eds).

  • Sharkovsky, A. N. 1964. “Coexistence of Cycles of a Continuous Map of the Line Into Itself.”Ukrain. Mat. Zh. 16, 61–71.

    Google Scholar 

  • —— 1965. “On Cycles and the Structure of a Continuous Map.”Ukrain. Mat. Zh. 17, 104–111.

    Google Scholar 

  • —— 1966. “The Behaviour of a Map in the Neighbourhood of an Attracting Set.”Ukrain. Mat. Zh. 18, 60–83.

    Article  Google Scholar 

  • Sharkovsky, A. N. 1981. “Oscillations of Relaxation and Turbulence Types: Differential-Difference Models.”Proc. IXth Int. Conference on Nonlinear Oscillations, Kiev.

  • —— 1982a. “Difference Equations and the Dynamics of Population Size.” Institute of Mathematics Preprint 82-18, Ukrainian Academy of Sciences, Kiev.

    Google Scholar 

  • Sharkovsky, A. N. 1982b. “Some Properties of Discrete Dynamical Systems.”Proc. Coll Int. Theorie de l'Iteration, Toulouse, May 1982.

  • Shaw, R. 1979. “Strange Attractors, Chaotic Behaviour and Information Flow.”Z. Naturforsch. 36A, 80–112.

    Google Scholar 

  • Shil'nikov, L. P. 1979. “On a New Type of Bifurcation and Multidimensional Dynamical Systems.”Sov. Math. Dok. 10, 1368–1371.

    Google Scholar 

  • Shiraiwa, K. and M. Kurata. 1982. “A Generalization of a Theorem of Marotto.”Nagoya math. J. 82, 83–97.

    MathSciNet  Google Scholar 

  • Siegberg, H. W. 1983. “Chaotic Difference Equations: Generic Aspects.”Trans. Am. math. Soc. 279, 205–213.

    Article  MATH  MathSciNet  Google Scholar 

  • Singer, D. 1978. “Stable Orbits and Bifurcations of Maps of the Interval.”SIAM J. appl. Math. 35, 260–267.

    Article  MATH  MathSciNet  Google Scholar 

  • Smale, S. 1967. “Differentiable Dynamical Systems.”Bull. Am. math. Soc. 73, 774–817.

    Article  MathSciNet  Google Scholar 

  • Smital, J. and K. Smitalova. 1982. “Structural Stability of Nonchaotic Difference Equations.”J. math. Analysis Applic.90, 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  • Sparrow, C. T. 1980. “Bifurcation and Chaotic Behaviour in Simple Feedback Systems.”J. theor. Biol. 83, 93–105.

    Article  MathSciNet  Google Scholar 

  • ——. 1981. “Chaotic Behaviour in a 3-Dimensional Feedback System.”J. math. Analysis Applic. 83, 273–291.

    Article  MathSciNet  Google Scholar 

  • —— 1982. “The Lorenz Equations: Bifurcations, Chaos and Strange Attractors.”Applied Mathematical Sciences, Vol. 41, New York: Springer.

    MATH  Google Scholar 

  • Sparrow, C. T. and H. P. F. Swinnerton-Dyer. 1982. “The Lorenz Equations for Larger.” Preprint.

  • Stefan, P. 1977. “A Theorem of Sarkovskii on the Existence of Periodic Orbits of Continuous Endomorphisms of the Real Line.”Commun. Math. Phys. 54, 237–248.

    Article  MATH  MathSciNet  Google Scholar 

  • Stein, P. R. and S. M. Ulam. 1964. “Nonlinear Transformation Studies on Electronic Computers.”Rozprawy Mat. 39, 1–66.

    MathSciNet  Google Scholar 

  • Straffin, P. D. 1978. “Period Points of Continuous Functions.”Math. Mag. 51, 99–105.

    Article  MATH  MathSciNet  Google Scholar 

  • Thyagaraja, A. 1979. “Recurrent Motions in Certain Dynamical Systems.”Phys. Fluids 22, 2093–2096.

    Article  MATH  Google Scholar 

  • Traub, R. D. 1982.Neuroscience 7, 1233–1242.

    Article  Google Scholar 

  • Tresser, R. D. 1982. “On Some Theorems of L. P. Shilnikov and Some Applications.” Preprint, Université de Nice.

  • Tyson, J. J. 1978. “On the Appearance of Chaos in a Model of the Belousov Reaction.”J. math. Biol. 5, 351–362.

    MATH  MathSciNet  Google Scholar 

  • Ulam, S. M. and J. von Neumann. 1947. “On Combinations of Stochastic and Deterministic Processes.”Bull. Am. math. Soc. 53, 1120.

    Google Scholar 

  • Veronis, G. 1965. “On Finite Amplitude Instability in Thermohaline Convection.”J. mar. Res. 23, 1–17.

    Google Scholar 

  • Vidal, C., J. C. Roux, S. Bachelart and A. Rossi. 1980. “Experimental Study of the Transition to Turbulence in the Belousov-Zhabotinski Reaction.”Ann. N.Y. Acad. Sci. 357, 377–396.

    Google Scholar 

  • Vul, E. B. and K. M. Khanin. 1982. “Instability of the Separatix of Feigenbaum's Fixed Point.”Uspekhi Mat. Nauk 37, 173–174.

    MATH  MathSciNet  Google Scholar 

  • Walther, H. O. 1981. “Homoclinic Solution and Chaos inx(t)=f(x(t−1)).”Nonlin. Analysis 5, 775–788.

    Article  MATH  MathSciNet  Google Scholar 

  • Wegman, K. and O. E. Rössler. 1978. “Different Kinds of Chaotic Oscillations in the Belousov-Zhabotinsky Reaction.”Z. Naturforsch. 33A, 1179–1183.

    Google Scholar 

  • Whitley, D. 1983. “Discrete Dynamical Systems in Dimensions One and Two.”Bull. Lond. math. Soc. 15, 177–217.

    MATH  MathSciNet  Google Scholar 

  • Wolf, A., J. B. Swift, H. L. Swinney and J. A. Vastano. 1985. “Determining Lyapunov Exponents from a Time Series.” Submitted toPhysica D.

  • Wong, S. 1984. “Newton's Method and Symbolic Dynamics.”Proc. Am. math. Soc. 91, 245–253.

    Article  MATH  Google Scholar 

  • Yamaguti, M. and S. Ushiki, 1980. “Discretisation et Chaos.”Cr. Acad. Sci. Paris 290A, 637–640.

    MathSciNet  Google Scholar 

  • Young, L. S. 1983. “Entropy, Lyapunov Exponents and Hausdorff Dimension in Differentiable Dynamical Systems.”IEEE Trans.CAS-30, 599–607.

    MATH  Google Scholar 

  • Zaslavskii, B. G. 1983. “An Investigation of the Quasihomoclinic Structure of a Semi-group Generating Operator on a Banach Space.”Sibirs. Matem. Zh. 23, 80–90.

    MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. P. E. Kloeden

    Present address: School of Mathematical and Physical Sciences, Murdoch University, 6150, Murdoch, Western Australia

Authors and Affiliations

  1. Institut für theoretische Physik der Universität Stuttgart, D-7000, Stuttgart 80, F.R.G.

    P. E. Kloeden

  2. Department of Mathematics, University of Western Australia, 6009, Nedlands, Western Australia

    A. I. Mees

Authors
  1. P. E. Kloeden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Mees
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kloeden, P.E., Mees, A.I. Chaotic phenomena. Bltn Mathcal Biology 47, 697–738 (1985). https://doi.org/10.1007/BF02469299

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02469299

Keywords

Search

Navigation