PUBLICATIONS

  • ISSN No. Print
  • ISSN No. Online
  • Registration No.
  • Periodicity
  • Print
  • Online
  • Website

Complexity Studies in Some Piecewise Continuous Dynamical Systems

DOI
10.15415/mjis.2016.51003

AUTHORS

Ashok K chitkara, Neha Kumra and l. M. Saha

ABSTRACT

The, “Complex systems” , stands as a broad term for many diverse disciplines of science and engineering including natural & medical sciences. Complexities appearing in various dynamical systems during evolution are now interesting subjects of studies. Chaos appearing in various dynamical systems can also be viewed as a form of complexity. For some cases nonlinearities within the systems and for other cases piecewise continuity property of the system are responsible for such complexity. Dynamical systems represented by mathematical models having piecewise continuous properties show strange complexity character during evolution. Interesting recent articles explain widely on complexities in various systems. Observable quantities for complexity are measurement of Lyapunov exponents (LCEs), topological entropies, correlation dimension etc. The present article is related to study of complexity in systems having piecewise continuous properties. Some mathematical models are considered here in this regard including famous Lozi map, a discrete mathematical model and Chua circuit, a continuous model. Investigations have been carried forward to obtain various attractors of these maps appearing during evolution in diverse and interesting pattern for different set of values of parameters and for different initial conditions. Numerical investigations extended to obtain bifurcation diagrams, calculations of LCEs, topological entropies and correlation dimension together with their graphical representation.

KEYWORDS

Chaos; Lyapunov Exponents; Topological Entropy; Bifurcation.

REFERENCES

  • Abarbanel HD, Brown R, & Kennel MB. Local Lyapunov exponents computed from observed data. Journal of Nonlinear Science 2, 343–365, (1992).
  • Adler RL, Konheim AG, & Mc&rew MH. Topological entropy. Transactions of the American Mathematical Society 114, 309–319, (1965).
  • Aziz-Alaoui M, Robert C, & Grebogi C. Dynamics of a Hénon–Lozi-type map. Chaos, Solitons & Fractals 12, 2323–2341, (2001).
  • Banerjee S, & Grebogi C. Border collision bifurcations in two-dimensional piecewise smooth maps. Physical Review E 59, 4052, (1999).
  • Baptista D, Severino R, & Vinagre S. The basin of attraction of Lozi mappings. International Journal of Bifurcation & Chaos 19, 1043–1049, (2009).
  • Beddington J, Free C, & Lawton J. Dynamic complexity in predator-prey models framed in difference equations. Transactions of the American Mathematical Society (1975).
  • Benettin G, Galgani L, Giorgilli A, & Strelcyn J-M. Lyapunov characteristic exponents for smooth dynamical systems & for Hamiltonian systems; a method for computing all of them. Part 1, Theory. Meccanica 15, 9–20, (1980).
  • Brown R, Bryant P, & Abarbanel HD. Computing the Lyapunov spectrum of a dynamical system from an observed time series. Physical Review A 43, 2787, (1991).
  • Bryant P, Brown R, & Abarbanel HD. Lyapunov exponents from observed time series. Physical Review Letters 65, 1523, (1990).
  • Bryant PH. Extensional singularity dimensions for strange attractors. Physics Letters A 179, 186–190, (1993).
  • Di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P, Nordmark AB, Tost GO, & Piiroinen PT. Bifurcations in nonsmooth dynamical systems. SIAM review 50, 629–701,(2008).
  • Grassberger P, & Procaccia I. Characterization of strange attractors. Physical review letters 50, 346, (1983).
  • Kaitala V, & Heino M. Complex non-unique dynamics in simple ecological interactions. Proceedings of the Royal Society of London B, Biological Sciences 263, 1011–1015, (1996).
  • Lozi R. Un attracteur étrange (?) du type attracteur de Hénon. Le Journal de Physique Colloques 39, C5-9-C5-10, (1978).
  • Lynch S. Dynamical systems with applications using MapleTM. ,(2009).
  • Martelli M. Introduction to discrete dynamical systems & chaos. John Wiley & Sons, (2011).
  • Rocha R, Andrucioli GL, & Medrano-T RO. Experimental characterization of nonlinear systems:a real-time evaluation of the analogous Chua’s circuit behavior. Nonlinear Dynamics 62, 237–251, (2010).
  • Ros J,Botella V, Castelo JM,Oteo JA. Bifurcations in Lozi map. Journal of Physics A: Mathematical & Theoretical 44, 1–14, (2011).
  • Skokos C. The Lyapunov characteristic exponents & their computation. In: Dynamics of Small Solar System Bodies & ExoplanetsSpringer, 63–135, (2010).
  • Tang S, & Chen L. A discrete predator-prey system with age-structure for predator & natural barriers for prey. ESAIM: Mathematical Modelling & Numerical Analysis-Modélisation Mathématique et Analyse Numérique 35, 675–690, (2001).
  • Xiao Y, Cheng D, & Tang S. Dynamic complexities in predator–prey ecosystem models with age-structure for predator. Chaos, Solitons & Fractals 14, 1403–1411, (2002).
  • Yang H-l, & Radons G. Comparison between covariant & orthogonal Lyapunov vectors. Physical Review E 82, 046204, (2010)
Call for Papers Publication Policy Instructions to the Authors Paper Submission Subscription Form Copyright Form Author Profile Format Sample paper Recommend to a Librarian Patrons & Leadership

Refereed Research Journal

Plagiarism Checked by

Member of CrossRef

Indexing

Call for Papers

Invites papers for next issue of Mathematical Journal of Interdisciplinary Sciences

Frequency

Mathematical Journal of Interdisciplinary Sciences is published Bi-Annually

Number-1 September
Number-2 March