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What is BunKi ?
Abstract
BunKi is a reading of a Japanese word Ê¬´ô, which means "Bifurcation." For a given dynamical system, one can observe sudden qualitative changes of limit sets, such as equilibria, fixed points, limit cycles, or periodic solutions, with slight change of a parameter value. This qualitative change of the limit set is called bifurcation. For instance, if a stable equilibrium point unexpectedly becomes an oscillating motion by changing a parameter value, you witness one of bifurcation phenomena. Bifurcation analysis means investigating these qualitative changes depending on the system parameters. The information obtained from the bifurcation analysis enables us to know fundamental properties of the system, e.g., a parameter range which the system behaves stably, the total behavior of the solution in the large, or transition mechanisms of the dynamic responses. Meanwhile, in general, obtaining exact solutions of nonlinear dynamical system is quite difficult, thus computer based numerical calculations take important roles for investigation of the bifurcation phenomena.
The software BunKi that is developed on the basis of the techniques proposed in the references [1][5] is an integrated environment dedicated to bifurcation analysis¡¥
Features
This software is based on the techniques proposed in the references [1][6]¡¥ The method proposed in the reference [1] in particular, has undergone many improvements up to now, and has been used over a large variety of systems in fields such as engineering, physics, biology and others [7][12]¡¥ However, none of the resulting softwares were neither usable nor generic enough, making it difficult for a general use.
The highlight feature of BunKi is the userfriendly analysis environment it offers. That is to say, by just entering the system equations, the program generates a set of analysis tools automatically. Moreover, it is possible to operate intuitively through a graphical user interface (GUI).
In addition, to describe the variational equation explicitly, this BunKi package is its ability to obtain precise bifurcation sets when compared with the results used various numerical differentiation methods. Thus our proposed method can reduce the influence of numerical error.
Some noteworthy features of the software BunKi can be summarized as follows:
 The availability of automatic generation of analytical tools with minimum description of the rules and equations of the system to be analyzed.
 Through a simulator, the dynamical behavior of the system can be visually checked directly.
 The user is able to obtain high quality results to perform advanced and high accuracy computations.
 Almost all operations can be run through a graphical user interface (GUI).
The following list enumerates the 3 tools composing this bifurcation analysis tool:
 Tool for Phase portrait
 It display visually the dynamical behavior of the system to trace the solution from the given initial states.
 Tool for Bifurcation point searching
 for some kinds of attractors (equilibrium point, Limit cycle, periodic point, and so on), it searches a bifurcation point that the stability of the solution changes by varying the system parameters.
 Tool for the computation of Bifurcation sets
 From the bifurcation points obtained with the bifurcation point search tool, a bifurcation set is computed in the given parameters plane.
References
 [1] H. Kawakami, Bifurcation of periodic responses in forced dynamic nonlinear circuits: computation of bifurcation values of the system parameters, IEEE Trans. Circuits and Systems. CAS31, pp.248260, 1984.
 [2] T. Yoshinaga and H. Kawakami, Bifurcation and chaotic state in forced oscillatory circuits containing saturable inductors, in: L. Pecora and T. Carroll, eds., Nonlinear Dynamics in Circuits, World Scientific, Singapore, pp.89119, 1995.
 [3] T. Yoshinaga and H. Kawakami, Codimension two bifurcation problems in forced nonlinear circuits, Trans. IEICE, vol.E73, no.6, pp.817824, 1990
 [4] H. Kawakami and T. Yoshinaga, Codimension two bifurcation and its computational algorithm, In J. Awrejcewicz, editor, Bifurcation and Chaos: Theory and Applications, pp.97132, SpringerVerlag Berlin Heidelberg, 1995
 [5] T.Ueta, M. Tsueike, H. Kawakami, T. Yoshinaga, and Y. Katsuta, A computation of bifurcation parameter values for limit cycles, IEICE Trans. Fundamentals. vol.E80A, no.9, pp.17251728, 1997.
 [6] T.Ueta and H. Kawakami, Numerical approaches to bifurcation analysis, In G. Chen and T. Ueta, editor, Chaos in circuits and systems, pp.593610, World Scientific, 2002.
 [7] T. Kousaka, T. Ueta and H. Kawakami. Bifurcation of Switched Nonlinear Dynamical Systems, IEEE Trans. Circuits and Systems II, vol.CAS46, no.7, pp.878–885, 1999
 [8] T. Yoshinaga, Y. Sano and H. Kawakami, A Method to Calculate Bifurcations in Synaptically Coupled HodgkinHuxley Equations, Int. J. Bifurcation and Chaos, vol.9, no.7, pp.14511458, 1999.
 [9] T. Ueta and G.R. Chen, Bifurcation analysis of Chen's equation, Int. J. Bifurcation and Chaos, vol.10, no.8, pp.19171931, 2000.
 [10] K. Tsumoto, T. Yoshinaga, H. Iida, H. Kawakami and K. Aihara, Bifurcations in a mathematical model for circadian oscillations of clock genes, J. of Theor. Biol, vol.239, no.1, pp.101122, 2006.
 [11] H. Kitajima and J. Kurths, Forced synchronization in MorrisLecar neurons, Int. J. of Bifurcation and Chaos, vol.17, no.10, pp.35233528, 2007.
 [12] S. Tsuji, T. Ueta and H. Kawakami, H. Fujii and K. Aihara, Bifurcations in twodimensional HindmarshRose type model, Int. J. of Bifurcation and Chaos, vol.17, no. 3, pp.985998, 2007.
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