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Papers by Yuri A . Kuznetsov

The Journal of Mathematical Neuroscience, 2017

Numerical Bifurcation Analysis of Maps, 2019

Applied Mathematical Sciences, 2004

In the previous two chapters we studied bifurcations of equilibria and fixed points in generic on... more In the previous two chapters we studied bifurcations of equilibria and fixed points in generic one-parameter dynamical systems having the minimum possible phase dimensions. Indeed, the systems we analyzed were either one- or two-dimensional. This chapter shows that these bifurcations occur in “essentially” the same way for generic n-dimensional systems. As we shall see, there are certain parameter-dependent one- or two-dimensional invariant manifolds on which the system exhibits the corresponding bifurcations, while the behavior off the manifolds is somehow “trivial,” for example, the manifolds may be exponentially attractive. Moreover, such manifolds (called center manifolds) exist for many dissipative infinite-dimensional dynamical systems. Below we derive quadratic approximations to the center manifolds in finite dimensions and for systems restricted to them at bifurcation parameter values. Using these results, we derive explicit invariant formulas for the critical normal form coefficients at all studied codimension 1 bifurcations of equilibria and fixed points. In Appendix A we consider a reaction-diffusion system on an interval to illustrate the necessary modifications of the technique to handle infinite-dimensional systems.

SIAM Journal on Applied Dynamical Systems, 2005

Mathematical Biosciences, 1998

Journal of Difference Equations and Applications, 2009

International Journal of Bifurcation and Chaos, 2009

In Part I of this paper we have discussed new methods for the numerical continuation of point-to-... more In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available.

International Journal of Bifurcation and Chaos, 1996

This paper presents extensions and improvements of recently developed algorithms for the numerica... more This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

… ScienceICCS 2005, 2005

Journal of Physics: Conference Series

Ijbc, 2005

Simple computational formulas are derived for the two-, three-, and four-order coefficients of th... more Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov–Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.

Http Dx Doi Org 10 1080 00207729408949324, Apr 26, 2007

... AUTO 97: Continuation And Bifurcation Software For Ordinary Differential Equations (with HomC... more ... AUTO 97: Continuation And Bifurcation Software For Ordinary Differential Equations (with HomCont) (0) [62 citations 13 self]. ... 5, Stability and multiplicity of solutions to discretizations of nonlinear ordinary differential equations Beyn, Doedel - 1981. ...

Acm Transactions on Mathematical Software, 2003

The study of differential equations requires good and powerful mathematical software. Also, flexi... more The study of differential equations requires good and powerful mathematical software. Also, flexibility and extendibility of the package are important. However, most of the existing software all have their own way of specifying the system or are written in a relatively low-level programming language, so it is hard to extend it. In 2000, A. Riet started the implementation of a

Lecture Notes in Computer Science, 2003