Sergey Gonchenko

Sergey Gonchenko
Ulyanova st. 10, 603005, Nizhny Novgorod, Russia
Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University


Gonchenko S. V., Safonov K. A., Zelentsov N. G.
We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism $T_1$ and an involution $h$, i.e., a map (diffeomorphism) such that $h^2 = Id$. We construct the desired reversible map $T$ in the form $T = T_1\circ T_2$, where $T_2 = h\circ T_1^{-1}\circ h$. We also discuss how this method can be used to construct normal forms of Poincaré maps near mutually symmetric pairs of orbits of homoclinic or heteroclinic tangencies in reversible maps. One of such normal forms, as we show, is a two-dimensional double conservative Hénon map $H$ of the form $\bar x = M + cx - y^2; \ y = M + c\bar y - \bar x^2$. We construct this map by the proposed method for the case when $T_1$ is the standard Hénon map and the involution $h$ is $h: (x,y) \to (y,x)$. For the map $H$, we study bifurcations of fixed and period-2 points, among which there are both standard bifurcations (parabolic, period-doubling and pitchfork) and singular ones (during transition through $c=0$).
Keywords: reversible diffeomorphism, parabolic bifurcation, period-doubling bifurcation
Citation: Gonchenko S. V., Safonov K. A., Zelentsov N. G.,  Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map, Regular and Chaotic Dynamics, 2022, vol. 27, no. 6, pp. 647-667
Delshams A., Gonchenko M. S., Gonchenko S. V.
We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable twodimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
Keywords: area-preserving map, non-orientable surface, elliptic point, homoclinic tangency, bifurcation
Citation: Delshams A., Gonchenko M. S., Gonchenko S. V.,  On Bifurcations of Area-preserving and Nonorientable Maps with Quadratic Homoclinic Tangencies, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 702-717
Afraimovich V. S., Gonchenko S. V., Lerman L. M., Shilnikov A. L., Turaev D. V.
Scientific Heritage of L.P. Shilnikov
2014, vol. 19, no. 4, pp.  435-460
This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.
Keywords: Homoclinic chaos, global bifurcations, spiral chaos, strange attractor, saddle-focus, homoclinic loop, saddle-node, saddle-saddle, Lorenz attractor, hyperbolic set
Citation: Afraimovich V. S., Gonchenko S. V., Lerman L. M., Shilnikov A. L., Turaev D. V.,  Scientific Heritage of L.P. Shilnikov, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 435-460
Gonchenko S. V., Gordeeva O. V., Lukyanov V. I., Ovsyannikov I. I.
We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.
Keywords: saddle-node, homoclinic tangency, Arnold tongues
Citation: Gonchenko S. V., Gordeeva O. V., Lukyanov V. I., Ovsyannikov I. I.,  On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 461-473
Gonchenko S. V., Ovsyannikov I. I., Tatjer J. C.
It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.
Keywords: Homoclinic tangency, rescaling, 3D Hénon map, bifurcation, Lorenz-like attractor
Citation: Gonchenko S. V., Ovsyannikov I. I., Tatjer J. C.,  Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 495-505
Gonchenko A. S., Gonchenko S. V., Kazakov A. O.
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Keywords: celtic stone, nonholonomic model, strange attractor, discrete Lorenz attractor, Shilnikov-like spiral attractor, mixed dynamics
Citation: Gonchenko A. S., Gonchenko S. V., Kazakov A. O.,  Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538
Gonchenko S. V., Gonchenko V. S., Shilnikov L. P.
On a homoclinic origin of Hénon-like maps
2010, vol. 15, nos. 4-5, pp.  462-481
We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.
Keywords: homoclinic tangency, Hénon-like maps, saddle-focus fixed point, wild-hyperbolic attractor
Citation: Gonchenko S. V., Gonchenko V. S., Shilnikov L. P.,  On a homoclinic origin of Hénon-like maps, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 4-5, pp. 462-481
Gonchenko S. V., Li M.
We study the hyperbolic dynamics of three-dimensional quadratic maps with constant Jacobian the inverse of which are again quadratic maps (the so-called 3D Hénon maps). We consider two classes of such maps having applications to the nonlinear dynamics and find certain sufficient conditions under which the maps possess hyperbolic nonwandering sets topologically conjugating to the Smale horseshoe. We apply the so-called Shilnikov’s crossmap for proving the existence of the horseshoes and show the existence of horseshoes of various types: (2,1)- and (1,2)-horseshoes (where the first (second) index denotes the dimension of stable (unstable) manifolds of horseshoe orbits) as well as horseshoes of saddle and saddle-focus types.
Keywords: quadratic map, Smale horseshoe, hyperbolic set, symbolic dynamics, saddle, saddlefocus
Citation: Gonchenko S. V., Li M.,  Shilnikov’s cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 165-184
Gonchenko M. S., Gonchenko S. V.
We study bifurcations of two-dimensional symplectic maps with quadratic homoclinic tangencies and prove results on the existence of cascade of elliptic periodic points for one and two parameter general unfoldings.
Keywords: symplectic map, homoclinic tangency, bifurcation, generic elliptic point, KAM-theory
Citation: Gonchenko M. S., Gonchenko S. V.,  On Cascades of Elliptic Periodic Points in Two-Dimensional Symplectic Maps with Homoclinic Tangencies, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 116-136
Gonchenko S. V., Shilnikov L. P., Turaev D. V.
We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransverse heteroclinic cycles. We show that bifurcations under consideration lead to the birth of wild-hyperbolic Lorenz attractors. These attractors can be viewed as periodically perturbed classical Lorenz attractors, however, they allow for the existence of homoclinic tangencies and, hence, wild hyperbolic sets.
Keywords: homoclinic tangency, strange attractor, Lorenz attractor, wild-hyperbolic attractor
Citation: Gonchenko S. V., Shilnikov L. P., Turaev D. V.,  On Global Bifurcations in Three-Dimensional Diffeomorphisms Leading to Wild Lorenz-Like Attractors, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 137-147
Gonchenko S. V., Gonchenko V. S., Tatjer J. C.
We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers $e^{\pm i \phi}$. On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.
Keywords: homoclinic tangency, rescaling, generalized Henon map, bifurcation
Citation: Gonchenko S. V., Gonchenko V. S., Tatjer J. C.,  Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 233-266
Gonchenko S. V., Lerman L. M., Turaev D. V.
Leonid Pavlovich Shilnikov. On his 70th birthday
2006, vol. 11, no. 2, pp.  139-140
In connection with the 70th birthday of Professor L.P. Shilnikov, an outstanding scientist and the leader of the famous Nizhny Novgorod Nonlinear Dynamics school, his colleagues and disciples organized the International Conference "Dynamics, Bifurcations and Chaos", which was held on January 31-February 4, 2005 in Nizhny Novgorod, Russia.
This special issue is a collection of research papers which were either contributed by participants of this conference or submitted in reply to a call for papers announced by Editorial Board of RCD in March 2005.
Citation: Gonchenko S. V., Lerman L. M., Turaev D. V.,  Leonid Pavlovich Shilnikov. On his 70th birthday , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 139-140
DOI: 10.1070/RD2006v011n02ABEH000340
Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.
We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(\lambda e^{i\varphi}, \lambda e^{-i\varphi}, \gamma)$, where $0<\lambda<1<|\gamma|$ and $|\lambda^2 \gamma|=1$. We show that in a three-parameter family, $g_\varepsilon$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\varepsilon=0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional Hénon-like map. This map possesses, in some parameter regions, a ''wild-hyperbolic'' Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional Hénon maps occupy in the class of three-dimensional quadratic maps with constant Jacobian
Keywords: saddle-focus fixed point, three-dimensional quadratic map, homoclinic bifurcation, strange attractor
Citation: Gonchenko S. V., Meiss J. D., Ovsyannikov I. I.,  Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 191-212
DOI: 10.1070/RD2006v011n02ABEH000345
Gonchenko S. V., Schneider K. R., Turaev D. V.
Quasiperiodic regimes in multisection semiconductor lasers
2006, vol. 11, no. 2, pp.  213-224
We consider a mode approximation model for the longitudinal dynamics of a multisection semiconductor laser which represents a slow-fast system of ordinary differential equations for the electromagnetic field and the carrier densities. Under the condition that the number of active sections $q$ coincides with the number of critical eigenvalues we introduce a normal form which admits to establish the existence of invariant tori. The case $q=2$ is investigated in more detail where we also derive conditions for the stability of the quasiperiodic regime
Keywords: multisection semiconductor laser, averaging, mode approximation, invariant torus, normal form, stability
Citation: Gonchenko S. V., Schneider K. R., Turaev D. V.,  Quasiperiodic regimes in multisection semiconductor lasers , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 213-224
DOI: 10.1070/RD2006v011n02ABEH000346
Gonchenko S. V., Shilnikov L. P., Turaev D. V.
We study bifurcations leading to the appearance of elliptic orbits in the case of four-dimensional symplectic diffeomorphisms (and Hamiltonian flows with three degrees of freedom) with a homoclinic tangency to a saddle-focus periodic orbit.
Citation: Gonchenko S. V., Shilnikov L. P., Turaev D. V.,  Elliptic Periodic Orbits Near a Homoclinic Tangency in Four-Dimensional Symplectic Maps and Hamiltonian Systems With Three Degrees of Freedom, Regular and Chaotic Dynamics, 1998, vol. 3, no. 4, pp. 3-26
Gonchenko S. V., Shilnikov L. P.
We consider two-dimensional analitical area-preserving diffeomorphisms that have structurally unstable symplest heteroclinic cycles. We find the conditions when diffeomorphisms under consideration possess a countable set of periodic elliptic points of stable type.
Citation: Gonchenko S. V., Shilnikov L. P.,  On two-dimensional analitical area-preserving diffeomorphisms with a countable set of elliptic periodic points of stable type, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 106-123

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