Leonid Shilnikov

10, Ulyanov Str, 603005 Nizhny Novgorod
Institute for Applied Mathematics & Cybernetics of Nizhny Novgorod University


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., 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., 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
Shilnikov L. P., Turaev D. V.
4D-Hamiltonian systems with discrete symmetries are studied. The symmetries under consideration are such that a system possesses two invariant sub-planes which intersect each other transversally at an equilibrium state. The equilibrium state is supposed to to be of saddle type; moreover, in each invariant sub-plane there are two homoclinic loops to the saddle. We establish the existence of stable and unstable invariant manifolds for the bouquet comprised by the four homoclinic trajectories at the Hamiltonian level corresponding to the saddle. These manifolds may intersect transversely along some orbit. We call such a trajectory a super-homoclinic one. We prove that the existence of a super-homoclinic orbit implies the existence of a countable set of multi-pulse homoclinic trajectories to the saddle.
Citation: Shilnikov L. P., Turaev D. V.,  Super-homoclinic orbits and multi-pulse homoclinic loops in Hamiltonian systems with discrete symmetries, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 126-138

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