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2013
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# Dmitry Turaev

South Kensington Campus, London SW7 2AZ, UK
Imperial College London

## Publications:

 Turaev D. V. Hyperbolic Sets near Homoclinic Loops to a Saddle for Systems with a First Integral 2014, vol. 19, no. 6, pp.  681-693 Abstract A complete description of dynamics in a neighborhood of a finite bunch of homoclinic loops to a saddle equilibrium state of a Hamiltonian system is given. Keywords: Hamiltonian system, nonintegrability and chaos, resonance crossing, Arnold diffusion Citation: Turaev D. V.,  Hyperbolic Sets near Homoclinic Loops to a Saddle for Systems with a First Integral, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 681-693 DOI:10.1134/S1560354714060069
 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 Abstract 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 DOI:10.1134/S1560354714040017
 Turaev D. V., Warner C., Zelik S. Energy Growth for a Nonlinear Oscillator Coupled to a Monochromatic Wave 2014, vol. 19, no. 4, pp.  513-522 Abstract A system consisting of a chaotic (billiard-like) oscillator coupled to a linear wave equation in the three-dimensional space is considered. It is shown that the chaotic behavior of the oscillator can cause the transfer of energy from a monochromatic wave to the oscillator, whose energy can grow without bound. Keywords: delayed equation, invariant manifold, normal hyperbolicity, billiard Citation: Turaev D. V., Warner C., Zelik S.,  Energy Growth for a Nonlinear Oscillator Coupled to a Monochromatic Wave, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 513-522 DOI:10.1134/S1560354714040078
 Lerman L. M., Turaev D. V. Breakdown of Symmetry in Reversible Systems 2012, vol. 17, no. 3-4, pp.  318-336 Abstract We review results on local bifurcations of codimension 1 in reversible systems (flows and diffeomorphisms) which lead to the birth of attractor-repeller pairs from symmetric equilibria (for flows) or periodic points (for diffeomorphisms). Keywords: reversible system, reversible diffeomorphism, bifurcation, symmetry, equilibrium state, periodic point Citation: Lerman L. M., Turaev D. V.,  Breakdown of Symmetry in Reversible Systems, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 318-336 DOI:10.1134/S1560354712030082
 Gelfreich V. G., Turaev D. V. Universal dynamics in a neighborhood of a generic elliptic periodic point 2010, vol. 15, no. 2-3, pp.  159-164 Abstract We show that a generic area-preserving two-dimensional map with an elliptic periodic point is $C^\omega$-universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows. Keywords: homoclinic tangency, wild hyperbolic set, Newhouse phenomenon, Hamiltonian system, area-preserving map, volume-preserving flow, exponentially small splitting, KAM theory Citation: Gelfreich V. G., Turaev D. V.,  Universal dynamics in a neighborhood of a generic elliptic periodic point, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 159-164 DOI:10.1134/S156035471002005X
 Gonchenko S. V., Shilnikov L. P., Turaev D. V. On Global Bifurcations in Three-Dimensional Diffeomorphisms Leading to Wild Lorenz-Like Attractors 2009, vol. 14, no. 1, pp.  137-147 Abstract 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 DOI:10.1134/S1560354709010092
 Gonchenko S. V., Lerman L. M., Turaev D. V. Leonid Pavlovich Shilnikov. On his 70th birthday 2006, vol. 11, no. 2, pp.  139-140 Abstract 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., Schneider K. R., Turaev D. V. Quasiperiodic regimes in multisection semiconductor lasers 2006, vol. 11, no. 2, pp.  213-224 Abstract 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. Elliptic Periodic Orbits Near a Homoclinic Tangency in Four-Dimensional Symplectic Maps and Hamiltonian Systems With Three Degrees of Freedom 1998, vol. 3, no. 4, pp.  3-26 Abstract 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 DOI:10.1070/RD1998v003n04ABEH000089
 Pisarevskii V., Shilnikov A. L., Turaev D. V. Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry 1998, vol. 3, no. 1, pp.  19-27 Abstract Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with $\mathbb{Z}_q$-symmetry are listed. Citation: Pisarevskii V., Shilnikov A. L., Turaev D. V.,  Asymptotic normal forms for equilibria with a triplet of zero characteristic exponents in systems with symmetry, Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 19-27 DOI:10.1070/RD1998v003n01ABEH000058
 Shilnikov L. P., Turaev D. V. Super-homoclinic orbits and multi-pulse homoclinic loops in Hamiltonian systems with discrete symmetries 1997, vol. 2, nos. 3-4, pp.  126-138 Abstract 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 DOI:10.1070/RD1997v002n04ABEH000053