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Lev Lerman

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


Gelfreich V. G., Lerman L. M.
Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation
2014, vol. 19, no. 6, pp.  635-655
We study the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a non-semisimple double zero one. It is well known that a one-parameter unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of single-round homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.
Keywords: Hamiltonian bifurcation, homoclinic orbit, separatrix splitting, asymptotics beyond all orders
Citation: Gelfreich V. G., Lerman L. M.,  Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 635-655
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
Lerman L. M., Turaev D. V.
Breakdown of Symmetry in Reversible Systems
2012, vol. 17, no. 3-4, pp.  318-336
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
Lerman L. M.
Breaking hyperbolicity for smooth symplectic toral diffeomorphisms
2010, vol. 15, no. 2-3, pp.  194-209
We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on $\pi_1$-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.
Keywords: symplectic diffeomorphism, torus, Anosov diffeomorphism, almost hyperbolic, degenerate saddle, bifurcation, homoclinic tangency, break of hyperbolicity, $\pi_1$-diffeomorphism
Citation: Lerman L. M.,  Breaking hyperbolicity for smooth symplectic toral diffeomorphisms, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 194-209
Lerman L. M., Markova A. P.
On Stability at the Hamiltonian Hopf Bifurcation
2009, vol. 14, no. 1, pp.  148-162
For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearization matrix, when some coefficient in the 4th order normal form is positive (the equilibrium is known to be unstable, if this coefficient is negative). Such the degenerate equilibrium is met generically in one-parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. Though the stability is known since 1977, proofs that were published are either incorrect or not complete. Our proof is based on the KAM theory and a work with the Weierstrass elliptic functions, estimates of power series and scaling.
Keywords: Hamiltonian Hopf Bifurcation, KAM theory, Lyapunov stability, normal form, action-angle variables, elliptic functions, scaling
Citation: Lerman L. M., Markova A. P.,  On Stability at the Hamiltonian Hopf Bifurcation, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 148-162
Lerman L. M.
Partial normal form near a saddle of a Hamiltonian system
2006, vol. 11, no. 2, pp.  291-297
For a smooth or real analytic Hamiltoniain vector field with two degrees of freedom we derive a local partial normal form of the vector field near a saddle equilibrium (two pairs of real eigenvalues $\pm \lambda_1$, $\pm \lambda_2$, $\lambda_1 > \lambda_2 > 0$). Only a resonance $\lambda_1 = n \lambda_2$ (if is present) influences on the normal form. This form allows one to get convenient almost linear estimates for solutions of the vector field using the Shilnikov's boundary value problem. Such technique is used when studying the orbit behavior near homoclinic orbits to saddle equilibria in a Hamiltonian system. The form obtained depends smoothly on parameters, if the vector field smoothly depends on parameters
Keywords: Hamiltonian, saddle, normal form, symplectic transformation, invariant manifold
Citation: Lerman L. M.,  Partial normal form near a saddle of a Hamiltonian system , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 291-297
DOI: 10.1070/RD2006v011n02ABEH000352
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
Lerman L. M.
Homo- and heteroclinic orbits, hyperbolic subsets in a one-parameter unfolding of a Hamiltonian system with heteroclinic contour with two saddle-foci
1997, vol. 2, nos. 3-4, pp.  139-155
We study a $1$-parametric family of the Hamiltonian systems with $2$ hyperbolic fixed points and analyze the structure and bifurcations of homoclinic and heteroclinic trajectories under the variation of the parameter and energy values.
Citation: Lerman L. M.,  Homo- and heteroclinic orbits, hyperbolic subsets in a one-parameter unfolding of a Hamiltonian system with heteroclinic contour with two saddle-foci, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 139-155

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