Lev Lerman
25/12 Bolshaya Pecherskaya Str., Nizhny Novgorod
Research University Higher School of Economics
Publications:
Kulagin N. E., Lerman L. M., Trifonov K. N.
Twin Heteroclinic Connections of Reversible Systems
2024, vol. 29, no. 1, pp. 4064
Abstract
We examine smooth fourdimensional vector fields reversible under some smooth
involution $L$ that has a smooth twodimensional submanifold of fixed points. Our main interest
here is in the orbit structure of such a system near two types of heteroclinic connections
involving saddlefoci and heteroclinic orbits connecting them. In both cases we found families
of symmetric periodic orbits, multiround heteroclinic connections and countable families of
homoclinic orbits of saddlefoci. All this suggests that the orbit structure near such connections
is very complicated. A nonvariational version of the stationary Swift – Hohenberg equation is
considered, as an example, where such structure has been found numerically.

Gelfreich V. G., Lerman L. M.
Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation
2014, vol. 19, no. 6, pp. 635655
Abstract
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 nonsemisimple double zero one. It is well known that a oneparameter 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 singleround 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.

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. 435460
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 saddlefocus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinitedimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicitytochaos transition, bluesky 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.

Lerman L. M., Turaev D. V.
Breakdown of Symmetry in Reversible Systems
2012, vol. 17, nos. 34, pp. 318336
Abstract
We review results on local bifurcations of codimension 1 in reversible systems (flows and diffeomorphisms) which lead to the birth of attractorrepeller pairs from symmetric equilibria (for flows) or periodic points (for diffeomorphisms).

Lerman L. M.
Breaking hyperbolicity for smooth symplectic toral diffeomorphisms
2010, vol. 15, nos. 23, pp. 194209
Abstract
We study a smooth symplectic 2parameter unfolding of an almost hyperbolic diffeomorphism on twodimensional 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 period2 point exists for all parameter values close to the codimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period2 point which occurs along a sequence of parameter values accumulating at the codimension 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 nonuniformly hyperbolic.

Lerman L. M., Markova A. P.
On Stability at the Hamiltonian Hopf Bifurcation
2009, vol. 14, no. 1, pp. 148162
Abstract
For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and nonsemisimple 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 oneparameter 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.

Lerman L. M.
Partial normal form near a saddle of a Hamiltonian system
2006, vol. 11, no. 2, pp. 291297
Abstract
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

Gonchenko S. V., Lerman L. M., Turaev D. V.
Leonid Pavlovich Shilnikov. On his 70^{th} birthday
2006, vol. 11, no. 2, pp. 139140
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 31February 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. 
Lerman L. M.
Homo and heteroclinic orbits, hyperbolic subsets in a oneparameter unfolding of a Hamiltonian system with heteroclinic contour with two saddlefoci
1997, vol. 2, nos. 34, pp. 139155
Abstract
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.
