144 results found
Clarke A, Turaev D, 2023, ARNOLD DIFFUSION IN MULTIDIMENSIONAL CONVEX BILLIARDS, DUKE MATHEMATICAL JOURNAL, Vol: 172, Pages: 1813-1878, ISSN: 0012-7094
Berger P, Turaev D, 2022, Generators of groups of Hamitonian maps, Publisher: arXiv
We prove that analytic Hamiltonian dynamics on tori, annuli, or Euclideanspace can be approximated by a composition of nonlinear shear maps where eachof the shears depends only on the position or only on the momentum.
Rom-Kedar V, Turaev D, 2022, Stable motions of high energy particles interacting via a repelling potential, Publisher: ArXiv
The motion of N particles interacting by a smooth repelling potential andconfined to a compact d-dimensional region is proved to be, under mildconditions, non-ergodic for all sufficiently large energies. Specifically,choreographic solutions, for which all particles follow approximately the samepath close to an elliptic periodic orbit of the single-particle system, areproved to be KAM stable in the high energy limit. Finally, it is proved thatthe motion of N repelling particles in a rectangular box is non-ergodic at highenergies for a generic choice of interacting potential: there exists aKAM-stable periodic motion by which the particles move fast only in onedirection, each on its own path, yet in synchrony with all the other parallelmoving particles. Thus, we prove that for smooth interaction potentials theBoltzmann ergodic hypothesis fails for a finite number of particles even in thehigh energy limit at which the smooth system appears to be very close to theBoltzmann hard-sphere gas.
Bakrani S, Lamb JSW, Turaev D, 2022, Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R4 with Z2-symmetry and integral of motion, Journal of Differential Equations, Vol: 327, Pages: 1-63, ISSN: 0022-0396
We consider a -equivariant flow in with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.
Valperga R, Webster K, Klein V, et al., 2022, Learning reversible symplectic dynamics, 4th Annual Learning for Dynamics and Control Conference, Publisher: PLMR
Time-reversal symmetry arises naturally as a structural property in manydynamical systems of interest. While the importance of hard-wiring symmetry isincreasingly recognized in machine learning, to date this has eludedtime-reversibility. In this paper we propose a new neural network architecturefor learning time-reversible dynamical systems from data. We focus inparticular on an adaptation to symplectic systems, because of their importancein physics-informed learning.
Li D, Li X, Shinohara K, et al., 2022, Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies, Publisher: ArXiv
We consider $C^r$ $(r=3,\dots,\infty,\omega)$ diffeomorphisms with a generichomoclinic tangency to a hyperbolic periodic point, where this point has atleast one complex (non-real) central multiplier and some explicit assumptionson central multipliers are satisfied so that the dynamics near the homoclinictangency is not effectively one-dimensional. We prove that $C^1$-robustheterodimensional cycles of co-index one appear in any generic two-parameter$C^r$-unfolding of such a tangency. These heterodimensional cycles also have$C^1$-robust homoclinic tangencies.
Yanchuk S, Wolfrum M, Pereira T, et al., 2022, Absolute stability and absolute hyperbolicity in systems with discrete time-delays, JOURNAL OF DIFFERENTIAL EQUATIONS, Vol: 318, Pages: 323-343, ISSN: 0022-0396
Kazakov A, Gonchenko S, Turaev D, et al., 2021, Leonid Shilnikov and mathematical theory of dynamical chaos, Publisher: arXiv
This Focus Issue Global Bifurcations, Chaos, and Hyperchaos Theory andApplications is dedicated to the 85th anniversary of the great mathematician,one of the founding fathers of dynamical chaos theory, Leonid PavlovichShilnikov.
Gonchenko AS, Gonchenko S, Turaev D, 2021, Doubling of invariant curves and chaos in three-dimensional diffeomorphisms, CHAOS, Vol: 31, ISSN: 1054-1500
Asaoka M, Shinohara K, Turaev D, 2021, Fast growth of the number of periodic points arising from heterodimensional connections, COMPOSITIO MATHEMATICA, Vol: 157, Pages: 1899-1963, ISSN: 0010-437X
Turaev D, 2021, A criterion for mixed dynamics in two-dimensional reversible maps, CHAOS, Vol: 31, ISSN: 1054-1500
Gonchenko S, Kazakov A, Turaev D, 2021, Wild pseudohyperbolic attractor in a four-dimensional Lorenz system, NONLINEARITY, Vol: 34, Pages: 2018-2047, ISSN: 0951-7715
Gonchenko S, Kaynov MN, Kazakov AO, et al., 2021, On methods for verification of the pseudohyperbolicity of strange attractors, IZVESTIYA VYSSHIKH UCHEBNYKH ZAVEDENIY-PRIKLADNAYA NELINEYNAYA DINAMIKA, Vol: 29, Pages: 160-185, ISSN: 0869-6632
Duca A, Joly R, Turaev D, 2020, Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall, JOURNAL OF MATHEMATICAL PHYSICS, Vol: 61, ISSN: 0022-2488
Li D, Turaev D, 2020, Persistent heterodimensional cycles in periodic perturbations of Lorenz-like attractors, NONLINEARITY, Vol: 33, Pages: 971-1015, ISSN: 0951-7715
Berger P, Turaev D, 2019, On Herman's positive entropy conjecture, Advances in Mathematics, Vol: 349, Pages: 1234-1288, ISSN: 0001-8708
We show that any area-preserving Cr-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be Cr-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every 1≤r≤∞. This proves a conjecture of Herman stating that the identity map of the disk can be C∞-perturbed to a conservative diffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be C∞-approximated by a conservative diffeomorphisms displaying a positive metric entropy (a weak version of Sinai's positive metric entropy conjecture). Finally, this sheds light onto a Herman's question on the density of Cr-conservative diffeomorphisms displaying a positive metric entropy: we show the existence of a dense set formed by conservative diffeomorphisms which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.
Clarke A, Turaev D, 2019, Arnold Diffusion in Multi-Dimensional Convex Billiards
Consider billiard dynamics in a strictly convex domain, and consider atrajectory that begins with the velocity vector making a small positive anglewith the boundary. Lazutkin proved that in two dimensions, it is impossible forthis angle to tend to zero along trajectories. We prove that such trajectoriescan exist in higher dimensions. Namely, using the geometric techniques ofArnold diffusion, we show that in three or more dimensions, assuming thegeodesic flow on the boundary of the domain has a hyperbolic periodic orbit anda transverse homoclinic, the existence of trajectories asymptoticallyapproaching the billiard boundary is a generic phenomenon in the real-analytictopology.
Neistadt A, Artemyev A, Turaev D, 2019, Remarkable charged particle dynamics near magnetic field null lines, Chaos, Vol: 29, ISSN: 1054-1500
The study of charged-particle motion in electromagnetic fields is a rich source of problems, models, and new phenomena for nonlinear dynamics. The case of a strong magnetic field is well studied in the framework of a guiding center theory, which is based on conservation of an adiabatic invariant—the magnetic moment. This theory ceases to work near a line on which the magnetic field vanishes—the magnetic field null line. In this paper, we show that the existence of these lines leads to remarkable phenomena which are new both for nonlinear dynamics in general and for the theory of charged-particle motion. We consider the planar motion of a charged particle in a strong stationary perpendicular magnetic field with a null line and a strong electric field. We show that particle dynamics switch between a slow guiding center motion and the fast traverse along a segment of the magnetic field null line. This segment is the same (in the principal approximation) for all particles with the same total energy. During the phase of a guiding center motion, the magnetic moment of particle’s Larmor rotation stays approximately constant, i.e., it is an adiabatic invariant. However, upon each traversing of the null line, the magnetic moment changes in a random fashion, causing the particle to choose a new trajectory of the guiding center motion. This results in a stationary distribution of the magnetic moment, which only depends on the particle’s total energy. The jumps in the adiabatic invariant are described by Painlevé II equation.The existence of adiabatic invariants—approximate conservation laws for systems with slow and fast motions—plays an important role in different physical theories. One of such theories is guiding center theory of motion of charged particles in a strong magnetic field. This theory is based on the adiabatic invariance of magnetic moment for the particle motion. Basic assumption of the guiding center approach is that t
Capiński MJ, Turaev D, Zgliczyński P, 2018, Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system, Nonlinearity, Vol: 31, Pages: 5410-5440, ISSN: 0951-7715
We prove that the Shimizu–Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.
Olszowiec C, Turaev D, 2018, On the non-transverse homoclinic channel of a center manifold
We consider a scenario when a stable and unstable manifolds of compact centermanifold of a saddle-center coincide. The normal form of the ODE governing thesystem near the center manifold is derived and so is the normal form of thereturn map to the neighbourhood of the center manifold. The limit dynamics ofthe return map is investigated by showing that it might take the form of aHenon-like map possessing a Lorenz-like attractor or satisfy 'cone-fieldcondition' resulting in partial hyperbolicity. We consider also motivatingexample from game theory.
Dettmann CP, Fain V, Turaev D, 2018, Splitting of separatrices, scattering maps, and energy growth for a billiard inside a time-dependent symmetric domain close to an ellipse, NONLINEARITY, Vol: 31, Pages: 667-700, ISSN: 0951-7715
Shah K, Turaev D, Gelfreich V, et al., 2017, Equilibration of energy in slow-fast systems, Proceedings of the National Academy of Sciences of the United States of America, Vol: 114, Pages: E10514-E10523, ISSN: 0027-8424
Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.
Turaev D, Gonchenko S, 2017, On three types of dynamics, and the notion of attractor, Proceedings of the Steklov Institute of Mathematics, Vol: 297, Pages: 116-137, ISSN: 0081-5438
We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.
Asaoka M, Shinohara K, Turaev D, 2017, Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics, Mathematische Annalen, Vol: 368, Pages: 1277-1309, ISSN: 0025-5831
We consider semigroup actions on the unit interval generated by strictlyincreasing $C^r$-maps. We assume that one of the generators has a pair of fixedpoints, one attracting and one repelling, and a heteroclinic orbit thatconnects the repeller and attractor, and the other generators form a robustblender, which can bring the points from a small neighborhood of the attractorto an arbitrarily small neighborhood of the repeller. This is a model settingfor partially hyperbolic systems with one central direction. We show that, under additional conditions on the non-linearity and theSchwarzian derivative, the above semigroups exhibit, $C^r$-generically for anyr, arbitrarily fast growth of the number of periodic points as a function ofthe period. We also show that a $C^r$-generic semigroup from the class underconsideration supports an ultimately complicated behavior called universaldynamics.
Gelfreich V, Turaev D, 2017, Arnold diffusion in a priori chaotic symplectic maps, Communications in Mathematical Physics, Vol: 353, Pages: 507-547, ISSN: 0010-3616
We assume that a symplectic real-analytic map has an invariant nor-mally hyperbolic cylinder and an associated transverse homoclinic cylinder. Weprove that generically in the real-analytic category the boundaries of the invariantcylinder are connected by trajectories of the map.
Li D, Turaev DV, 2017, Existence of heterodimensional cycles near shilnikov loops in systems with a Z(2) symmetry, Discrete and Continuous Dynamical Systems - Series A, Vol: 37, Pages: 4399-4437, ISSN: 1078-0947
We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a Z2 symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.
Gonchenko AS, Gonchenko SV, Kazakov AO, et al., 2017, On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled rotators, Physica D: Nonlinear Phenomena, Vol: 350, Pages: 45-57, ISSN: 0167-2789
A one-parameter family of time-reversible systems on three-dimensional torus is considered. It is shown that the dynamics is not conservative, namely the attractor and repeller intersect but not coincide. We explain this as the manifestation of the so-called mixed dynamics phenomenon which corresponds to a persistent intersection of the closure of the stable periodic orbits and the closure of the completely unstable periodic orbits. We search for the stable and unstable periodic orbits indirectly, by finding non-conservative saddle periodic orbits and heteroclinic connections between them. In this way, we are able to claim the existence of mixed dynamics for a large range of parameter values. We investigate local and global bifurcations that can be used for the detection of mixed dynamics.
Ovsyannikov II, Turaev D, 2016, Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model, Nonlinearity, Vol: 30, Pages: 115-137, ISSN: 1361-6544
We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich–Morioka–Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof for the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an open set of parameter values in a 4-parameter family of 3D Henon-like diffeomorphisms.
Giles W, Lamb J, Turaev D, 2016, On homoclinic orbits to center manifolds of elliptic-hyperbolic equilibria in Hamiltonian systems, Nonlinearity, Vol: 29, ISSN: 1361-6544
We consider a Hamiltonian system which has an elliptic-hyperbolic equilibriumwith a homoclinic loop. We identify the set of orbits which are homoclinic tothe center manifold of the equilibrium via a Lyapunov- Schmidt reductionprocedure. This leads to the study of a singularity which inherits certainstructure from the Hamiltonian nature of the system. Under non-degeneracyassumptions, we classify the possible Morse indices of this singularity,permitting a local description of the set of homoclinic orbits. We alsoconsider the case of time-reversible Hamiltonian systems.
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