We study the behavior of linear nonstationary shallow water waves generated by aninstantaneous localized source as they propagate over and become trapped by elongatedunderwater banks or ridges. To find the solutions of the corresponding equations, we usean earlier-developed asymptotic approach based on a generalization of Maslov’s canonicaloperator, which provides a relatively simple and efficient analytic-numerical algorithmfor the wave field computation. An analysis of simple examples (where the bank and sourceshapes are given by certain elementary functions) shows that the appearance and dynamicsof trapped wave trains is closely related to a cascade of bifurcations of space-timecaustics, the bifurcation parameter being the bank length-to-width ratio.

We prove the existence of a global bifurcation branch of 2π-periodic,smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset ofsolutions in the global branch contains a sequence which converges uniformly to somesolution of Hölder class Cα, α < 1/2. Bifurcation formulas are given, as well as some properties along theglobal bifurcation branch. In addition, a spectral scheme for computing approximations tothose waves is put forward, and several numerical results along the global bifurcationbranch are presented, including the presence of a turning point and a ‘highest’, cuspedwave. Both analytic and numerical results are compared to traveling-wave solutions of theKdV equation.

We review recent results on stability of traveling waves in partly parabolicreaction-diffusion systems with stable or marginally stable equilibria. We explain howattention to what are apparently mathematical technicalities has led to theorems thatallow one to convert spectral calculations, which are used in the sciences and engineeringto study stability of a wave, into detailed, theoretically-based information about thebehavior of perturbations of the wave.

We display a gallery of Lorenz-like attractors that emerge in a class ofthree-dimensional maps. We review the theory of Lorenz-like attractors for diffeomorphisms(as opposed to flows), define various types of such attractors, and find sufficientconditions for three-dimensional Henon-like maps to possess pseudohyperbolic Lorenz-likeattractors. The numerically obtained scenarios of the creation and destruction of theseattractors are also presented.

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransversal heteroclinic cycles. We show that bifurcations under consideration lead to the birth of Lorenz-like attractors. They can be viewed as attractors in the Poincare map for periodically perturbed classical Lorenz attractors and hence they can allow for the existence of homoclinic tangencies and wild hyperbolic sets.

We are interested in the stability of the localized stationary solutions of a three-component reaction-diffusion system with one activator and two inhibitors. We show that depending on control parameters, solutions in form of moving and breathing localized structures can be observed in the vicinity of the codimension-two bifurcation point. We analyze this situation performing multiple scale perturbation expansion in the vicinity of the bifurcation point and derive a set of order parameter equations, explicitly describing the dynamics of the single localized structure. Numerical simulations are carried out, showing good agreement with the analytical predictions.

In this paper we study the complex dynamics of predator-prey systems with nonmonotonicfunctional response and harvesting. When the harvesting is constant-yield for prey, it isshown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopfbifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. Theexistence of two limit cycles and a homoclinic loop is established by numericalsimulations. When the harvesting is seasonal for both species, sufficient conditions forthe existence of an asymptotically stable periodic solution and bifurcation of a stableperiodic orbit into a stable invariant torus of the model are given. Numerical simulationsare carried out to demonstrate the existence of bifurcation of a stable periodic orbitinto an invariant torus and transition from invariant tori to periodic solutions,respectively, as the amplitude of seasonal harvesting increases.

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

The bifurcation to one-dimensional weakly subcritical periodic patterns is described bythe cubic-quintic Ginzburg-Landau equation

       At = µA + Axx + i(a1|A|2Ax + a2A2Ax*) + b|A|2A - |A|4A.

These periodic patterns may in turn become unstable through one of two differentmechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamicsnear the instability threshold in each of these cases using the corresponding modulationequations and compare the results with those obtained from direct numerical simulation ofthe equation. We also study the stability properties and dynamical evolution of differenttypes of fronts present in the protosnaking region of this equation. The results providenew predictions for the dynamical properties of generic systems in the weakly subcriticalregime.

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equilibriumof the saddle-center type can have a sequence of small parameter values for which aone-round homoclinic orbit to this equilibrium exists. This contrasts with the well-knownfindings by Amick and McLeod and others that solutions of such type do not exist inanalytic Hamiltonian systems, and that the separatrices are split by the exponentiallysmall quantity. We also discuss existence of homoclinic trajectories to small periodicorbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinicconnection. Our further result, illustrated by simulations, concerns the complicatedstructure of orbits related to passage through a non-smooth bifurcation of a periodicorbit.

We study the destabilization mechanism in a unidirectional ring of identical oscillators,perturbed by the introduction of a long-range connection. It is known that for ahomogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivialequilibrium undergoes a sequence of Hopf bifurcations eventually leading to thecoexistence of multiple stable periodic states resembling the Eckhaus scenario. We showthat this destabilization scenario persists under small non-local perturbations. In thiscase, the Eckhaus line is modulated according to certain resonance conditions. In the casewhen the shortcut is strong, we show that the coexisting periodic solutions split up intotwo groups. The first group consists of orbits which are unstable for all parametervalues, while the other one shows the classical Eckhaus behavior.

Building on earlier work, we have given in [29] aproof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model ofsuperconductivity and shown that the triangular lattice gives the lowest energy perlattice cell. After [29] was published, we realizedthat it proves a stronger result than was stated there. This result is recorded in thepresent paper. The proofs remain the same as in [29], apart from some streamlining.

The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to ageneral class of reaction-diffusion (RD) system with and without sub-diffusion isanalyzed. For both the case of regular and sub-diffusion, the method of matched asymptoticexpansions is used to derive an ODE characterizing the spike locations in the absence ofany 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fastinstabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem(NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibriumsolution. For a particular sub-class of the reaction kinetics, it is shown that thediscrete spectrum of this NLEP is determined by the roots of some simple transcendentalequations. From a rigorous analysis of these transcendental equations, explicit sufficientconditions are given to predict the occurrence of either Hopf bifurcations or competitioninstabilities of the two-spike quasi-equilibrium solution. The theory is illustrated forseveral specific choices of the reaction kinetics.