American Institute of Mathematical Sciences

Actual research concerning, in particular, the occurrence of "gap-solitons" bifurcating from the continuous spectrum confirms that this part of Bifurcation Theory that started around 40 years ago flourishes. In this lecture we review the origins of "Bifurcation from the continuous spectrum" with regard to the achievements of Jürgen Scheurle and sketch how the early results dealing with the bifurcation of singular solutions have prepared the ground for present and further developments.

The four Galilean moons of Jupiter were discovered by Galileo in the early 17th century, and their motion was first seen as a miniature solar system. Around 1800 Laplace discovered that the Galilean motion is subjected to an orbital \begin{document}$ 1{:}2{:}4 $\end{document}-resonance of the inner three moons Io, Europa and Ganymedes. In the early 20th century De Sitter gave a mathematical explanation for this in a Newtonian framework. In fact, he found a family of stable periodic solutions by using the seminal work of Poincaré, which at the time was quite new. In this paper we review and summarize recent results of Broer, Hanßmann and Zhao on the motion of the entire Galilean system, so including the fourth moon Callisto. To this purpose we use a version of parametrised Kolmogorov-Arnol'd-Moser theory where a family of multi-periodic isotropic invariant three-dimensional tori is found that combines the periodic motions of De Sitter and Callisto. The \begin{document}$ 3 $\end{document}-tori are normally elliptic and excite a family of invariant Lagrangean \begin{document}$ 8 $\end{document}-tori that project down to librational motions. Both the \begin{document}$ 3 $\end{document}- and the \begin{document}$ 8 $\end{document}-tori occur for an almost full Hausdorff measure set in the product of corresponding dimension in phase space and a parameter space, where the external parameters are given by the masses of the moons.

We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.

The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

This paper presents the continuous and discrete variational formulations of simple thermodynamical systems whose configuration space is a (finite dimensional) Lie group. We follow the variational approach to nonequilibrium thermodynamics developed in [12,13], as well as its discrete counterpart whose foundations have been laid in [14]. In a first part, starting from this variational formalism on the Lie group, we perform an Euler-Poincaré reduction in order to obtain the reduced evolution equations of the system on the Lie algebra of the configuration space. We obtain as corollaries the energy balance and a Kelvin-Noether theorem. In a second part, a compatible discretization is developed resulting in discrete evolution equations that take place on the Lie group. Then, these discrete equations are transported onto the Lie algebra of the configuration space with the help of a group difference map. Finally we illustrate our framework with a heavy top immersed in a viscous fluid modeled by a Stokes flow and proceed with a numerical simulation.

The forward \begin{document}$ \omega $\end{document}-limit set \begin{document}$ \omega_{\mathcal{B}} $\end{document} of a nonautonomous dynamical system \begin{document}$ \varphi $\end{document} with a positively invariant absorbing family \begin{document}$ \mathcal{B} $\end{document} \begin{document}$ = $\end{document} \begin{document}$ \{ B(t), t \in \mathbb{R}\} $\end{document} of closed and bounded subsets of a Banach space \begin{document}$ X $\end{document} which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if \begin{document}$ \varphi $\end{document} is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for a \begin{document}$ \varphi $\end{document}-invariant family \begin{document}$ \mathcal{A} $\end{document} \begin{document}$ = $\end{document} \begin{document}$ \left\{A(t), t \in \mathbb{R}\right\} $\end{document} in \begin{document}$ \mathcal{B} $\end{document} of nonempty compact subsets of \begin{document}$ X $\end{document} to be a forward attractor is generalised to this context.

In this paper we define an action of a Lie algebra on a smooth manifold. We get nearly the same results as those for group actions, when the flows of the symmetry vector fields are complete. We show that the orbit space of a Lie algebra action is a differential space. We discuss differential spaces occuring in the reduction of symmetries in integrable Hamiltonian systems.

We answer here a question posed by F. Diacu in 2012 that asked whether there exist relative equilibria on \begin{document}$ \mathbb S^2 $\end{document} and \begin{document}$ \mathbb H^2 $\end{document} that move in a plane non-perpendicular to the rotation axis. For 3-body non-geodesic ordinary central configurations on \begin{document}$ \mathbb S^2 $\end{document} and \begin{document}$ \mathbb H^2 $\end{document}, we find all relative equilibria that move in a plane perpendicular to the rotation axis. We also show that the set of shapes of 3-body non-geodesic ordinary central configurations on \begin{document}$ \mathbb S^2 $\end{document} and \begin{document}$ \mathbb H^2 $\end{document} is a 3-dimensional manifold. Then we conclude that almost all 3-body relative equilibria move in planes non-perpendicular to the rotation axis.

We study scalar delay equations

\begin{document}$ \dot{x} (t) = \lambda f(x(t-1)) + b^{-1} (x(t) + x(t -p/2)) $\end{document}

with odd nonlinearity \begin{document}$ f $\end{document}, real nonzero parameters \begin{document}$ \lambda, \, b $\end{document}, and two positive time delays \begin{document}$ 1, \ p/2 $\end{document}. We assume supercritical Hopf bifurcation from \begin{document}$ x \equiv 0 $\end{document} in the well-understood single-delay case \begin{document}$ b = \infty $\end{document}. Normalizing \begin{document}$ f' (0) = 1 $\end{document}, branches of constant minimal period \begin{document}$ p_k = 2\pi/\omega_k $\end{document} are known to bifurcate from eigenvalues \begin{document}$ i\omega_k = i(k+\tfrac{1}{2})\pi $\end{document} at \begin{document}$ \lambda_k = (-1)^{k+1}\omega_k $\end{document}, for any nonnegative integer \begin{document}$ k $\end{document}. The unstable dimension of these rapidly oscillating periodic solutions is \begin{document}$ k $\end{document}, at the local branch \begin{document}$ k $\end{document}. We obtain stabilization of such branches, for arbitrarily large unstable dimension \begin{document}$ k $\end{document}, and for, necessarily, delicately narrow regions \begin{document}$ \mathcal{P} $\end{document} of scalar control amplitudes \begin{document}$ b < 0 $\end{document}.

For \begin{document}$ p $\end{document}: = \begin{document}$ p_k $\end{document} the branch \begin{document}$ k $\end{document} of constant period \begin{document}$ p_k $\end{document} persists as a solution, for any \begin{document}$ b\neq 0 $\end{document}. Indeed the delayed feedback term controlled by \begin{document}$ b $\end{document} vanishes on branch \begin{document}$ k $\end{document}: the feedback control is noninvasive there. Following an idea of Pyragas [30], we seek parameter regions \begin{document}$ \mathcal{P} = (\underline{b}_k, \overline{b}_k) $\end{document} of controls \begin{document}$ b \neq 0 $\end{document} such that the branch \begin{document}$ k $\end{document} becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for \begin{document}$ \mathcal{P} $\end{document} in the limit of large \begin{document}$ k $\end{document}. Our analysis is based on a 2-scale covering lift for the slow and rapid frequencies involved.

These results complement earlier results in [8] which required control terms

\begin{document}$ b^{-1} (x(t-\vartheta) + x(t-\vartheta -p/2)) $\end{document}

with a third delay \begin{document}$ \vartheta $\end{document} near 1.

We analyze a simplistic model for run-and-tumble dynamics, motivated by observations of complex spatio-temporal patterns in colonies of myxobacteria. In our model, agents run with fixed speed either left or right, and agents turn with a density-dependent nonlinear turning rate, in addition to diffusive Brownian motion. We show how a very simple nonlinearity in the turning rate can mediate the formation of self-organized stationary clusters and fronts. Phenomenologically, we demonstrate the formation of barriers, where high concentrations of agents at the boundary of a cluster, moving towards the center of a cluster, prevent the agents caught in the cluster from escaping. Mathematically, we analyze stationary solutions in a four-dimensional ODE with a conserved quantity and a reversibility symmetry, using a combination of bifurcation methods, geometric arguments, and numerical continuation. We also present numerical results on the temporal stability of the solutions found here.

I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.

We review opportunities for stochastic geometric mechanics to incorporate observed data into variational principles, in order to derive data-driven nonlinear dynamical models of effects on the variability of computationally resolvable scales of fluid motion, due to unresolvable, small, rapid scales of fluid motion.

The field of sub-Riemannian geometry has flourished in the past four decades through the strong interactions between problems arising in applied science (in areas such as robotics) and questions of a pure mathematical character about the nature of space. Methods of control theory, such as controllability properties determined by Lie brackets of vector fields, the Hamilton equations associated to the Maximum Principle of optimal control, Hamilton-Jacobi-Bellman equation etc. have all been found to be basic tools for answering such questions. In this paper, we find a useful role for the vantage point of sub-Riemannian geometry in attacking a problem of interest in non-equilibrium statistical mechanics: how does one create rules for operation of micro- and nano-scale systems (heat engines) that are subject to fluctuations from the surroundings, so as to be able to do useful things such as converting heat into work over a cycle of operation? We exploit geometric optimal control theory to produce such rules selected for maximal efficiency. This is done by working concretely with a model problem, the stochastic oscillator. Essential to our work is a separation of time scales used with great efficacy by physicists and justified in the linear response regime.

We investigate a singularly perturbed, non-convex variational problem arising in material science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters. This allows us to observe the influence of the parameters on the behavior of the orbits and to study their interplay in the minimization process. Our results confirm previous analytical results such as the asymptotics of the period of minimizers predicted by Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits.

The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dynamics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area has been recently called "Dynamical Topology". As an illustration, some topological properties of the space of all transitive interval maps are described. For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". A Slovak space, as a dynamical analogue of a rigid space, is a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in the theory of dynamical systems and means the collection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness is that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

We discuss the occurrence of Poincaré-Andronov-Hopf bifurcations in parameter dependent ordinary differential equations, with no a priori assumptions on special coordinates. The first problem is to determine critical parameter values from which such bifurcations may emanate; a solution for this problem was given by W.-M. Liu. We add a few observations from a different perspective. Then we turn to the second problem, viz., to compute the relevant coefficients which determine the nature of the Hopf bifurcation. As shown by J. Scheurle and co-authors, this can be reduced to the computation of Poincaré-Dulac normal forms (in arbitrary coordinates) and subsequent reduction, but feasibility problems quickly arise. In the present paper we present a streamlined and less computationally involved approach to the computations. The efficiency and usefulness of the method is illustrated by examples.

We study a class of scalar differential equations on the circle \begin{document}$ S^1 $\end{document}. This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes \begin{document}$ \mathbb R_+ $\end{document} and \begin{document}$ \mathbb R_+ $\end{document}. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.

We consider the equation

\begin{document}$\Delta u+{{u}_{yy}}+f(x,u) = 0,\quad (x,y)\in {{\mathbb{R}}^{N}}\times \mathbb{R}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$ \end{document}

where \begin{document}$ f $\end{document} is sufficiently regular, radially symmetric in \begin{document}$ x $\end{document}, and \begin{document}$ f(\cdot,0)\equiv 0 $\end{document}. We give sufficient conditions for the existence of solutions of (1) which are quasiperiodic in \begin{document}$ y $\end{document} and decaying as \begin{document}$ |x|\to\infty $\end{document} uniformly in \begin{document}$ y $\end{document}. Such solutions are found using a center manifold reduction and results from the KAM theory. A required nondegeneracy condition is stated in terms of \begin{document}$ f_u(x,0) $\end{document} and \begin{document}$ f_{uu}(x,0) $\end{document}, and is independent of higher-order terms in the Taylor expansion of \begin{document}$ f(x,\cdot) $\end{document}. In particular, our results apply to some quadratic nonlinearities.

We consider the dynamics of a Hamiltonian particle forced by a rapidly oscillating potential in \begin{document}$ m $\end{document}-dimensional space. As alternative to the established approach of averaging Hamiltonian dynamics by reformulating the system as Hamilton-Jacobi equation, we propose an averaging technique via reformulation using the Maupertuis principle. We analyse the result of these two approaches for one space dimension. For the initial value problem the solutions converge uniformly when the total energy is fixed. If the initial velocity is fixed independently of the microscopic scale, then the limit solution depends on the choice of subsequence. We show similar results hold for the one-dimensional boundary value problem. In the higher dimensional case we show a novel connection between the Hamilton-Jacobi and Maupertuis approaches, namely that the sets of minimisers and saddle points coincide for these functionals.

The purpose of this article is to discuss two basic ideas of Henri Poincaré in the theory of dynamical systems. The first one, the recurrence theorem, got at first a lot of attention but most scientists lost interest when finding out that long timescales were involved. We will show that recurrence can be a tool to find complex dynamics in resonance zones of Hamiltonian systems; this is related to the phenomenon of quasi-trapping. To demonstrate the use of recurrence phenomena we will explore the \begin{document}$ 2:2:3 $\end{document} Hamiltonian resonance near stable equilibrium. This will involve interaction of low and higher order resonance. A second useful idea is concerned with the characteristic exponents of periodic solutions of dynamical systems. If a periodic solution of a Hamiltonian system has more than two zero characteristic exponents, this points at the existence of an integral of motion besides the energy. We will apply this idea to examples of two and three degrees-of-freedom (dof), the Hénon-Heiles (or Braun's) family and the \begin{document}$ 1:2:2 $\end{document} resonance.