Variational integrators of higher order for constrained dynamical systems

The variational integrators presented in [5] are applied to systems with holonomic constraints, yielding constrained higher order variational integrators that are an extension of the constrained Galerkin methods in [4]. The construction of the integrators bases on a discrete version of Hamilton's principle. The inheritance of qualitative properties associated to the solution of the dynamical system to the discrete solution is analysed. Furthermore, the convergence order of the integrators and the computational efficiency is investigated numerically. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hopf bifurcations in dynamical systems

The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

     

Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form

Discretizing dynamical systems with generalized Hopf bifurcations

We consider the discretizations of parameter-dependent, continuous-time dynamical systems. We show that the general one-step methods shift a generalized Hopf bifurcation and turn it into a generalized Neimark–Sacker point. We analyze the effect of discretization methods on the emanating Hopf curve. In particular, we obtain estimates for the eigenvalues of the discretized system along this curve. A detailed analysis of the discretized first Lyapunov coefficient is also given. The results are illustrated by a numerical example. Dynamical consequences are discussed.     

On flow switching bifurcations in discontinuous dynamical systems

In paper, the sliding dynamics on the separation boundary is discussed based on the set-valued vector field theory. From vector fields in the neighborhood of a specific separation boundary, the passability of the flow from the one domain into another one is further discussed. The switching bifurcation conditions from the passable boundary to the non-passable boundary are developed. The sliding flow fragmentation on the separation boundary surface is also presented. The normal vector product field function is introduced to determine the switching bifurcation and sliding fragmentation.     

Linear dynamical systems of higher genus

A class of linear systems which after ordinary linear systems are in a certain sense the simplest ones, is associated with every algebraic function field. From the standpoint developed in this paper ordinary linear systems are associated with the rational function field.     

Construction and analysis of higher order variational integrators for dynamical systems with holonomic constraints

In this work, variational integrators of higher order for dynamical systems with holonomic constraints are constructed and analyzed. The construction is based on approximating the configuration and the Lagrange multiplier via different polynomials. The splitting of the augmented Lagrangian in two parts enables the use of different quadrature formulas to approximate the integral of each part. Conditions are derived that ensure the linear independence of the higher order constrained discrete Euler-Lagrange equations and stiff accuracy. Time reversibility is investigated for the discrete flow on configuration level only as for the flow on configuration and momentum level. The fulfillment of the hidden constraints plays an important role for the time reversibility of the presented integrators. The order of convergence is investigated numerically. Order reduction of the momentum and the Lagrange multiplier compared to the order of the configuration occurs in general, but can be avoided by fulfilling the hidden constraints in a simple post processing step. Regarding efficiency versus accuracy a numerical analysis yields that higher orders increase the accuracy of the discrete solution substantially while the computational costs decrease. A comparison to the constrained Galerkin methods in Marsden and West (Acta Numerica 10, 357–514 2001) and the symplectic SPARK integrators of Jay (SIAM Journal on Numerical Analysis 45(5), 1814–1842 2007) reveals that the approach presented here is more general and thus allows for more flexibility in the design of the integrator.     

Stability radii of higher order linear difference systems under multi-perturbations

We study stability radii of higher order linear difference systems under multi-perturbations. A formula for complex stability radius of higher order linear difference systems under multi-perturbations is given. Then, for the class of positive systems, we prove that the complex stability radius and real stability radius of the system under multi-perturbations coincide and they are computed via a simple formula. These are extensions of corresponding results of Hinrichsen and Son, Hinrichsen et al., Ngoc and Son, and Pappas and Hinrichsen. An example is given to illustrate the obtained results.     

Nonsmooth bifurcations of equilibria in planar continuous systems

In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.     

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.     

First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

We generalize the proof of Karamata’s Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of uniform dual ergodicity for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series which, to our knowledge, has not previously been covered.     

Stability radii of positive higher order difference systems under fractional perturbations

In this paper we study stability radii of positive higher order difference systems under fractional perturbations and affine perturbations of the coefficient matrices. It is shown that real and complex stability radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

Hypoellipticity and higher order Levi conditions

We study the C^∞$C^{\infty }$-hypoellipticity for a class of double characteristic operators with symplectic characteristic manifold, in the case the classical condition of minimal loss of derivatives is violated.     

Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity

We investigate the KdV like equation with higher order nonlinearity ut + a(1 +bun)unux + uxxx = 0with n ≥ 1, a, b ∈ R and α≠ 0. The bifurcations and explicit expressions of solitary wave solutions for theequation are discussed by using the bifurcation method and qualitative theory of dynamical systems. Thebifurcation diagrams, existence and number of the solitary waves are given.     

Strong approximation of partial sums under dependence conditions with application to dynamical systems

In this paper, we obtain precise rates of convergence in the strong invariance principle for stationary sequences of real-valued random variables satisfying weak dependence conditions including strong mixing in the sense of Rosenblatt (1956) [30] as a special case. Applications to unbounded functions of intermittent maps are given.     

Local existence for quasilinear parabolic systems under nonlinear boundary conditions

Summary We prove local solvability of quasilinear parabolic systems by means of classical techniques based upon a priori estimates, without assuming any growth condition.     

Local singularities of dynamical systems with shock interactions

Local (qualitative) singularities of a special class of dynamical systems with shock interactions are classified. For the first four of the specified types of local singularities, certain properties of the qualitative structure are described and used to establish the topological equivalence of the corresponding singularities.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 531–542, October, 1998.