Systematic Derivation of Amplitude Equations and Normal Forms for Dynamical Systems

We present a systematic approach to deriving normal forms and related amplitude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general, explicit recurrence relation that completely determines the amplitude equation and the associated transformation from amplitudes to physical space. At any order, the relation provides explicit expressions for all the nonvanishing coefficients of the amplitude equation together with straightforward linear equations for the coefficients of the transformation. The recurrence relation therefore provides all the machinery needed to solve a given physical problem in physical terms through an amplitude equation. The new result applies to any local bifurcation of a flow or map for which all the critical eigenvalues are semisimple i.e. have Riesz index unity). The method is an efficient and rigorous alternative to more intuitive approaches in terms of multiple time scales. We illustrate the use of the method by deriving amplitude equations and associated transformations for the most common simple bifurcations in flows and iterated maps. The results are expressed in tables in a form that can be immediately applied to specific problems.Comment: 40 pages, 1 figure, 4 tables. Submitted to Chaos. Please address any correspondence by email to [email protected]

Regional Growth Policy in Denmark - An Assessment of the Role of Innovation As an Instrument in Regional Policy

A recent study for the Danish government has identified innovation as one of the major drivers of regional competitiveness in Denmark. Innovation and the capacity to innovate are crucial factors in the development of a firm and its ability to adapt to changes in the external environment. In particular changes in the international production system with increasing out-sourcing of physical production from Western Europe has highlighted the need for an alternative economic base in many regions. As a consequence, attention has been on the role of innovation policy in economic policy in general and regional development in particular. The aim of this paper is to analyze the interaction between the actors in the innovative environment (i.e. the firm, advisory and research institution) and the external environment as a part of a broader network of innovative relations covering intra-firm as well as extra-firm relations and processes. The project covers the following aspects: • In the first part of the paper concepts and policies of innovation are discussed with regard to their ability to move the economy toward higher growth. • The second section provides a brief overview of regional convergence and disparities in Denmark in the last decade, and compares with the trends in a broader European perspective. • The next section summarizes the findings of a recent study of the regional system of innovation in Western Denmark, and provides a critical review of the role of innovation in the process of economic restructuring in the perspective of growing internationalization in many branches. Based on this assessment the future perspectives of regional policy in Denmark are discussed on the background of the ongoing reorganization of local and regional government in general, and the introduction of five regional growth-forums in particular.

Regional impacts of economic transition: From manufacturing to service and knowledge based development: Long term trends and recent Danish experiences in the Wind Energy industry.

Most mature industrial economies faced the challenge of severe structural changes in the last decades. Traditional manufacturing moved from the metropolitan and central regions to the periphery. Later the value chain in most industries changed toward knowledge intensive and service activities. This tendency has been reinforced by the trend toward globalization and recently by the impacts of the financial crisis. The aim of the paper is to sketch the long term trends of regional and industrial development in Denmark; and to identify the drivers of economic and geographical change in the production system Denmark belongs to. The starting hypothesis is that the long term transition is driven by a combination of domestic and international factors. The first part provides an overview of the Danish economy in an international perspective focusing on structural change (i.e. the relative size of primary, secondary and tertiary activities) and the impacts of the internationalization. The second section provides an analysis of the regional structural change in Denmark with special attention on the impacts of globalization and the changes in the international production system. The main focus is on the fact that the global financial crisis seemingly has been a catalyst of a process of change, with probably irreversible impacts on the geography of economic activities in Denmark. The third part digs deeper into the processes of regional and functional transformation based on a case study of the Danish wind power industry facing fierce global competition and the challenges of international relocation, with heavy regional consequences for employment. Regional impacts in particular in Western Denmark have been significant both with regards to employment and value-added. The functional and competence profile of the industry in Denmark has transformed, reducing the content of physical production. Currently two tendencies are identified; the move from a production to a skill and knowledge based industry integrated in a global value chain, and an increasing importance of construction and service provision for production capacities in offshore locations. Keywords: International interdependence – regional and structural change – a new functional division of production – toward a service based economy

Radial sine-Gordon kinks as sources of fast breathers

We consider radial sine-Gordon kinks in two, three and higher dimensions. A full two dimensional simulation showing that azimuthal perturbations remain small allows to reduce the problem to the one dimensional radial sine-Gordon equation. We solve this equation on an interval $[r_0,r_1]$ and absorb all outgoing radiation. Before collision the kink is well described by a simple law derived from the conservation of energy. In two dimensions for $r_0 \le 2$, the collision disintegrates the kink into a fast breather while for $r_0 \ge 4$ we obtain a kink-breather meta-stable state where breathers are shed at each kink "return". In three and higher dimensions $d$ a kink-pulson state appears for small $r_0$. The three states then exist as shown by a study of the $(d,r_0)$ parameter space. On the application side, the kink disintegration opens the way for new types of terahertz microwave generators

Control of Multi-level Voltage States in a Hysteretic SQUID Ring-Resonator System

In this paper we study numerical solutions to the quasi-classical equations of motion for a SQUID ring-radio frequency (rf) resonator system in the regime where the ring is highly hysteretic. In line with experiment, we show that for a suitable choice of of ring circuit parameters the solutions to these equations of motion comprise sets of levels in the rf voltage-current dynamics of the coupled system. We further demonstrate that transitions, both up and down, between these levels can be controlled by voltage pulses applied to the system, thus opening up the possibility of high order (e.g. 10 state), multi-level logic and memory.Comment: 8 pages, 9 figure

Switching between dynamic states in intermediate-length Josephson junctions

The appearance of zero-field steps (ZFS’s) in the current-voltage characteristics of intermediate-length overlap-geometry Josephson tunnel junctions described by a perturbed sine-Gordon equation (PSGE) is associated with the growth of parametrically excited instabilities of the McCumber background curve (MCB). A linear stability analysis of a McCumber solution of the PSGE in the asymptotic linear region of the MCB and in the absence of magnetic field yields a Hill’s equation which predicts how the number, locations, and widths of the instability regions depend on the junction parameters. A numerical integration of the PSGE in terms of truncated series of time-dependent Fourier spatial modes verifies that the parametrically excited instabilities of the MCB evolve into the fluxon oscillations characteristic of the ZFS’s. An approximate analysis of the Fourier mode equations in the presence of a small magnetic field yields a field-dependent Hill’s equation which predicts that the major effect of such a field is to reduce the widths of the instability regions. Experimental measurements on Nb-NbxOy-Pb junctions of intermediate length, performed at different operating temperatures in order to vary the junction parameters and for various magnetic field values, verify the physical existence of switching from the MCB to the ZFS’s. Good qualitative, and in many cases quantitative, agreement between analytic, numerical, and experimental results is obtained

Close binary evolution. III. Impact of tides, wind magnetic braking, and internal angular momentum transport

Massive stars with solar metallicity lose important amounts of rotational angular momentum through their winds. When a magnetic field is present at the surface of a star, efficient angular momentum losses can still be achieved even when the mass-loss rate is very modest, at lower metallicities, or for lower-initial-mass stars. In a close binary system, the effect of wind magnetic braking also interacts with the influence of tides, resulting in a complex evolution of rotation. We study the interactions between the process of wind magnetic braking and tides in close binary systems. We discuss the evolution of a 10 M$_\odot$ star in a close binary system with a 7 M$_\odot$ companion using the Geneva stellar evolution code. The initial orbital period is 1.2 days. The 10 M$_\odot$ star has a surface magnetic field of 1 kG. Various initial rotations are considered. We use two different approaches for the internal angular momentum transport. In one of them, angular momentum is transported by shear and meridional currents. In the other, a strong internal magnetic field imposes nearly perfect solid-body rotation. The evolution of the primary is computed until the first mass-transfer episode occurs. The cases of different values for the magnetic fields and for various orbital periods and mass ratios are briefly discussed. We show that, independently of the initial rotation rate of the primary and the efficiency of the internal angular momentum transport, the surface rotation of the primary will converge, in a time that is short with respect to the main-sequence lifetime, towards a slowly evolving velocity that is different from the synchronization velocity. (abridged).Comment: 11 pages, 13 figures, accepted for publication in Astronomy and Astrophysic