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Institute
Partial synchronous states exist in systems of coupled oscillators between full synchrony and asynchrony. They are an important research topic because of their variety of different dynamical states. Frequently, they are studied using phase dynamics. This is a caveat, as phase dynamics are generally obtained in the weak coupling limit of a first-order approximation in the coupling strength. The generalization to higher orders in the coupling strength is an open problem. Of particular interest in the research of partial synchrony are systems containing both attractive and repulsive coupling between the units. Such a mix of coupling yields very specific dynamical states that may help understand the transition between full synchrony and asynchrony. This thesis investigates partial synchronous states in mixed-coupling systems. First, a method for higher-order phase reduction is introduced to observe interactions beyond the pairwise one in the first-order phase description, hoping that these may apply to mixed-coupling systems. This new method for coupled systems with known phase dynamics of the units gives correct results but, like most comparable methods, is computationally expensive. It is applied to three Stuart-Landau oscillators coupled in a line with a uniform coupling strength. A numerical method is derived to verify the analytical results. These results are interesting but give importance to simpler phase models that still exhibit exotic states. Such simple models that are rarely considered are Kuramoto oscillators with attractive and repulsive interactions. Depending on how the units are coupled and the frequency difference between the units, it is possible to achieve many different states. Rich synchronization dynamics, such as a Bellerophon state, are observed when considering a Kuramoto model with attractive interaction in two subpopulations (groups) and repulsive interactions between groups. In two groups, one attractive and one repulsive, of identical oscillators with a frequency difference, an interesting solitary state appears directly between full and partial synchrony. This system can be described very well analytically.
Selfsustained oscillations are some of the most commonly observed phenomena in biological systems. They emanate from non-linear systems in a heterogeneous environment and can be described by the theory of dynamical systems. Part of this theory considers reduced models of the oscillator dynamics by means of amplitudes and a phase variable. Such variables are highly attractive for theoretical and experimental studies. Theoretically these variables correspond to an integrable linearization of the generally non-linear system. Experimentally, there exist well established approaches to extract phases from oscillator signals. Notably, one can define phase models also for networks of oscillators. One highly active field examines effects of non-local coupling among oscillators, which is thought to play a key role in networks with strong coupling. The dissertation introduces and expands the knowledge about high-order phase coupling in networks of oscillators. Mathematical calculations consider the Stuart-Landau oscillator. A novel phase estimation scheme for direct observations of an oscillator dynamics is introduced based on numerics. A numerical study of high-order phase coupling applies a Fourier fit for the Stuart-Landau and for the van-der-Pol oscillator. The numerical approach is finally tested on observation-based phase estimates of the Morris-Lecar neuron. A popular approach for the construction of phases from signals is based on phase demodulation by means of the Hilbert transform. Generally, observations of oscillations contain a small and generic variation of their amplitude. The work presents a way to quantify how much the variations of signal amplitude spoil a phase demodulation procedure. For the ideal case of phase modulated signals, amplitude modulations vanish. However, the Hilbert transform produces artificial variations of the reconstructed amplitude even in this case. The work proposes a novel procedure called Iterative Hilbert Transform Embedding to obtain an optimal demodulation of signals. The text presents numerous examples and tests of application for the method, covering multicomponent signals, observables of highly stable limit cycle oscillations and noisy phase dynamics. The numerical results are supported by a spectral theory of convergence for weak phase modulations.
The Earth's inner magnetosphere is a very dynamic system, mostly driven by the external solar wind forcing exerted upon the magnetic field of our planet. Disturbances in the solar wind, such as coronal mass ejections and co-rotating interaction regions, cause geomagnetic storms, which lead to prominent changes in charged particle populations of the inner magnetosphere - the plasmasphere, ring current, and radiation belts. Satellites operating in the regions of elevated energetic and relativistic electron fluxes can be damaged by deep dielectric or surface charging during severe space weather events. Predicting the dynamics of the charged particles and mitigating their effects on the infrastructure is of particular importance, due to our increasing reliance on space technologies.
The dynamics of particles in the plasmasphere, ring current, and radiation belts are strongly coupled by means of collisions and collisionless interactions with electromagnetic fields induced by the motion of charged particles. Multidimensional numerical models simplify the treatment of transport, acceleration, and loss processes of these particles, and allow us to predict how the near-Earth space environment responds to solar storms. The models inevitably rely on a number of simplifications and assumptions that affect model accuracy and complicate the interpretation of the results. In this dissertation, we quantify the processes that control electron dynamics in the inner magnetosphere, paying particular attention to the uncertainties of the employed numerical codes and tools.
We use a set of convenient analytical solutions for advection and diffusion equations to test the accuracy and stability of the four-dimensional Versatile Electron Radiation Belt (VERB-4D) code. We show that numerical schemes implemented in the code converge to the analytical solutions and that the VERB-4D code demonstrates stable behavior independent of the assumed time step. The order of the numerical scheme for the convection equation is demonstrated to affect results of ring current and radiation belt simulations, and it is crucially important to use high-order numerical schemes to decrease numerical errors in the model.
Using the thoroughly tested VERB-4D code, we model the dynamics of the ring current electrons during the 17 March 2013 storm. The discrepancies between the model and observations above 4.5 Earth's radii can be explained by uncertainties in the outer boundary conditions. Simulation results indicate that the electrons were transported from the geostationary orbit towards the Earth by the global-scale electric and magnetic fields.
We investigate how simulation results depend on the input models and parameters. The model is shown to be particularly sensitive to the global electric field and electron lifetimes below 4.5 Earth's radii. The effects of radial diffusion and subauroral polarization streams are also quantified.
We developed a data-assimilative code that blends together a convection model of energetic electron transport and loss and Van Allen Probes satellite data by means of the Kalman filter. We show that the Kalman filter can correct model uncertainties in the convection electric field, electron lifetimes, and boundary conditions. It is also demonstrated how the innovation vector - the difference between observations and model prediction - can be used to identify physical processes missing in the model of energetic electron dynamics.
We computed radial profiles of phase space density of ultrarelativistic electrons, using Van Allen Probes measurements. We analyze the shape of the profiles during geomagnetically quiet and disturbed times and show that the formation of new local minimums in the radial profiles coincides with the ground observations of electromagnetic ion-cyclotron (EMIC) waves. This correlation indicates that EMIC waves are responsible for the loss of ultrarelativistic electrons from the heart of the outer radiation belt into the Earth's atmosphere.