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]

Membrane mediated aggregation of curvature inducing nematogens and membrane tubulation

The shapes of cell membranes are largely regulated by membrane associated, curvature active, proteins. We use a numerical model of the membrane with elongated membrane inclusions, recently developed by us, which posses spontaneous directional curvatures that could be different along and perpendicular to its long axis. We show that, due to membrane mediated interactions these curvature inducing membrane nematogens can oligomerize spontaneously, even at low concentrations, and change the local shape of the membrane. We demonstrate that for a large group of such inclusions, where the two spontaneous curvatures have equal sign, the tubular conformation and sometime the sheet conformation of the membrane are the common equilibrium shapes. We elucidate the factors necessary for the formation of these {\it protein lattices}. Furthermore, the elastic properties of the tubes, like their compressional stiffness and persistence length are calculated. Finally, we discuss the possible role of nematic disclination in capping and branching of the tubular membranes.Comment: 15pages, 8 figure

Finite Wavelength Instabilities in a Slow Mode Coupled Complex Ginzburg-Landau Equation

In this Letter, we discuss the effect of slow real modes in reaction-diffusion systems close to a supercritical Hopf bifurcation. The spatiotemporal effects of the slow mode cannot be captured by traditional descriptions in terms of a single complex Ginzburg-Landau equation (CGLE). We show that the slow mode coupling to the CGLE introduces a novel set of finite wavelength instabilities not present in the CGLE. For spiral waves, these instabilities highly affect the location of regions for convective and absolute instability. These new instability boundaries are consistent with transitions to spatiotemporal chaos found by simulation of the corresponding coupled amplitude equations

Composition of volatile compounds in bovine milk heat treated by instant infusion pasteurization and correlation to sensory analysis

Volatile compounds in skim milk and nonstandardised milk subjected to instant infusion pasteurisation at 80°C, 100°C and 120°C were compared with raw milk, high temperature short time pasteurised milk and milk pasteurised at 85°C/30 s. The composition of volatile compounds differed between infusion pasteurisation treated samples and the reference pasteurisations. The sensory properties of skim milk subjected to instant infusion pasteurisation were described by negative attributes, such as cardboard sour and plastic flavours, which are not associated normally with fresh milk. Partial least squares modelling showed good correlation between the volatile compounds and the sensory properties, indicating the predictive and possible causal importance of the volatile compounds for the sensory characteristics

Investigation of the XCAT phantom as a validation tool in cardiac MRI tracking algorithms.

PURPOSE: To describe our magnetic resonance imaging (MRI) simulated implementation of the 4D digital extended cardio torso (XCAT) phantom to validate our previously developed cardiac tracking techniques. Real-time tracking will play an important role in the non-invasive treatment of atrial fibrillation with MRI-guided radiosurgery. In addition, to show how quantifiable measures of tracking accuracy and patient-specific physiology could influence MRI tracking algorithm design. METHODS: Twenty virtual patients were subjected to simulated MRI scans that closely model the proposed real-world scenario to allow verification of the tracking technique's algorithm. The generated phantoms provide ground-truth motions which were compared to the target motions output from our tracking algorithm. The patient-specific tracking error, ep, was the 3D difference (vector length) between the ground-truth and algorithm trajectories. The tracking errors of two combinations of new tracking algorithm functions that were anticipated to improve tracking accuracy were studied. Additionally, the correlation of key physiological parameters with tracking accuracy was investigated. RESULTS: Our original cardiac tracking algorithm resulted in a mean tracking error of 3.7 ± 0.6 mm over all virtual patients. The two combinations of tracking functions demonstrated comparable mean tracking errors however indicating that the optimal tracking algorithm may be patient-specific. CONCLUSIONS: Current and future MRI tracking strategies are likely to benefit from this virtual validation method since no time-resolved 4D ground-truth signal can currently be derived from purely image-based studies