Hopf bifurcations in plasma layers between rigid planes in thermal magnetohydrodynamics, via a simple formula

AbstractThe phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let $$m_0$$ m 0 be the thermal conduction rest state and let $$P_r, P_m, R, Q$$ P r , P m , R , Q , be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of $$m_0$$ m 0 via Hopf bifurcation can occur only in a plasma with $$P_m>P_r$$ P m > P r , in this paper it is shown that the Hopf bifurcations occur if and only if $$\begin{aligned} Q>Q_c=\displaystyle \frac{4\pi ^2[1+P_r(\mu /2\pi )^4]}{P_m-P_r}, \end{aligned}$$ Q > Q c = 4 π 2 [ 1 + P r ( μ / 2 π ) 4 ] P m - P r , with $$\mu =7.8532$$ μ = 7.8532 . Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be $$\displaystyle \frac{1}{4}$$ 1 4 of the previous value

Stability and absorbing set of parabolic chemotaxis model of Escherichia coli

This paper is devoted to model (1) for escherichia coli, introduced in [1]. Based on the experimental observations of Budrene and Berg [2, 3], Tyson and coworkers derived (1) with n cell density, c chemotrattactant concentration and s stimulant concentration. Our aim is to study the stability of constant meaning full solution and ultimately boundedness of the solutions. Precisely: (i) linear and nonlinear stability is proved by using a peculiar Lyapunov function, (ii) the ultimately boundedness of the solutions in the L2-norm is obtained, (iii) conditions guaranteeing the global stability are also obtained

Oscillation thresholds via the novel MBR method with application to oncolytic virotherapy

Oncolytic virotherapy is a therapy for the treatment of malignant tumours. In some undesirable cases, the injection of viral particles can lead to stationary oscillations, thus preventing the full destruction of the tumour mass. We investigate the oscillation thresholds in a model for the dynamics of a tumour under treatment with an oncolytic virus. To this aim, we employ the minimum bifurcation roots (MBR) method, which is a novel approach to determine the existence and location of Hopf bifurcations. The application to oncolytic virotherapy confirms how this approach may be more manageable than classical methods based on the Routh–Hurwitz criterion. In particular, the MBR method allows to explicitly identify a range of values in which the oscillation thresholds fall

Event-related potential brain correlates of episodic and semantic memory in adults with autism spectrum disorder

The aim of this thesis was to refine current models of memory in Autism Spectrum Disorder (ASD) and elucidate similarities and differences between memory in ASD and that of Typically Developing (TD) individuals. This thesis built upon recently established behavioural patterns of preserved semantic and diminished episodic memory in ASD, by linking these with established patterns of electrophysiological activity in different cortical regions of the brain in TD individuals. Event-Related Potential (ERP) Old-New effects (enhanced positivity in the brain for correctly recognised Old stimuli compared to correctly rejected New stimuli) were recorded for episodic and semantic memory judgments (the parietal Old-New effect 400-800 ms, and the mid-frontal Old-New effect 300-500 ms respectively) in ASD, and were compared to TD individuals in terms of (1) temporal and (2) topographical similarities and differences. A series of five experimental studies were conducted using nameable line drawings, non-nameable kaleidoscope images and words as stimuli. A combination of two experimental paradigms were used; the Remember/Know paradigm and the Inclusion/Exclusion paradigm. The current thesis demonstrates that recognition memory Old-New effects are attenuated in ASD for word stimuli and nameable line drawings, but that the topography remains comparable between groups. The current findings suggest that although episodic memory is quantitatively diminished in ASD during behavioural tests, residual episodic memory judgements are phenomenologically similar to TD individuals. Despite demonstrating quantitatively preserved semantic memory, different patterns of neurophysiology are observed for these judgements in ASD, suggesting that different processes are engaged in this group. Finally, although individuals with ASD demonstrated diminished Old-New effects for nameable line drawing and word stimuli compared to TD individuals, enhanced Old-New effects were observed for non-nameable and novel kaleidoscope images in ASD. These findings have important implications for our understanding of verbal mediation and its relation to thought in ASD. The findings are interpreted within a theoretical account of diminished episodic memory and verbal processing difficulties in ASD.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Su alcuni recenti risultati di stabilità di moti fluidi

Nella presente tesi sono contenuti alcuni recenti risultati di stabilità per moti fluidi fisicamente significativi. Essa si compone di due parti. Laprima (Capitoli I,II,III) è dedicata all'esposizione delle teorie introduttive. La seconda (Capitoli IV, V,VI,VII) é dedicata alla descrizione dei contributi che in tale ambito sono stati ottenuti in alcuni lavori originali

A new approach to nonlinear L^2-stability of double diffusive convection in porous media: necessary and sufficient conditions for global stability via a linearization principle

AbstractA new approach to nonlinear L2-stability for double diffusive convection in porous media is given. An auxiliary system Σ of PDEs and two functionals V, W are introduced. Denoting by L and N the linear and nonlinear operators involved in Σ, it is shown that Σ-solutions are linearly linked to the dynamic perturbations, and that V and W depend directly on L-eigenvalues, while (along Σ) dVdt and dWdt not only depend directly on L-eigenvalues but also are independent of N. The nonlinear L2-stability (instability) of the rest state is reduced to the stability (instability) of the zero solution of a linear system of ODEs. Necessary and sufficient conditions for general, global L2-stability (i.e. absence of regions of subcritical instabilities for any Rayleigh number) are obtained, and these are extended to cover the presence of a uniform rotation about the vertical axis

Heat and mass transfer by convection in multicomponent Navier-Stokes mixtures: absence of subcritical instabilities and global nonlinear stability via the Auxiliary System Method

Because of its great geophysical relevance (engineering geology, volcanism, subsurface fluid motions) and the frequent applications (industrial processes, crystal growth, thermal engineering, air and water pollution) in the past as nowadays, the heat and mass transfer by convection in horizontal layers has attracted the attention of many scientists. In the present paper, this problem is investigated in the general case of a horizontal layer L-filled by a Navier-Stokes multicomponent fluid mixture-heated from below and salted (partly from below and partly from above) by m a N salts S1; S2; Sm. Generalizing the Auxiliary System Method (AS Method), recently introduced for the Darcy fluid mixtures in porous layers [32]-[34], it is shown that: i) for each Fourier component of the perturbation fields there exists an own nonlinear evolution system (auxiliary system); ii) via the auxiliary system, a linearization principle can be obtained; iii) the absence of subcritical instabilities and the property of the linear stability conditions to guarantee also the global nonlinear L2-stability hold; iv) the Routh-Hurwitz stability conditions are characterized Em a N and handled for ma2; v) the looking for hidden symmetries and skew-symmetries allows to guarantee- via simple algebraic conditions in closed form-the global nonlinear stability