Stability Analysis Of Stationary Solutions Research Articles

Nonlinear Stability Analysis of Stationary Solutions for a Special Class of Reaction-Diffusion Systems with Respect to Small Perturbations

Nonlinear Stability Analysis of Stationary Solutions for a Special Class of Reaction-Diffusion Systems with Respect to Small Perturbations

Emergence of traveling waves and their stability in a free boundary model of cell motility

We consider a 2D free boundary model of cell motility, inspired by the 1D contraction-driven cell motility model due to P. Recho, T. Putelat, and L. Truskinovsky [Phys. Rev. Lett. 111 (2013), p. 108102]. The key ingredients of the model are the Darcy law for overdamped motion of the acto-myosin network, coupled with the advection-diffusion equation for myosin density. These equations are supplemented with the Young-Laplace equation for the pressure and no-flux condition for the myosin density on the boundary, while evolution of the boundary is subject to the acto-myosin flow at the edge. The focus of the work is on stability analysis of stationary solutions and translationally moving traveling wave solutions. We study stability of radially symmetric stationary solutions and show that at some critical radius a pitchfork bifurcation occurs, resulting in emergence of a family of traveling wave solutions. We perform linear stability analysis of these latter solutions with small velocities and reveal the type of bifurcation (sub- or supercritical). The main result of this work is an explicit asymptotic formula for the stability determining eigenvalue in the limit of small traveling wave velocities.

A POD Reduced Order Method Applied to an Instability Problem of Rayleigh–Bénard Convection

The application of a proper orthogonal decomposition (POD) reduced order method to a two-dimensional instability problem of Rayleigh–Bénard convection is described in this work. The partial differential equations that model the problem along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions are numerically solved for different values of the bifurcation parameter, i.e. the Rayleigh number [Formula: see text]. The POD reduced order method considers spaces derived from different stationary solutions depending on [Formula: see text] as snapshots. The eigenvalue with the largest real part and its corresponding eigenfunction are calculated for the POD solutions. The resulting matrices of the eigenvalue problems are low-dimensional. There is a drastic reduction in the computational cost.

Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay

<p style='text-indent:20px;'>This paper is concerned with the pullback dynamics and asymptotic stability for a 3D Brinkman-Forchheimer equation with infinite delay. The well-posedness of weak solution to the 3D Brinkman-Forchheimer flow with infinite delay is investigated in the weighted space <inline-formula><tex-math id="M1">\begin{document}$ C_\kappa(H) $\end{document}</tex-math></inline-formula> firstly, then the pullback attractors are presented for the process of weak solution. Moreover, the existence of global attractor and the exponential stability analysis of stationary solutions are shown, which is based on the estimate of corresponding steady state equation.

Reduced Basis Method Applied to Eigenvalue Problems from Convection

The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems.

Inexact matrix exponential preconditioner for implicitly restarted Arnoldi method in fluid dynamics stability problems for parallel heterogeneous architecture

We wish to solve the problem of linear stability analysis of stationary solutions for a fluid dynamics problem governed by 3D Navier Stokes equations. The discrete problem is of order O(106) - O(107) so we use parallel computations to perform linear analysis. We aim at the heterogeneous computational architecture based on central (CPU) and graphical (GPU) processing units. In the paper we use the implicitly restarted Arnoldi method (IRAM) to recover the most dangerous eigenvalues. The IRAM has poor convergence without specific transformation of the matrix spectrum. Standard methods require either long computations or inverse of the transformed matrix. This can be troublesome in fluid dynamics problems where matrix is not available explicitly and is non-symmetric, ill-conditioned and can be fully filled. We propose the usage of the inexact matrix exponential with shift-inverse transformation using only matrix-vector product. The inverse is performed using Krylov methods using only matrix-vector product. We prove convergence bounds for the GMRES method and show that this type of matrix transformation accelerates GMRES convergence. We demonstrate application of the method in recovering eigenvalues close to the imaginary axis in some model problems on some matrices from the Matrix Market. We also demonstrate the application of the method for some fluid dynamics problems. We demonstrate convergence, acceleration and efficiency on the multiple GPU cluster.

Stability results for 2D Navier–Stokes equations with unbounded delay

Some results related to 2D Navier–Stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed. First, the existence and uniqueness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solution is then established by means of the Lax–Milgram theorem and the Schauder fixed point theorem. The local stability analysis of stationary solutions is studied by several different methods: the classical Lyapunov function method, the Razumikhin–Lyapunov technique and by constructing appropriate Lyapunov functionals. Finally, we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay. Exponential stability in this infinite delay setting remains as an open problem.

Faraday wave patterns on a square cell network

We present the experimental observations of the Faraday instability when the vibrated liquid is contained in a network of small square cells for exciting frequencies in the range $$10\le F\le 24$$ Hz. A sweep of the parameter space has been performed to investigate the amplitudes and frequencies of the driving force for which different patterns form over the network. Regular patterns in the form of square lattices are observed for driving frequencies in the range $$10\le F<14$$ Hz, while ordered matrices of oscillons are formed for $$14<F\le 23$$ Hz. At $$F>23$$ Hz, disordered periodic patterns appear within individual cells for a small range of amplitudes. In this case, the wave field is dominated by oscillating blobs that interact on the capillary–gravity scale. A Pearson correlation analysis of the recorded videos shows that for all ordered patterns, the surface waves are periodic and correspond to Faraday waves of dominant frequency equal to half the excitation frequency (i.e., $$f=F/2$$ ). In contrast, the oscillons formed for $$14<F\le 23$$ Hz are at the first subharmonic ( $$f=F/2$$ ) and first harmonic ( $$f=F$$ ) response frequencies, with higher harmonics being negligible or absent as in most cases. The disordered wave fields forming at $$F>23$$ Hz are not subharmonic and correspond to periodic harmonic waves with $$f=nF/2$$ (for $$n=2,4,\ldots $$ ). We find that the experimentally determined minimum forcing necessary to destabilize the rest state and generate surface waves is consistent with a recent stability analysis of stationary solutions as derived from a new dispersion relation for time-periodic waves with nonzero forcing and dissipation.

Stability of stationary solutions for semilinear heat equations with concave nonlinearity

This paper is concerned with the stability analysis of stationary solutions of the Cauchy–Dirichlet problem for some semilinear heat equation with concave nonlinearity. The instability of sign-changing solutions is verified under some variational assumption. Moreover, the exponential stability of the positive stationary solution at an optimal rate is proved by exploiting a super–subsolution method as well as the parabolic regularity theory. The base of our analysis relies on the linearization of the equation at each stationary solution and spectral analysis of the corresponding linearized operator. The main difficulties reside in the singularity of the linearized operator due to the concave nonlinearity.

Domain of oscillatory growth in directional solidification of dilute binary alloys

The oscillatory growth of a dilute binary alloy has recently been described by a nonlinear oscillator equation that applies to small temperature gradients and large growth velocities in the setup of directional solidification. Based on a one-dimensional stability analysis of stationary solutions of this equation, we explore in the present paper the complete region where the solidification front propagates in an oscillatory way. The boundary of this region is calculated exactly, and the nature of the oscillations is evaluated numerically in several segments of the region.

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either D A ( θ ) or D A ( θ , ∞ ) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ * then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ < γ * then this stationary solution is unstable.

Extended Lindstedt–Poincare method for non-stationary resonances of dynamical systems with cubic nonlinearities

This paper presents the extended Lindstedt–Poincare (EL–P) method, which applies multiple time variables to treat non-stationary oscillations arising in dynamical systems with cubic nonlinearities due to the slowly varied excitation parameters. The method is applied extensively in research of non-stationary vibrations of clamped-hinged beams. Recognizing the aperiodic nature of non-stationary oscillations, the new formulation is presented by adding an additional, slow time scale beside time scales of the nonlinear system, which generally correspond to the incommensurate nonlinear frequencies of the response. Using this concept, a generalized approach of the study to the passage through fundamental, superharmonic and subharmonic resonances is presented in the paper. Effects of slowly varying excitation frequency and slowly varying excitation amplitude on the non-stationary oscillations are studied with the computation of deviations from the stationary response. Although the method is formulated for N-dof dynamical systems having weak cubic nonlinearities, it is applied for non-stationary vibrations, where two-mode shape approximation of damped and undamped clamped-hinged beam, respectively, is used and the simultaneous appearance of internal resonance is taken into account. Stability analysis of stationary solutions is performed and comparisons of stationary resonance curves by results obtained with the incremental harmonic balance (IHB) method show good agreement. The passage through the fundamental resonance of damped and undamped clamped-hinged beam, respectively, is investigated in detail.

Linearized Stability Analysis of Stationary Solutions for Surface Diffusion with Boundary Conditions

The linearizedstability of stationary solutions to the surface diffusion flow with angle conditions and no-flux conditions as boundary conditions is studied. We perform a linearized stability analysis in which the H-1 -gradient flow structure plays a key role. As a byproduct our analysis also gives a criterion for the stability of critical points of the length functional of curves which come into contact with the outer boundary. Finally, we study the linearized stability of several examples.

A bifurcation study of convective heat transfer in porous media. Part II: Effect of tilt on stationary and nonstationary solutions*

In the present work the investigation of Weinitschke et al. [Phys. Fluids A 2, 912 (1990)] on the bifurcation structure of stationary, two-dimensional solutions of the Darcy–Oberbeck–Boussinesq model equations, which governs the convection heat transfer in a porous medium, is extended. The effect of imposing a symmetry breaking geometrical perturbation, viz., a tilt, on the unfolding of the bifurcation structure, is investigated first. The symmetry breaking bifurcation points that are found at zero tilt are structurally unstable to even a slight degree of tilt, and they unfold into limit points that coalesce with the neighboring limit points as the degree of tilt is increased. Two such limit points disappear through the formation of a double limit point at very small angles of tilt. On the fold curve of such limit points are found origins of paths of Hopf points, also known as the B-point singularity. Several such B points are located on the fold curves of limit points. This is helpful in pointing to regions of parameter space, where interesting dynamical behavior with oscillatory and chaotic flow structures are possible. The dynamical behavior is explored through simulation of the governing time-dependent equations after suitable spatial discretization through the Arakawa scheme. Linear stability analysis of stationary solutions indicate that such solutions remain stable over an increasing range of Rayleigh number (Ra) as the degree of tilt is increased. The first limit point (L1), below which there is a unique solution, tends toward infinite Rayleigh number as the tilt approaches 45°. At 45° tilt, a reflective symmetry about the diagonal of a square cell is restored and the stationary solutions computed by the continuation method remain stable for Ra as high as 200 000. Results from the dynamic simulation confirm this behavior. For tilts of 5° and 10° stationary solutions lose stability at Rayleigh numbers in the range of 4000–5000. Periodic solutions interspersed with regions of chaotic behavior are observed as Ra is increased continuously. For 20°, however, periodic solutions begin only above Ra≊10 000.

Algorithms for qualitative analysis of complex systems

Algorithms are proposed that construct first integrals and Lyapunov functions for stability analysis of stationary solutions of differential equations describing mechanical systems of linked bodies.