Branches Of Stationary Solutions Research Articles

From a vortex gas to a vortex crystal in instability-driven two-dimensional turbulence

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal.

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth.

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.

Stability analysis of dual viscous flows saturating a vertical porous pipe

The linear stability analysis of a mixed convection viscous flow in a vertical porous pipe is here investigated. The contribution of viscous heating is assumed to be non negligible. A fully developed flow regime is assumed for the basic state. The local balance equations for this state display dual stationary solutions. The dual branches of stationary solutions are determined numerically. Since the pipe is characterised by an isothermal lateral surface, the viscous heating is the sole cause of the buoyancy force. In order to investigate the stability of the basic dual solutions, small amplitude disturbances with the form of normal modes are superposed to the basic state. The solution of the eigenvalue problem obtained allows one to determine the growth rate associated to both the basic solution branches. The sign of the growth rate determines whether the particular basic solution is stable or unstable.

Computer assisted proof of branches of stationary and periodic solutions, and Hopf bifurcations, for dissipative PDEs

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.

A Hopf Bifurcation in the Planar Navier\u2013Stokes Equations

We consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

Arbitrary-angle rotation of the polarization of a dipolar Bose-Einstein condensate

We have employed the theory of harmonically trapped dipolar Bose-Einstein condensates to examine the influence of a uniform magnetic field that rotates at an arbitrary angle to its own orientation. This is achieved by semi-analytically solving the dipolar superfluid hydrodynamics of this system within the Thomas-Fermi approximation and by allowing the body frame of the condensate's density profile to be tilted with respect to the symmetry axes of the nonrotating harmonic trap. This additional degree of freedom manifests itself in the presence of previously unknown stationary solution branches for any given dipole tilt angle. We also find that the tilt angle of the stationary state's body frame with respect to the rotation axis is a nontrivial function of the trapping geometry, rotation frequency, and dipole tilt angle. For rotation frequencies of at least an order of magnitude higher than the in-plane trapping frequency, the stationary state density profile is almost perfectly equivalent to the profile expected in a time-averaged dipolar potential that effectively vanishes when the dipoles are tilted along the ``magic angle'' 54.7 deg. However, by linearizing the fully time-dependent superfluid hydrodynamics about these stationary states, we find that they are dynamically unstable against the formation of collective modes, which we expect would result in turbulent decay.

Stationary states, dynamical stability, and vorticity of Bose-Einstein condensates in tilted rotating harmonic traps

We theoretically investigate a Bose-Einstein condensate confined by a rotating harmonic trap whose rotation axis is not aligned with any of its principal axes. The principal axes of the Thomas-Fermi density profiles of the resulting stationary solutions are found to be tilted with respect to those of the rotating trap, representing an extra degree of freedom that is associated with the existence of additional branches of stationary solutions for any given rotation axis alignment. By linearizing the time-dependent theory about the stationary states, we obtain a semi-analytical prediction of their dynamical instability at high rotation frequencies against collective modes arising from environmental perturbations. Comparing the stationary states to direct simulations of the Gross-Pitaevskii equation, we predict the nucleation of quantum vortices in the dynamically unstable rotational regime. These vortex lines are aligned along the rotation axis despite the tilting of the rotating trap although the background density profile is tilted with respect to the trapping and rotation axes.

Bifurcation branch of stationary solutions in a general predator–prey system with prey-taxis

In this paper, a general reaction–diffusive predator–prey system with prey-taxis subject to the homogeneous Neumann boundary condition is considered. Firstly, we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique. And then, choosing the prey-tactic sensitivity coefficient as bifurcation parameter, we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey-tactic sensitivity is repulsive. Moreover, we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions. Our results show that prey-taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns.

Vortex modes supported by spin–orbit coupling in a laser with saturable absorption

We introduce a system of two component two-dimensional (2D) complex Ginzburg–Landau equations with spin–orbit-coupling (SOC) describing a wide-aperture microcavity laser with saturable gain and absorption. We report families of two-component self-trapped dissipative laser solitons in this system. The SOC terms are represented by the second-order differential operators, which sets the difference, , between the vorticities of the two components. We have found stable solitons of two types: vortex–antivortex (VAV) and semi-vortex (SV) bound states, featuring vorticities and , respectively. In previous works, 2D localized states of these types were found only in models including a trapping potential, while we are dealing with the self-trapping effect in the latteraly unconfined (free-space) model. The SV states are stable in a narrow interval of values of the gain coefficients. The stability interval is broader for VAV states, and it may be expanded by making SOC stronger (although the system without SOC features a stability interval too). We have found three branches of stationary solutions of both VAV and SV types, two unstable and one stable. The latter one is an attractor, as the unstable states spontaneously transform into the stable one, while retaining vorticities of their components. Unlike previously known 2D localized states, maintained by the combination of the trapping potential and SOC, in the present system the VAV and SV complexes are stable in the absence of diffusion. In contrast with the bright solitons in conservative models, chemical potentials of the dissipative solitons reported here are positive.

Poisson–Nernst–Planck equations with steric effects — non-convexity and multiple stationary solutions

We study the existence and stability of stationary solutions of Poisson–Nernst–Planck equations with steric effects (PNP-steric equations) with two counter-charged species. We show that within a range of parameters, steric effects give rise to multiple solutions of the corresponding stationary equation that are smooth. The PNP-steric equation, however, is found to be ill-posed at the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP–Cahn–Hilliard model, show that it is well-posed and that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.

The index bundle and multiparameter bifurcation for discrete dynamical systems

We develop a K-theoretic approach to multiparameter bifurcation theory of homoclinic solutions of discrete non-autonomous dynamical systems from a branch of stationary solutions. As a byproduct we obtain a family index theorem for asymptotically hyperbolic linear dynamical systems which is of independent interest. In the special case of a single parameter, our bifurcation theorem weakens the assumptions in previous work by Pejsachowicz and the first author.

Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions

We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram $\left( \lambda ,{{\left\| {{\mu }_{\lambda }} \right\|}_{\infty }} \right)$ we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

Stability and sustained oscillations in a ventricular cardiomyocyte model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. The associated stationary problem was reduced to a nonlinear system in only two unknowns, the transmembrane potential V and the intracellular calcium concentration [Ca]( i ). By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For a variable steady depolarizing or hyperpolarizing current (I (st)), the corresponding projection of the static bifurcation diagram in the (I (st), V) plane is complex, featuring three branches of stationary solutions joined by two limit points. On the upper branch oscillations can occur, being either damped at a stable focus or diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle, thus resulting in early after-depolarizations (EADs). The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K(+) current maximal conductances (g (si) and g (K)), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances sustained oscillations tending to a limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods, leading to identification of Hopf bifurcation points, limit points of cycles and period doubling bifurcations. In particular settings, e.g. one-fifth-of-normal g (si), EADs and sustained high amplitude oscillations due to an unstable resting state may occur simultaneously.

Gap solitons in nonlinear periodicχ(2)media

The existence, spatial properties, and stability of localized modes in nonlinear quadratic materials with periodically modulated linear refractive index and quadratic nonlinearity (the latter having the zero mean value along the structure) are investigated. The branches of stationary solutions existing in the semi-infinite and three higher gaps are computed. It is shown that modes may possess different symmetries of the fundamental field (i.e., profiles described either by even or odd functions of the transverse coordinate), while requiring the second harmonic to be symmetric. We found that only small-amplitude gap solitons can be stable. The respective modes can bifurcate only from an edge of a total gap when it coincides with the band edge of the linear spectrum of the fundamental field. Moreover, in this case stable modes can be excited with their centers belonging to a slab with either higher or lower refractive index. This simultaneous existence of different stable localized modes can be viewed as a bistability of gap solitons. Examples of the dynamics and excitations of gap solitons are also given.

Stability and Self-Sustained Oscillations in a Ventricular Cardiomyocyte Model

The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For different values of a steady depolarizing/hyperpolarizing current (Ist), the corresponding projection of the static bifurcation diagram in the (Ist, V) plane is complex, featuring three branches of stationary solutions delimited by two saddle-node bifurcation points. In addition, on the upper branch oscillations can occur within an Ist range [−4.45, −0.51 μA/cm2] where the Jacobian of the linearized system features two complex conjugated eigenvalues. Oscillations are either damped at a stable focus or amplified until the system trajectory is diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle. The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K+ current maximal conductances (gsi and gK), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances self-sustained oscillations tending to a stable limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods based on numerical continuation algorithms. Both unsustained oscillations, resulting in early after-depolarizations, and sustained oscillations may trigger dangerous ventricular arrhythmias by multiple mechanisms. In particular settings, e.g. for a normal gK and one-fifth-of-normal gsi, these two arrhythmia-threatening conditions may occur simultaneously.

Dynamics of particle trajectories in a Rayleigh–Bénard problem

Fluid particle trajectories for the Rayleigh–Bénard problem in a cube with perfectly conducting lateral walls have been investigated. The velocity and temperature fields of the stationary flow solutions have been obtained by means of a parameter continuation procedure based on a Galerkin spectral method. The rich dynamics of the resulting fluid particle paths has been studied for three branches of stationary solutions and different values of the Rayleigh number within the range 10 4 ⩽ Ra ⩽ 1.5 × 10 5 at a Prandtl number equal to 130. The stability properties and bifurcations of fixed points, which play a key role in the global dynamics, have been analyzed. Main periodic orbits and their stability character have also been determined. Poincaré maps reveal that regions of chaotic motion and regions of regular motion coexist inside the cavity. The boundaries of these three-dimensional regions have been determined. The metric entropy gives an indication of the mixing properties of the large chaotic zone.

Bifurcation branch of stationary solutions for a Lotka–Volterra cross-diffusion system in a spatially heterogeneous environment

This paper is concerned with the following Lotka–Volterra cross-diffusion system in a spatially heterogeneous environment (SP) { Δ [ ( 1 + k ρ ( x ) v ) u ] + u ( a − u − c ( x ) v ) = 0 in Ω , Δ v + v ( b + d ( x ) u − v ) = 0 in Ω , ∂ ν u = ∂ ν v = 0 on ∂ Ω . Here Ω is a bounded domain in R N ( N ≤ 3 ) , a and k are positive constants, b is a real constant, c ( x ) > 0 and d ( x ) ≥ 0 are continuous functions and ρ ( x ) > 0 is a smooth function with ∂ ν ρ = 0 on ∂ Ω . From a viewpoint of the mathematical ecology, unknown functions u and v , respectively, represent stationary population densities of prey and predator which interact and migrate in Ω . Hence, the set Γ p of positive solutions (with bifurcation parameter b ) forms a bounded line in a spatially homogeneous case that ρ , c and d are constant. This paper proves that if a and | b | are small and k is large, a spatial segregation of ρ ( x ) and d ( x ) causes Γ p to form a ⊂ -shaped curve with respect to b . A crucial aspect of the proof involves the solving of a suitable limiting system as a , | b | → 0 and k → ∞ by using the bifurcation theory and the Lyapunov–Schmidt reduction.

Morphology changes in the evolution of liquid two-layer films

We consider a thin film consisting of two layers of immiscible liquids on a solid horizontal (heated) substrate. Both the free liquid-liquid and the liquid-gas interface of such a bilayer liquid film may be unstable due to effective molecular interactions relevant for ultrathin layers below 100-nm thickness, or due to temperature-gradient-caused Marangoni flows in the heated case. Using a long-wave approximation, we derive coupled evolution equations for the interface profiles for the general nonisothermal situation allowing for slip at the substrate. Linear and nonlinear analyses of the short- and long-time film evolution are performed for isothermal ultrathin layers, taking into account destabilizing long-range and stabilizing short-range molecular interactions. It is shown that the initial instability can be of a varicose, zigzag, or mixed type. However, in the nonlinear stage of the evolution the mode type, and therefore the pattern morphology, can change via switching between two different branches of stationary solutions or via coarsening along a single branch.

Transcritical bifurcation with O(3) symmetry

Bifurcation from spherical symmetry occurs transcritically when the degree l of the spherical harmonics is even. In this case the leading-order bifurcation equations are completely determined by the symmetry. Several new results are presented concerning the existence or non-existence of solutions with dihedral symmetry in two-dimensional subspaces. For large l, there is an alternating arrangement of existence and non-existence of such solutions. Although all bifurcating branches of stationary solutions are unstable, a preferred solution is identified using a variational criterion; this solution also has only one positive eigenvalue. It is shown that the axisymmetric state is never the preferred solution according to this criterion. Results on the existence and stability of solution branches are given for even values of l up to l = 18, including all solutions in subspaces of dimension three or lower. For l = 6, 10 and 12, the preferred solution has icosahedral symmetry.

Branching and stability of stationary solutions in multi-equation systems

Steady states of many lumped parameter systems are described by equations that cannot be reduced to a single equation. Existing works on the bifurcation diagrams and steady-state multiplicity either assume that the equations are reducible to one equation or use some mathematical techniques (e.g. Lyapounov reduction) to reduce them. In an earlier communication (Inamdar and Karimi (Chem. Eng. Sci. 56(12) (2001) 3915), we introduced the use of a new and powerful method, called reductive perturbation method (RPM), which deals with multiple equations as they are, without using any reduction. In this work, we apply RPM to fully study the branching and stability of steady states near a singularity in an n-equation system. Analogous to those known for the one-equation system, we identify several types of singularities and branching patterns for the n-equation system, including some that have not appeared so far in the chemical engineering literature. We present explicit, analytical conditions and expressions for the occurrence of these singularities, the stationary solution branches and their stability. Our work easily generalizes and complements many of the branching and multiplicity results available for the one-equation system and is the first to give explicit, analytical results on branch stability. As illustrated using a literature example, it also facilitates computation and construction of solution diagrams.