Showing papers in "Journal of Nonlinear Science in 2015"

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

Abstract: The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005) Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions

Optimal Harvesting of a Stochastic Logistic Model with Time Delay

Abstract: This note is concerned with the optimal harvesting of a stochastic logistic model with time delay. The classical optimal harvesting question of this type of model is difficult because it is very difficult to obtain the explicit solution of the corresponding delay Fokker–Planck equation. The main aim of this note was to find a new approach to overcome this problem. In this note, using the ergodic method, sufficient and necessary criteria for the existence of optimal harvesting policy of our model are obtained. At the same time, the optimal harvesting effort and the maximum of harvesting yield are given. This method provides a new approach to study the optimal harvesting problem of stochastic population models, which can be also applied to investigate stochastic multi-species models.

Dimension Reduction for the Landau-de Gennes Model in Planar Nematic Thin Films

Abstract: We use the method of $$\Gamma $$ -convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role, and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic Q-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.

Lissajous and Halo Orbits in the Restricted Three-Body Problem

Abstract: We study the dynamics near the collinear Lagrangian points of the spatial, circular, restricted three-body problem. Following a standard procedure, we reduce the system to the center manifold and we analyze the Lissajous orbits as well as the halo orbits, the latter ones arising from bifurcations of the planar Lyapunov family of periodic orbits. To obtain the Lissajous orbits, we perform a classical perturbation theory and we provide a formal approximate solution under suitable non-degeneracy and non-resonance conditions. As for the halo orbits, we construct a normal form adapted to the synchronous resonance: introducing a detuning, measuring the displacement from the resonance, and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place. Except for a particular case, the analytical values obtained after a second order resonant perturbation theory are in very good agreement (in some cases up to the fourth decimal digit) with the numerical values found in the literature.

Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems

Abstract: We prove results that enable realization of heteroclinic networks in coupled homogeneous and heterogeneous systems of identical cells. We also consider various models for network dynamics, which allow variation in the number of inputs to identical cells.

Uniqueness and Non-uniqueness of Limit Cycles for Piecewise Linear Differential Systems with Three Zones and No Symmetry

Abstract: Some techniques for proving the existence and uniqueness of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems with two and three zones and no symmetry. For planar systems with three linearity zones, the existence of two limit cycles surrounding the only equilibrium point at the origin is rigorously shown for the first time. The usefulness of the achieved analytical results is illustrated by considering non-symmetric memristor-based electronic oscillators.

Periodic and Chaotic Orbits of Plane-Confined Micro-rotors in Creeping Flows

Abstract: We explore theoretically the complex dynamics and emergent behaviors of spinning spheres immersed in viscous fluid. The particles are coupled to each other via the fluid in which they are suspended: Each particle disturbs the surrounding fluid with a rotlet field and that fluid flow affects the motion of the other particles. We notice the emergence of intricate periodic or chaotic trajectories that depend on the rotors initial position and separation. The point-rotor motions confined to a plane bear similarities to the classic 2D point-vortex dynamics. Our analyses highlight the complexity of the interaction between just a few rotors and suggest richer behavior in denser populations. We discuss how the model gives insight into more complex systems and suggest possible extensions for future theoretical studies.

Swimming Dynamics Near a Wall in a Weakly Elastic Fluid

Abstract: We present a fully resolved solution of a low-Reynolds-number two- dimensionalmicroswimmerinaweaklyelasticfluidnearano-slipsurface.Theresults illustrate that elastic properties of the background fluid dramatically alter the swim- ming hydrodynamics and, depending on the initial position and orientation of the microswimmer, its residence time near the surface can increase by an order of magni- tude. Elasticity of the extracellular polymeric substance secreted by microorganisms canthereforeenhancetheiradhesionrate.Thedynamicalsystemisexaminedthrougha phase portrait in the swimming orientation and distance from the wall for four types of self-propulsionmechanisms,namely:neutralswimmers,pullers,pushers,andstirrers. The time-reversibility of the phase portraits breaks down in the presence of polymeric materials. The elasticity of the fluid leads to the emergence of a limit cycle for pullers and pushers and the change in type of fixed points from center to unstable foci for a microswimmer adjacent to a no-slip boundary.

Global Smooth Solutions to the n-Dimensional Damped Models of Incompressible Fluid Mechanics with Small Initial Datum

Abstract: In this paper, we consider the $$n$$ -dimensional ( $$n\ge 2$$ ) damped models of incompressible fluid mechanics in Besov spaces and establish the global (in time) regularity of classical solutions provided that the initial data are suitable small.

Multiscale Geometry of the Olsen Model and Non-classical Relaxation Oscillations

Abstract: We study the Olsen model for the peroxidase–oxidase reaction. The dynamics is analyzed using a geometric decomposition based on multiple timescales. The Olsen model is four-dimensional, not in a standard form required by geometric singular perturbation theory and contains multiple small parameters. These three obstacles are the main challenges we resolve by our analysis. Scaling and the blow-up method are used to identify several subsystems. The results presented here provide a rigorous analysis for two oscillatory modes. In particular, we prove the existence of non-classical relaxation oscillations in two cases. The analysis is based on desingularization of lines of transcritical and submanifolds of fold singularities in combination with an integrable relaxation phase. In this context, our analysis also explains an assumption that has been utilized, based purely on numerical reasoning, in a previous bifurcation analysis by Desroches et al. (Discret Contin Dyn Syst S 2(4):807–827, 2009). Furthermore, the geometric decomposition we develop forms the basis to prove the existence of mixed-mode and chaotic oscillations in the Olsen model, which will be discussed in more detail in future work.

Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

Abstract: This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray–Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time \(T>0\)) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

Stability and Bifurcation in a Predator-Prey Model with Age Structure and Delays

Abstract: This paper investigates a predator–prey model with age structure and two delays. By formulating the age-structured model with delays as a non-densely defined Cauchy problem and applying the theory of integrated semigroup and recently established Hopf bifurcation theory for abstract Cauchy problems with non-dense domain, we show that Hopf bifurcation occurs in the model. This also shows the sensitivity of the model dynamics on the threshold $$\tau $$ which might be taken as a measure of a biological maturation period and a time lag between conception and birth. Numerical simulations are performed to illustrate the obtained results and a summary is given.

Noisy Lagrangian Tracers for Filtering Random Rotating Compressible Flows

Abstract: The recovery of a random turbulent velocity field using Lagrangian tracers that move with the fluid flow is a practically important problem. This paper studies the filtering skill of $$L$$ -noisy Lagrangian tracers in recovering random rotating compressible flows that are a linear combination of random incompressible geostrophically balanced (GB) flow and random rotating compressible gravity waves. The idealized random fields are defined through forced damped random amplitudes of Fourier eigenmodes of the rotating shallow-water equations with the rotation rate measured by the Rossby number $$\varepsilon $$ . In many realistic geophysical flows, there is fast rotation so $$\varepsilon $$ satisfies $$\varepsilon \ll 1$$ and the random rotating shallow-water equations become a slow–fast system where often the primary practical objective is the recovery of the GB component from the Lagrangian tracer observations. Unfortunately, the $$L$$ -noisy Lagrangian tracer observations are highly nonlinear and mix the slow GB modes and the fast gravity modes. Despite this inherent nonlinearity, it is shown here that there are closed analytical formulas for the optimal filter for recovering these random rotating compressible flows for any $$\varepsilon $$ involving Ricatti equations with random coefficients. The performance of the optimal filter is compared and contrasted through mathematical theorems and concise numerical experiments with the performance of the optimal filter for the incompressible GB random flow with $$L$$ -noisy Lagrangian tracers involving only the GB part of the flow. In addition, a sub-optimal filter is defined for recovering the GB flow alone through observing the $$L$$ -noisy random compressible Lagrangian trajectories, so the effect of the gravity wave dynamics is unresolved but effects the tracer observations. Rigorous theorems proved below through suitable stochastic fast-wave averaging techniques and explicit formulas rigorously demonstrate that all these filters have comparable skill in recovering the slow GB flow in the limit $$\varepsilon \rightarrow 0$$ for any bounded time interval. Concise numerical experiments confirm the mathematical theory and elucidate various new features of filter performance as the Rossby number $$\varepsilon $$ , the number of tracers $$L$$ and the tracer noise variance change.

Unimodularity and Preservation of Volumes in Nonholonomic Mechanics

Abstract: The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, which unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.

On critical behaviour in systems of Hamiltonian partial differential equations

Abstract: We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painleve-I (P\(_I\)) equation or its fourth-order analogue P\(_I^2\). As concrete examples, we discuss nonlinear Schrodinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.

An Information-Theoretic Framework for Improving Imperfect Dynamical Predictions Via Multi-Model Ensemble Forecasts

Abstract: This work focuses on elucidating issues related to an increasingly common technique of multi-model ensemble (MME) forecasting The MME approach is aimed at improving the statistical accuracy of imperfect time-dependent predictions by combining information from a collection of reduced-order dynamical models Despite some operational evidence in support of the MME strategy for mitigating the prediction error, the mathematical framework justifying this approach has been lacking Here, this problem is considered within a probabilistic/stochastic framework which exploits tools from information theory to derive a set of criteria for improving probabilistic MME predictions relative to single-model predictions The emphasis is on a systematic understanding of the benefits and limitations associated with the MME approach, on uncertainty quantification, and on the development of practical design principles for constructing an MME with improved predictive performance The conditions for prediction improvement via the MME approach stem from the convexity of the relative entropy which is used here as a measure of the lack of information in the imperfect models relative to the resolved characteristics of the truth dynamics It is also shown how practical guidelines for MME prediction improvement can be implemented in the context of forced response predictions from equilibrium with the help of the linear response theory utilizing the fluctuation–dissipation formulas at the unperturbed equilibrium The general theoretical results are illustrated using exactly solvable stochastic non-Gaussian test models

Frustrated Ferromagnetic Spin Chains: A Variational Approach to Chirality Transitions

Abstract: We study the energy per particle of a one-dimensional ferromagnetic/antiferromagnetic frustrated spin chain with nearest and next-to-nearest interactions close to the helimagnet/ferromagnet transition point as the number of particles diverges. We rigorously prove the emergence of chiral ground states, and we compute, by performing the $$\Gamma $$ -limits of proper renormalizations and scalings, the energy for a chirality transition.

Nonlinear Network Modes in Cyclic Systems with Applications to Connected Vehicles

Abstract: In this paper, we propose a novel technique to decompose networked sys- tems with cyclic structure into nonlinear modes and apply these ideas to a system of connected vehicles. We perform linear and nonlinear transformations that exploit the network structure and lead to nonlinear modal equations that are decoupled. Each mode can be obtained by solving a small set of algebraic equations without deriving the coefficients for any other mode. By focusing on the mode that is loosing stability, bifurcation analysis can be carried out. The techniques developed are applied to eval- uate the impact of connected cruise control on the nonlinear dynamics of a connected vehicle system.

Finite Dimensions of the Global Attractor for 3D Primitive Equations with Viscosity

Abstract: For a set of non-periodic boundary conditions, we prove the uniform boundedness of the \(H^2\) norms of the solutions of the 3D Primitive Equations with viscosity. An absorbing set of the solutions in \(H^2\) is also obtained. As an application of this result, we prove also the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solutions of the 3D Primitive Equations with viscosity. Our results also improve the existing results for the case with periodic boundary conditions.

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

Abstract: The Kuramoto model of coupled phase oscillators on complete, Paley, and Erdős–Renyi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the Kuramoto model on these graphs can be qualitatively different. Specifically, we identify twisted states, steady-state solutions of the Kuramoto model on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the initial value problems for the Kuramoto model on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the Kuramoto model on Cayley and random graphs.

Emergent Behaviors of Quantum Lohe Oscillators with All-to-All Coupling

Abstract: We study an emergent synchronous behavior for an ensemble of Lohe qubit oscillators whose quantum states are described by $$2\times 2$$ unitary matrices. The quantum Lohe model can be regarded as a non-abelian and quantum generalization of the Kuramoto model for classical oscillators. For the interacting qubit system, the Lohe model can be recast as a coupled ODE system. We provide several explicit sufficient conditions for the complete synchronization of Lohe qubit oscillators in terms of the initial condition and coupling strength. We also show that for identical qubit oscillators, the Lohe model for interacting qubits satisfies an asymptotic completeness property. Our analytical results confirm the numerical results from Lohe (J Phys A Math Theor 43:465301, 2010).

Asymptotic Dynamics of Inertial Particles with Memory

Abstract: Recent experimental and numerical observations have shown the significance of the Basset–Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. Here, we prove that the observed algebraic decay is a universal property of the Maxey–Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is $$\mathcal {O}(\epsilon )$$ -close to the fluid velocity, where $$0<\epsilon \ll 1$$ is proportional to the square of the ratio of the particle radius to the fluid characteristic length scale. These results follow from a sharp analytic upper bound that we derive for the particle velocity. For completeness, we also present a first proof of the global existence and uniqueness of mild solutions to the Maxey–Riley equation, a nonlinear system of fractional differential equations.

Sufficient Conditions for Global Minimality of Metastable States in a Class of Non-convex Functionals: A Simple Approach Via Quadratic Lower Bounds

Abstract: We consider mass-constrained minimizers for a class of non-convex energy functionals involving a double-well potential. Based upon global quadratic lower bounds to the energy, we introduce a simple strategy to find sufficient conditions on a given critical point (metastable state) to be a global minimizer. We show that this strategy works well for the one exact and known metastable state: the constant state. In doing so, we numerically derive an almost optimal lower bound for both the order–disorder transition curve of the Ohta–Kawasaki energy and the liquid–solid interface of the phase-field crystal energy. We discuss how this strategy extends to non-constant computed metastable states, and the resulting symmetry issues that one must overcome. We give a preliminary analysis of these symmetry issues by addressing the global optimality of a computed lamellar structure for the Ohta–Kawasaki energy in one (1D) and two (2D) space dimensions. We also consider global optimality of a non-constant state for a spatially in-homogenous perturbation of the 2D Ohta–Kawasaki energy. Finally we use one of our simple quadratic lower bounds to rigorously prove that for certain values of the Ohta–Kawasaki parameter and aspect ratio of an asymmetric torus, any global minimizer $$v(x)$$ for the 1D problem is automatically a global minimizer for the 2D problem on the asymmetric torus.

Coordinated Speed Oscillations in Schooling Killifish Enrich Social Communication

Abstract: We examine the spatial dynamics of individuals in small schools of banded killifish (Fundulus diaphanus) that exhibit rhythmic, oscillating speed, typically with sustained, coordinated, out-of-phase speed oscillations as they move around a shallow water tank. We show that the relative motion among the fish yields a periodically time-varying network of social interactions that enriches visually driven social communication. The oscillations lead to the regular making and breaking of occlusions, which we term “switching.” We show that the rate of convergence to consensus (biologically, the capacity for individuals in groups to achieve effective coordinated motion) governed by the switching outperforms static alternatives, and performs as well as the less practical case of every fish sensing every other fish. We show further that the oscillations in speed yield oscillations in relative bearing between fish over a range that includes the angles previously predicted to be optimal for a fish to detect changes in heading and speed of its neighbors. To investigate systematically, we derive and analyze a dynamic model of interacting agents that move with oscillatory speed. We show that coordinated circular motion of the school leads to systematic cycling of spatial ordering of agents and possibilities for enriched spatial density of measurements of the external environment. Our results highlight the potential benefits of dynamic communication topologies in collective animal behavior, and suggest new, useful control laws for the distributed coordination of mobile robotic networks.

Spontaneous Flows in Suspensions of Active Cyclic Swimmers

Abstract: Many swimming cells rely on periodic deformations to achieve locomotion. We introduce in this work a theoretical model and numerical simulations in order to elucidate the impact of these cyclic strokes on the emergence of mesoscale structures and collective motion in swimmer suspensions. The model extends previous kinetic theories for populations of identical swimmers to the case of self-propelled particles undergoing transitions between pusher and puller states, and is applied to quantify how the unsteadiness of the hydrodynamic velocity field, to which each swimmer population contributes, affects the onset and characteristics of spontaneous flows. A linear stability analysis reveals that the sign of the population-averaged dipole determines the stability of the uniform isotropic state, with suspensions dominated by pushers being subject to growing nematic bend fluctuations. Stochastic transitions, however, are also seen to provide an additional damping mechanism. To investigate the population dynamics above the instability threshold, we also perform direct particle simulations based on a slender-body model, where the growth or decay of the active power generated by the swimmers is found to be a robust measure of the structural and dynamical instability.

Butterfly Catastrophe for Fronts in a Three-Component Reaction--Diffusion System

Abstract: We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically.

Orbital-Free Density Functional Theory of Out-of-Plane Charge Screening in Graphene

Abstract: We propose a density functional theory of Thomas–Fermi–Dirac–von Weizsacker type to describe the response of a single layer of graphene resting on a dielectric substrate to a point charge or a collection of charges some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers, both in the case of free graphene layers and under back-gating. We further provide conditions under which those minimizers are unique and correspond to configurations consisting of inhomogeneous density profiles of charge carrier of only one type. The associated Euler–Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. In addition, a bifurcation from zero to nonzero response at a finite threshold value of the external charge is proved.

Hessian Operators on Constraint Manifolds

Abstract: On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group $$\mathbf{O}(n)$$ by using only Euclidean coordinates on $${\mathbb {R}}^{n^2}$$ . An optimization problem on $$\mathbf{SO}(3)$$ is completely carried out. Its applications to nonlinear stability problems are also analyzed.

Collective Dynamics in the Vicsek and Vectorial Network Models Beyond Uniform Additive Noise

Abstract: In this work, we analyze the coordination of interacting individuals in two nonlinear dynamical models that are subject to a new form of noise. Specifically, we propose extensions both to the classical Vicsek model, whereby each individual averages the orientation of its geographically proximal neighbors, and to the vectorial network model, in which the selection of neighbors is random and independent of the group geometric configuration. In the traditional forms of these models, the update rule for the individuals’ orientations is affected by additive uniform noise. Motivated by biological groups in which individuals’ turn rates exhibit sporadic and large changes, we extend the uniform additive noise model to a turn rate stochastic process. Through comprehensive numerical simulations, we demonstrate the impact of such occasional large deviations (intensity and frequency), along with the role of the neighbors’ selection process, on the coordination of the group. In addition, we establish a closed-form expression for the group polarization for the vectorial network model in the vicinity of an ordered state.

Monotonicity of Spatial Critical Points Evolving Under Curvature-Driven Flows

Abstract: We describe the variation of the number $$N(t)$$ of spatial critical points of smooth curves (defined as a scalar distance $$r$$ from a fixed origin $$O$$ ) evolving under curvature-driven flows. In the latter, the speed $$v$$ in the direction of the surface normal may only depend on the curvature $$\kappa $$ . Under the assumption that only generic saddle-node bifurcations occur, we show that $$N(t)$$ will decrease if the partial derivative $$v_{\kappa }$$ is positive and increase if it is negative (Theorem 1). Justification for the genericity assumption is provided in Sect. 5. For surfaces embedded in 3D, the normal speed $$v$$ under curvature-driven flows may only depend on the principal curvatures $$\kappa , \lambda $$ . Here we prove the weaker (stochastic) Theorem 2 under the additional assumption that third-order partial derivatives can be approximated by random variables with zero expected value and covariance. Theorem 2 is a generalization of a result by Kuijper and Florack for the heat equation. We formulate a Conjecture for the case when the reference point coincides with the centre of gravity and we motivate the Conjecture by intermediate results and an example. Since models for collisional abrasion are governed by partial differential equations with $$v_{\kappa },v_{\lambda }>0$$ , our results suggest that the decrease of the number of static equilibrium points is characteristic of some natural processes.