Zero Solution Research Articles

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.

On the stability of the zero solution of a differential equation of the second order under periodic perturbation of a center

On the stability of the zero solution of a differential equation of the second order under periodic perturbation of a center

Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation

A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.

On the Stability of the Zero Solution of a Second-Order Differential Equation under a Periodic Perturbation of the Center

Small periodic perturbations of the oscillator \(\ddot x + {x^{2n}}\) sgn x = Y(t, x, \(\dot x\)) are considered, where n < 1 is a positive integer and the right-hand side is a small perturbation periodic in t, which is an analytic function in \(\dot x\) and x in a neighborhood of the origin. New Lyapunov-type periodic functions are introduced and used to investigate the stability of the equilibrium position of the given equation. Sufficient conditions for asymptotic stability and instability are given.

New conditions for the exponential stability of fractionally perturbed ODEs

The aim of this paper is to present some results on the exponential stability of the zero solution for a class of fractionally perturbed ordinary differential equations, whose right-hand sides involve the Riemann–Liouville substantial fractional integrals of different orders and we assume that they are polynomially bounded. In their proofs we apply a method recently developed by Rigoberto Medina. We also prove an existence result for this type of equations.

A note on explosion suppression for nonlinear delay differential systems by polynomial noise

Liu and Shen discussed the role of stochastic suppression on the explosive solution by a polynomial noise for a deterministic differential system satisfying a general polynomial growth condition. They further showed that the global solution of the corresponding perturbed system grows at most polynomially. However, the estimation of the asymptotic property of polynomial growth is rough, and we see the necessity to develop a more accurate estimation which is the main motivation of the present paper. As to the existence of time delays, we aim to discuss the stochastic roles of the polynomial noise for a deterministic delay differential system with the general polynomial growth condition. We show that a properly chosen polynomial stochastic noise not only can guarantee the existence and uniqueness of the global solution of the stochastically perturbed delay differential system, but also can make almost every sample path of the global solution grow at most with polynomial rate and even decay to the zero solution exponentially.

Stability of a budworm growth model with random perturbations

A budworm growth model perturbed by both white noises and regime switchings is proposed and analyzed. It is proven that there is a threshold. If this threshold is positive, then the model has a unique ergodic stationary distribution; if this threshold is negative, then the zero solution of the model is stable. The results show that both white noises and regime switchings can change the stability of the model greatly. Several numerical simulations based on realistic data are also introduced to illustrate the main results.

Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming

We study the global asymptotic stability in probability of the zero solution of linear stochastic differential equations with constant coefficients. We develop a sum-of-squares program that verifies whether a parameterized candidate Lyapunov function is in fact a global Lyapunov function for such a system. Our class of candidate Lyapunov functions are naturally adapted to the problem. We consider functions of the form $V(\mathbf{x}) = \|\mathbf{x}\|_Q^p: = (\mathbf{x}^\top Q\mathbf{x})^{\frac{p}{2}}$, where the parameters are the positive definite matrix $Q$ and the number $p&gt;0$. We give several examples of our proposed method and show how it improves previous results.

Stability results for nonlinear functional differential equations using fixed point methods

We present new conditions for stability of the zero solution for three distinct classes of scalar nonlinear delay differential equations. Our approach is based on fixed point methods and has the advantage that our conditions neither require boundedness of delays nor fixed sign conditions on the coefficient functions. Our work extends and improves a number of recent stability results for nonlinear functional differential equations in a unified framework. A number of examples are given to illustrate our main results.

Mixed Order Fractional Differential Equations

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

A new mechanism for buoyancy driven convection in pulsating viscous flows: A theoretical study

A new mechanism for the onset of thermal convection is proposed. This mechanism is the result of the interaction between a pulsating flow, viscous dissipation, and buoyancy within a channel. The study considers a Newtonian fluid moving inside an infinitely wide horizontal channel bounded by impermeable, rigid plates. The basic flow is characterized by a pulsating pressure gradient. Viscous dissipation acts as an internal heat source which produces a potentially unstable basic temperature gradient. The heat source has a vertical non uniform distribution inside the channel. This configuration is investigated with respect to the onset of buoyancy driven convection. The basic state fields are solved analytically by expanding them in series as functions of the pulsating frequency. In order to perform the linear stability analysis, an arbitrarily small perturbation is superimposed upon the basic state order zero solution. The normal mode method is employed and an ordinary differential eigenvalue problem is obtained. The perturbations were found to have zero angular frequency and thus the resonance phenomena between the basic flow and the perturbations can be neglected. The critical values of the governing parameter are obtained by solving the eigenvalue problem numerically. A growth rate analysis of the possibly unstable configurations relative to the most unstable mode is performed. The present study proves, theoretically, that a pulsating flow can undergo thermal convection. A future experimental study is suggested to validate the proposed instability mechanism.

Stability in nonlinear neutral Levin-Nohel integro-differential equations

In this paper we use the Krasnoselskii-Burton's fixed point theorem to obtain asymptotic stability and stability results about the zero solution for the following nonlinear neutral Levin-Nohel integro-differential equation $$ x^{\prime }(t)+\int_{t-\tau (t)}^{t}a(t,s)g\left( x(s)\right) ds+c(t)x^{\prime }(t-\tau (t))=0. $$ The results obtained here extend the work of Mesmouli, Ardjouni and Djoudi [20].

Stability Analysis of Linear Distributed Order Fractional Systems With Distributed Delays

Abstract In the present work we study linear systems with distributed delays and distributed order fractional derivatives based on Caputo type single fractional derivatives, with respect to a nonnegative density function. For the initial problem of this kind of systems, existence, uniqueness and a priory estimate of the solution are proved. As an application of the obtained results, we establish sufficient conditions for global asymptotic stability of the zero solution of the investigated types of systems.

Stability analysis of the zero solution of a relay system of ordinary differential equations with two relays

We consider a relay system of ordinary differential equations whose right-hand sides are sums of linear functions and two discontinuous functions. We analyze the stability of the zero solution of a relay system of this form for the case in which the system parameters satisfy some equality-type constraints.

Hidden Symmetry in a Kuramoto–Sivashinsky Initial-Boundary Value Problem

We investigate the bifurcation structure of the Kuramoto–Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and [Formula: see text] symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov–Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical continuation, and show that they lose stability in pitchfork, transcritical or Hopf bifurcations. Tracking the corresponding secondary branches reveals an interval of the viscosity parameter in which up to four stable equilibria and time-periodic solutions coexist. Since the study of the extended problem is indispensable for the explanation of the bifurcation structure, the Kuramoto–Sivashinsky problem with Dirichlet boundary conditions provides an elegant manifestation of hidden symmetry.

Distributions of zeros of solutions to first order delay dynamic equations

This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences.

Existence, uniqueness, stability and bifurcation of periodic patterns for a seasonal single phytoplankton model with self-shading effect

We study the existence, uniqueness, global attractivity and bifurcation of time-periodic patterns for a seasonal phytoplankton model with self-shading effect. By the comparison principle, we obtain the globally asymptotical stability of the zero solution when the principal eigenvalue λ1 is less than zero. When λ1>0, by transforming the model into a new system, we successfully prove the conjecture in previous studies on the uniqueness and attractivity of the positive periodic solution. The positive periodic pattern bifurcating from the zero solution is a very interesting phenomenon. Here we apply the Crandall and Rabinowiz's theory to prove rigorously the existence of a bifurcation point. By way of asymptotic analysis, we derive an asymptotic formula for the positive periodic pattern. Based on the solution formula, we find the linear stability of this positive pattern. Finally, we provide a numerical scheme for the calculations of the principal eigenvalue and the simulations of the solution. The simulations corroborate our theoretical analysis.

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.

Nonlinear Dynamic Inequalities and Stability Quasilinear Systems on Time Scales

In this paper a novel nonlinear integral inequality on time scale is proposed. This inequality is applied to analyze stability of zero solution of quasilinear dynamic equations on time scale. Also, stability conditions are established for a wide class of nonlinearities in the system of dynamic equations.

Estimation of the Attraction Domain for Nonlinear Single-Input System with Constrained Control

Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of construction of an estimate of the attraction domain of zero solution is studied. A method for constructing an ellipsoidal estimate that is based on results of the absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.