Trivial Solution Research Articles

Equilibrium Solutions of a Modified SIR Model with Vaccination and Several Levels of Immunity

We consider a system of ordinary differential equations which extends the well-known SIR model for the dynamics of an epidemic. The main feature is that the population is divided in several subgroups according to their immunity level, which has as a consequence different infection rates. The maximum level of immunity can be achieved either by recovering from an infection, or by possible vaccination. We consider the cases that the vaccination rate is independent on the size of infected population, or that it depends also on this value by a power law. In addition, we assume that the immunity level can decay in time. The goal of this paper is to analyze the existence and uniqueness of equilibrium solutions, which can be either a trivial (disease-free) equilibrium, with no infections, or an endemic equilibrium, with a certain amount of infected individuals. Moreover, we give conditions for the local asymptotic stability of the unique trivial equilibrium solution. It will turn out that, if this is the case, then there exists no endemic equilibrium, which means that the epidemic can be eradicated, by arriving at herd immunity. On the other hand, if the trivial equilibrium is unstable, then we prove the existence of an endemic equilibrium which, under natural conditions, turns out to be unique. The stability of the endemic equilibrium remains still an open problem.

Fixed-Time Stabilization of a Class of Stochastic Nonlinear Systems

This paper investigates an improved fixed-time stability theory together with a state feedback controller for a class of nonlinear stochastic systems. First, a delicate transformation is performed, and next, a Gamma function is utilized to directly derive the value of the integral function, which ultimately yields a fixed-time stabilization theorem with a higher precision upper bound for the settling time. Unlike the existing estimation process of amplifying twice, we only performed one amplification, which weakens the effect of amplification. Then, a state feedback controller is constructed for stochastic systems by the method of adding a power integrator. Utilizing the proposed stochastic fixed-time stability theory, simulations show that the intended controller ensures that the trivial solution of the suggested system is fixed-time stable in probability. The results of the simulation demonstrate that the suggested control scheme is meaningful.

Some remark on the functional equation

The appearance of function equations can be traced back to a long time ago. The oldest and perhaps the best-known function equation is the Cauchy function equation. The trivial solution of Cauchy equation is simple, but the general solution is very complex. Later several function equations appeared, which either arose during some practical problem or became known from a theoretical perspective. The function equation Hosszú is also such a function equation, introduced by Miklós Hosszú based on theoretical considerations. We further generalize the Hosszú-type functional equation and give its solution.

The stability of a class of 2D non-newtonian fluid equations with unbounded delays

ABSTRACT We address the stability of stationary solutions to a class of 2D non-newtonian fluid equations, when the external force contains hereditary characteristics involving unbounded delays. Firstly, when the unbounded variable delay is driven by a continuously differential function, we establish the stability of nontrivial weak stationary solutions and the asymptotic stability of trivial stationary solution. Then when the general unbounded delay is continuous with respect to time, the stability of nontrivial strong stationary solutions is also obtained. Eventually, when the proportional delay is considered, the polynomial stability of trivial stationary solution is verified.

State-Spatial Viewpoint of Stress in Thermally Postbuckled and Nonlinearly Bent FG Saturated Poroelastic Circular Plates

A dimensionless discrete state-space mathematical model is proposed for stress analysis of an axially-symmetric saturated poroelastic circular plate being loaded by temperature rise of one of the flat faces. Computation of stress by means of this model is free from a need for differentiation. The curves of thermal postbuckling, thermal nonlinear bending, radial stress, circumferential stress and resultant radial force have been achieved by means of six state-space variables and differential quadrature procedure. The effects of loading temperature, boundary conditions, porosity, pore distribution and fluid pore pressure on the thermal post buckling and nonlinear bending phenomena have been investigated. The critical postbuckling thermal load has been determined by means of a numerical algorithm that does not converge to trivial solution because it starts from a relatively great numerical thermal load. Contrary to the governing equation in usual form, the governing equations in state space together with the state equations equalize the number of the governing assembled difference equations and that of the unknowns, and eventually do not require to apply the so-called auxiliary extra boundary conditions at the nodes adjacent to the boundary nodes. As another advantage, in state-spatial viewpoint, there is not any necessity for computing the coefficients of derivatives with order higher than unity.

Elastic Lateral Torsional Buckling of Beams Under External Moments: A New Type of Nonconservative Problem

The important character of the applied moment with a fixed line of direction as implied in deriving the governing differential equations of LTB for elastic beams, however, is not represented in stating the boundary conditions based on conventional energy approaches. This paper sets out to consider the boundary conditions derived by conventional potentials in order to identify the ignored second-order moments that need to be added for maintaining the initial direction of the applied moment during LTB. By taking into account the non-zero work done by these ignored moments for modifying original potentials, the nonconservative nature of the elastic LTB of beam under external moments is clarified in two aspects — by discovering the path-dependent property of the non-zero work and by demonstrating that only trivial solution of LTB mode is expected from the analytical solution of the governing differential equations. Using the concept of dynamic instability, an analysis procedure combining the numerical solution of an eigen-value problem and the bisection method is proposed to identify the first appearance of imaginary or conjugate complex frequencies and to determine the value of the critical moment. Demonstrative examples are considered for illuminating the elastic LTB behavior using conventional energy methods and revised virtual work equation. It has been found that the critical moments given by conventional potentials are generally not consistent with each other unless the work done by the ignored second-order moments vanishes. However, consistent critical moment, under which the free vibrating amplitude of elastic beam is increased unboundedly with time, can be obtained by the kinetic approach based on any revised virtual work equation, indicating the necessity of considering the nonconservative nature of non-follower applied moment.

Improved Initial Orbit Determination Based on the Gooding Method of Low Earth Orbit Space Debris Using Space-Based Observations

Initial orbit determination (IOD), as a basis for initial orbit association and accurate orbit determination (OD), has a crucial role in the process of obtaining space debris orbit information. Among the traditional methods, the Gooding method has better convergence and stability. In this study, the Gooding method is enhanced to solve the issues discovered. A novel initial orbit determination (IOD) method is developed using the proposed improvement measures of the single-parameter initial value determination (SIVD) method, the fitted-curve noise suppression method, the restricted corrective value solution method, the removal of trivial solutions, etc. The experimental results verify the effectiveness of the improved method. The success rate of the initial orbit determination reached 99%, and the accuracy of the solved orbit parameters was significantly improved, especially the semi-major axis (SMA) error of less than 50 km, accounting for 88% of the total. It can be seen that the method meets the demand of space-based space debris cataloging for initial orbit association and can serve in the field of space situational awareness, which has important practical significance and application potential.

Results on relative controllability for nonlinear system with multi‐delays in control

This study uses a delayed matrix exponential to illustrate the solution of both linear and nonlinear delay problems using permutable matrices. We outline several prerequisites that are necessary to ensure the exponential stability of the trivial solution of delay systems. The conditions that are necessary and sufficient for the linear system to possess relative controllability are established by introducing a delayed Grammian matrix‐like criterion. A sufficient condition is obtained by Krasnoselskii's fixed point theorem for the relative controllability of nonlinear systems. Numerical examples provided at the end of the paper demonstrate the validity of our theoretical findings.

Applied Lyapunov Stability for Some Nonlinear Stochastic Differential Equations

In this paper, we applied and explain the stability to some linear and non-linear stochastic differential equations by using the Lyapunov direct second method, after finding the stochastic differential equation which obtained by applying the (Ito-integrated formula) and the quadratic Lyapunov function be taken, we use the Lyapunov theorems to find and explain if the trivial (zero) solution are stochastically stabile (p-stable, mean square stable and stochastically asymptotically stable in the large ), then we explain the methods by some examples.

Exact solution for capillary waves on the surface of a liquid of finite depth

Using the Schwartz function method, we have obtained a new exact solution for the problem of stationary capillary waves of finite amplitude on the surface of a liquid that has a finite depth. The reliability of the solution was confirmed by the results of numerical verification of the main boundary equation. The obtained solution of the problem is general in the sense that for any Weber number one can find the corresponding wave configuration. Parametric analysis showed a nonmonotonic dependence of the wave-length and its amplitude on the Weber number. The fact that the problem has one more branch of the solution (the trivial solution) indicates the possibility of the existence of other branches. The Schwartz function method cannot guarantee finding all solutions of the problem even from the specified class of functions. Therefore, the question of reproducing the known exact solution of W. Kimmersley for this problem and its reliability remains open. Note that for the parameter ß included in the main boundary equation, W. Kimmersley preliminarily laid down assumption ß = 1. The found exact solution has the property that ß > 1 and cannot coincide with W. Kimmersley’s solution.

Dynamical analysis of a two-dimensional discrete predator–prey model

In this paper, we explore the existence of periodic solutions, local dynamics at equilibrium solutions, chaos and bifurcations of a discrete prey–predator model with Michaelis–Menten type functional response. More specifically, we explore local dynamical characteristics at equilibrium solutions, and existence of periodic solutions for the under consideration model. It is also studied the existence of bifurcations at equilibrium solutions, and investigated that at semitrivial and trivial equilibrium solutions model does not undergo flip bifurcation, but at positive equilibrium solution it undergoes flip and Neimark–Sacker bifurcations when parameters goes through the certain curves. It is also investigated that fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by center manifold theorem and bifurcation theory. We also studied chaos by feedback control method. The theoretical results are confirmed numerically. Furthermore, we use 0–1 chaos test in order to quantify whether chaos exists in the model or not.

Expansion-compression fixed point theorem of Leggett-Williams type for the sum of two operators and applications for some classes of BVPs

"The purpose of this work is to establish an extension of a Leggett- Williams type expansion-compression fixed point theorem for a sum of two operators. As illustration, our approach is applied to prove the existence of non trivial nonnegative solutions for two-point BVPs and three-point BVPs."

Finite-time stochastic integral input-to-state stability and its applications

In this paper, finite-time stabilization is investigated in depth for stochastic nonlinear systems with stochastic inverse dynamics. The contributions of this work are characterized by the following novel features: (i) Motivated by the concept of stochastic integral input-to-state stability (SiISS) and the existing stochastic finite-time stability theorem, a new concept of finite-time stochastic integral input-to-state stability (FT-SiISS) using Lyapunov function is first introduced, and an inclusion relationship among three types of stochastic stability, i.e., FT-SiISS, FT-SISS and SISS, is rigorously established. On the basis of FT-SiISS, we develop two new FT-SiISS small-gain conditions and discuss their relationship. Then, to analyze finite-time stability of stochastic nonlinear systems with FT-SiISS inverse dynamics, a stochastic finite-time stability theorem is improved. (ii) As the application of (i), we propose a unified finite-time control method, which can simultaneously handle stochastic nonlinear systems with FT-SiISS or FT-SISS inverse dynamics, and guarantee that the closed-loop system has an almost surely continuous solution, all the closed-loop signals are bounded almost surely, and its trivial solution is stochastically finite-time stable.

A note on positive solutions of Lichnerowicz equations involving the $\Delta_\lambda$-Laplacian

In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ v_t-\Delta_\lambda v=v^{-p-2}-v^p,\quad v> 0, \quad \mbox{ in }\mathbb R^N\times\mathbb R, $$ where $p> 0$ and $\Delta_\lambda$ is a sub-elliptic operator of the form $$ \Delta_\lambda=\sum_{i=1}^N\partial_{x_i}\big(\lambda_i^2\partial_{x_i}\big). $$ Under some general assumptions of $\lambda_i$ introduced by A.E. Kogoj and E. Lanconelli in Nonlinear Anal. {\bf 75} (2012), no.\ 12, 4637-4649, we shall prove a uniform lower bound of positive solutions of the equation provided that $p> 0$. Moreover, in the case $p> 1$, we shall show that the equation has only the trivial solution $v=1$. As a consequence, when $v$ is independent of the time variable, we obtain the similar results for the elliptic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ -\Delta_\lambda u=u^{-p-2}-u^p,\quad u> 0,\quad \mbox{in }\mathbb R^N. $$

Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem

Physics-informed neural networks (PINNs) leverage data and knowledge about a problem. They provide a nonnumerical pathway to solving partial differential equations by expressing the field solution as an artificial neural network. This approach has been applied successfully to various types of differential equations. A major area of research on PINNs is the application to coupled partial differential equations in particular, and a general breakthrough is still lacking. In coupled equations, the optimization operates in a critical conflict between boundary conditions and the underlying equations, which often requires either many iterations or complex schemes to avoid trivial solutions and to achieve convergence. We provide empirical evidence for the mitigation of bad initial conditioning in PINNs for solving one-dimensional consolidation problems of porous media through the introduction of affine transformations after the classical output layer of artificial neural network architectures, effectively accelerating the training process. These affine physics-informed neural networks (AfPINNs) then produce nontrivial and accurate field solutions even in parameter spaces with diverging orders of magnitude. On average, AfPINNs show the ability to improve the {mathscr {L}}_2 relative error by 64.84% after 25,000 epochs for a one-dimensional consolidation problem based on Biot’s theory, and an average improvement by 58.80% with a transfer approach to the theory of porous media.

Multiple modes of bursting phenomena in a vector field of triple Hopf bifurcation with two time scales

For the development of the theoretical framework of slow–fast dynamics, it is essential to study the mechanism of bursting oscillations with high co-dimensional bifurcations in the high-dimensional vector field. This paper is aimed to explore the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at the origin. We establish an eight-dimension system with the slow–fast effect, in which two time scales exist. The fast subsystem is represented as a six-dimensional vector field with triple Hopf bifurcation at the origin, while the slow subsystem is adopted to adjust the variation of parameters in the fast subsystem, which may lead to abundant dynamic behaviors. To better understand the qualitatively different behaviors in the fast subsystem, the trivial solution, single-mode, two-mode, and three-mode movements are introduced based on the projections of trajectory on the three sub-planes. As a result, several types of bursting phenomena, such as Hopf/Hopf bursting oscillations with single-mode, Hopf/Hopf/fold bursting oscillations with single-mode, Hopf/Hopf bursting oscillations with two-mode have been revealed, the mechanism of which is obtained upon the superposition of equilibrium branches together with bifurcations in the fast subsystem as well as transformed phase portrait. In addition, for bursting oscillations on the whole phase space, the amplitude of spiking oscillations may change according to two different limit cycles, which leads to an abrupt change of the oscillating amplitude on the sub-plane, corresponding to different projections of Poincaré. It can be observed the state variables on the different sub-planes alternate between synchronized and non-synchronized states in the fast subsystem, which may be accounted for by the bifurcations.

On a principle for mass sensing using self-excited template dynamics of coupled oscillators and root-finding algorithms

In this work, we analyze a mass sensing mechanism that relies on self-excited template dynamics of a simple network of two coupled oscillators, one linear and one nonlinear, to detect changes in the mass of the linear oscillator with tunable sensitivity. The resultant shift in the ratio of oscillation amplitudes is predicted using perturbation analysis and verified against numerical results obtained via parameter continuation. A physical realization of the mass sensing mechanism is proposed, in which the linear oscillator is represented by a periodically excited actuator/microcantilever assembly, the nonlinear oscillator is simulated in silico on a finite interval of time while driven by the periodic steady-state response of the linear oscillator, and a nonlinear root-finding algorithm is used to identify the input to the actuator that reproduces the sought, coupled, self-excited dynamics. The analysis shows that the sensor gain can be adjusted by appropriate tuning of the coupling stiffness without modifications to the physical components. Numerical results for two different actuator models demonstrate rapid convergence of the root-finding algorithm from a trivial initial solution guess over ranges of values of model parameters and added mass ratios.

Toward an Optimal Countermeasure for Cache Side-Channel Attacks

In the last 15 years, we have witnessed a never ending arm’s race between the attacker and the defender with respect to cache-based side-channel attacks. We have seen a slew of attacks, countermeasures (CMs), counterattacks, counter-countermeasures and so on. We analyze the evolution of this area, propose three necessary conditions for designing a successful CM, and then analyze timing and address-based CMs for popular algorithms such as AES and PRESENT. We show that an optimal yet trivial solution for timing-based CMs is possible. Furthermore, address-based CMs are inferior to timing-based CMs, and they can be broken in O(nlog(log(n))) time.

Codesign for Resilience and Performance

We present two optimization approaches to minimize the impact of sensor falsification attacks in linear time invariant systems controlled by estimate-based feedback. We accomplish this by finding observer and controller gain matrices that minimize outer ellipsoidal bounds that contain the reachable set of attack-induced states. To avoid trivial solutions, we involve a covariance-based <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Vert H\Vert _{2}$</tex-math></inline-formula> closed-loop performance constraint. This exposes a trade-off between system security and closed-loop performance and demonstrates that only small concessions in performance can lead to large gains in our reachability-based security metric. We provide both a nonlinear optimization based on geometric sums and a fully convexified approach formulated with linear matrix inequalities. We demonstrate the effectiveness of these tools on two numerical case studies.

Self-Supervised Global Spatio-Temporal Interaction Pre-Training for Group Activity Recognition

This paper focuses on exploring distinctive spatio-temporal representation in a self-supervised manner for group activity recognition. Firstly, previous networks treat spatial- and temporal-aware information as a whole, limiting their abilities to represent complex spatio-temporal correlations for group activity. Here, we propose the Spatial and Temporal Attention Heads (STAHs) to extract spatial- and temporal-aware representations independently, which generate complementary contexts for boosting group activity understanding. Then, we propose the Global Spatio-Temporal Contrastive (GSTCo) loss to aggregate these two kinds of features. Unlike previous works focusing on the individual temporal consistency while overlooking the correlations between actors, i.e., in a local perspective, we explore the global spatial and temporal dependency. Moreover, GSTCo could effectively avoid the trivial solution faced in contrastive learning by achieving the right balance between spatial and temporal representations. Furthermore, our method imports affordable overhead during pre-training, without additional parameters or computational costs in inference, guaranteeing efficiency. By evaluating on widely-used datasets for group activity recognition, our method achieves good performance. State-of-the-art performance is achieved when applying our pre-trained backbone to existing networks. Extensive experiments verify the generalizability of our method.