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Successful performance in long distance race requires both high efficiency and stability. Previous research has demonstrated the high running efficiency of trained runners, but no prior study quantitatively addressed their orbital stability. In this study, we evaluated the efficiency and orbital stability of 8 professional long-distance runners and compared them with those of 8 novices. We calculated the cost of transport and normalized mechanical energy to assess physiological and mechanical running efficiency, respectively. We quantified orbital stability using Floquet Multipliers, which assess how fast a system converges to a limit cycle under perturbations. Our results show that professional runners run with significantly higher physiological and mechanical efficiency but with weaker orbital stability compared to novices. This finding is consistent with the inevitable trade-off between efficiency and stability; increase in orbital stability necessitates increase in energy dissipation. We suggest that professional runners have developed the ability to exploit inertia beneficially, enabling them to achieve higher efficiency partly at the cost of sacrificing orbital stability.

Consideration herein is the stability issue of peaked solitary wave solution for the modified Camassa-Holm-Novikov equation, which is derived from the shallow water theory. This wave configuration accommodates the ordered trains of the modified Camassa-Holm-Novikov-peaked solitary solution. With the application of conservation laws and the monotonicity property of the localized energy functionals, we prove the orbital stability of this wave profile in the H1(R){H}^{1}\left({\mathbb{R}}) energy space according to the modulation argument.

In this paper, we generalized the results of the following chemotaxis model with the nonlinear degenerate viscosity $ \begin{equation*} \begin{cases} u_{t} -\chi (uv)_{x} = D(u^{m})_{xx}, \\ v_{t} -u_{x} = 0, \end{cases} \end{equation*} $ by introducing the following general initial perturbation $ \begin{equation*} \begin{split} \int_{-\infty}^{+\infty}\kappa(Z_0|\tilde{Z})dx where $ \kappa $ is the relative entropy function defined in Eq (2.24). We further employed the relative entropy method by choosing the specific shift function. According to the estimates with the cutoff version, and overcoming the complexity caused by the porous media diffusion, the nonlinear orbital stability of traveling waves was established under small amplitude and general perturbations.

In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $ L^{2} $-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $ p_{2} = 1+\frac{2+\alpha}{N} $, we explore the limiting dynamics of blow-up solutions to the $ L^{2} $-critical Choquard equation with $ L^{2} $-subcritical perturbation, including the $ L^{2} $-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $ L^{2} $-subcritical perturbation, focusing $ L^{2} $-critical perturbation and defocusing $ L^{2} $-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.

In this article, we consider the global existence and stability issues of the nonlinear Schrödinger equation with partial confinement. First, by establishing some new cross-invariant manifolds and variational problems, a new sharp criterion of global existence is derived in the $ L^{2} $-critical and $ L^{2} $-supercritical cases. Then, the existence of orbitally stable standing waves is obtained in the $ L^{2} $-subcritical and $ L^{2} $-critical cases by taking advantage of the profile decomposition technique. Our work extends and complements some earlier results.

In this work, we show some results concerning the orbital stability of dnoidal wave solutions to some Benjamin-Bona-Mahony equations (BBM equations henceforth). First, by the standard argument, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type. Then, we show that this type of solutions are orbitally stable by perturbations with the same period L. The major tools to obtain these results are the Grillaks, Shatah and Strauss' general theory in the periodic case. The results in the present paper extend some previous stability results for the BBM equations.

This paper investigates the orbital stability of periodic standing waves for the following coupled Klein-Gordon-Zakharov equations $ \begin{equation*} \left\{ \begin{aligned} &u_{tt}-u_{xx}+u+\alpha uv+\beta|u|^{2}u = 0, \ &v_{tt}-v_{xx} = (|u|^{2})_{xx}, \end{aligned} \right. \end{equation*} $ where $\alpha>0$ and $\beta$ are two real numbers and $\alpha>\beta$. Under some suitable conditions, we show the existence of a smooth curve positive standing wave solutions of dnoidal type with a fixed fundamental period L for the above equations. Further, we obtain the stability of the dnoidal waves for the coupled Klein-Gordon-Zakharov equations by applying the abstract stability theory and combining the detailed spectral analysis given by using Lam\'{e} equation and Floquet theory. When period $L\rightarrow\infty$, dnoidal type will turn into sech-type in the sense of limit. In such case, we can obtain stability of sech-type standing waves. In particular, $\beta = 0$ is advisable, we still can show the the stability of the dnoidal type and sech-type standing waves for the classical Klein-Gordon-Zakharov equations.

For the double power one dimensional nonlinear Schrödinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.

Vertical transmission refers to the process in which a mother transmits bacteria or viruses to her offspring through childbirth, and this phenomenon takes place commonly in nature. This paper formulates an SIR epidemic model where the impact of vertical transmission and two state-dependent pulse controls are both taken into consideration. Using the Poincare´map, an analogue of Poincare´ criterion and the method of related qualitative analysis, the existence and the stability of a positive order-1 or order-2 periodic solution for the epidemic model are proved. Furthermore, phase diagrams are demonstrated by means of numerical simulations, illustrating the feasibility and correctness of our main results. It can be further implied that the epidemic can be controlled to a certain extent, with vertical transmission reduced and timely state-dependent pulse controls carried out.

In this article, we deal with orbital stability and Zhukovskiǐ quasi-stability of periodic or recurrent orbits in an impulsive dynamical system defined in the n-dimensional Euclidean space Rn{{\mathbb{R}}}^{n}. We show that for a periodic orbit of an impulsive system, its asymptotically orbital stability is equivalent to the asymptotically Zhukovskiǐ quasi-stability, and for a recurrent orbit, the orbital stability is equivalent to the Zhukovskiǐ quasi-stability.

We study the existence and orbital stability of normalized solutions of the biharmonic equation with the mixed dispersion and a general nonlinear term $ \begin{equation*} \gamma\Delta^2u-\beta\Delta u+\lambda u = f(u), \quad x\in\mathbb{R}^N \end{equation*} $ with a priori prescribed $ L^2 $-norm constraint $ S_a: = \left\{u\in H^2(\mathbb{R}^N):\int_{\mathbb{R}^N}|u|^2dx = a\right\}, $ where $ a > 0 $, $ \gamma > 0, \beta\in\mathbb{R} $ and the nonlinear term $ f $ satisfies the suitable $ L^2 $-subcritical assumptions. When $ \beta\geq0 $, we prove that there exists a threshold value $ a_0\geq0 $ such that the equation above has a ground state solution which is orbitally stable if $ a > a_0 $ and has no ground state solution if $ a < a_0 $. However, for $ \beta < 0 $, this case is more involved. Under an additional assumption on $ f $, we get the similar results on the existence and orbital stability of ground state. Finally, we consider a specific nonlinearity $ f(u) = |u|^{p-2}u+\mu|u|^{q-2}u, 2 < q < p < 2+8/N, \mu < 0 $ under the case $ \beta < 0 $, which does not satisfy the additional assumption. And we use the example to show that the energy in the case $ \beta < 0 $ exhibits a more complicated nature than that of the case $ \beta\geq0 $.

The aim of this paper is to study the existence of stable standing waves for the following nonlinear Schrödinger type equation with mixed power-type and Choquard-type nonlinearities $ i\partial_t \psi+\Delta \psi+\lambda | \psi|^q \psi+\frac{1}{|x|^\alpha}\left(\int_{\mathbb{R}^N}\frac{| \psi|^p}{|x-y|^\mu|y|^\alpha}dy\right)| \psi|^{p-2} \psi = 0, $ where $ N\geq3 $, $ 0 < \mu < N $, $ \lambda > 0 $, $ \alpha\geq0 $, $ 2\alpha+\mu\leq{N} $, $ 0 < q < \frac{4}{N} $ and $ 2-\frac{2\alpha+\mu}{N} < p < \frac{2N-2\alpha-\mu}{N-2} $. We firstly obtain the best constant of a generalized Gagliardo-Nirenberg inequality, and then we prove the existence and orbital stability of standing waves in the $ L^2 $-subcritical, $ L^2 $-critical and $ L^2 $-supercritical cases by the concentration compactness principle in a systematic way.

In this paper, we are concerned with the nonlinear fractional Schrödinger equation. We extend the result of Guo and Huo and prove that the Cauchy problem of the nonlinear fractional Schrödinger equation is global well-posed in H32−γ(R) with 12≤γ<1. In view of the complexity of the nonlinear fractional Schrödinger equation itself, the local smoothing effect and maximal function estimates are not enough for presenting the global well-posedness for the nonlinear fractional Schrödinger equation. In this paper, we use a suitably iterative scheme and complete the global well-posed result for Equation (R). Moreover, we obtain the orbital stability of standing waves for the above equations via establishing the profile decomposition of bounded sequences in Hs(RN) (0

In this paper, we investigate the existence of stable standing waves for the nonlinear Schr\"{o}dinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities \[ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi=0,~~(t,x)\in [0,T^\star)\times \mathbb{R}^N. \] By using concentration compactness principle, when one nonlinearity is focusing and $L^2$-critical, the other is defocusing and $L^2$-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper \cite {13} to the $L^2$-critical and $L^2$-supercritical nonlinearities.

In this paper we will continue the analysis of two dimensional Schr?dinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

Abstract In this paper, we investigate the orbital stability of solitary waves for the following generalized long-short wave resonance equations of Hamiltonian form: 0.1 { i u t + u x x = α u v + γ | u | 2 u + δ | u | 4 u , v t + β | u | x 2 = 0 . $$ \textstyle\begin{cases} iu_{t}+u_{{xx}}=\alpha uv+\gamma \vert u \vert ^{2}u+\delta \vert u \vert ^{4}u, \\ v_{t}+\beta \vert u \vert ^{2}_{x}=0. \end{cases} $$ We first obtain explicit exact solitary waves for Eqs. (0.1). Second, by applying the extended version of the classical orbital stability theory presented by Grillakis et al., the approach proposed by Bona et al., and spectral analysis, we obtain general results to judge orbital stability of solitary waves. We finally discuss the explicit expression of det ( d ″ ) $\det (d^{\prime \prime })$ in three cases and provide specific orbital stability results for solitary waves. Especially, we can get the results obtained by Guo and Chen with parameters α = 1 $\alpha =1$ , β = − 1 $\beta =-1$ , and δ = 0 $\delta =0$ . Moreover, we can obtain the orbital stability of solitary waves for the classical long-short wave equation with γ = δ = 0 $\gamma =\delta =0$ and the orbital instability results for the nonlinear Schrödinger equation with β = 0 $\beta =0$ .

An impulsive system with a linear continuous-time part and a nonlinear discrete-time part is considered. A criterion for exponential orbital stability of its periodic solutions is obtained. The proof is based on linearization by the first approximation of an auxiliary discrete-time system. The formulation of the criterion depends significantly on a number of impulses per period of the solution. The paper provides a mathematical rationale for some results previously examined in mathematical biology by computer simulations.

In the past few years, distant retrograde orbits (DROs) have become increasingly popular due to their conspicuous stability. Nevertheless, it is this characteristic that results in the challenge to the design of transfer orbits into/out of DROs. This paper investigates the DROs around Ganymede in order to utilize their dynamical characteristics for Jupiter system exploration. In particular, the DRO family is calculated by numerical integration and numerical continuation, higher-period orbits near the DROs are detected using bifurcation theory, and characteristics including orbital stability and invariant manifolds of these orbits are investigated through stability indices and manifold theory. The stability of DROs and the higher-period orbits are first investigated in the circular restricted three-body problem and are then verified in a third-body gravitation perturbation model. The results show that the strong stability of DROs makes it possible to observe the Galilean moons for long periods and that the higher-period orbits that bifurcate from the DROs offer additional insight into the motion of probes approaching/departing from the vicinities of the DROs. Further investigation of the invariant manifolds around higher-period orbits reveals the feasibility of utilizing the DRO family and the nearby unstable structures for multi-target exploration and low-energy transfer between the Galilean moons.

Peakons and periodic peakons are two kinds of special symmetric traveling wave solutions, which have important applications in physics, optical fiber communication, and other fields. In this paper, we study the orbital stability of peakons and periodic peakons for a generalized Camassa–Holm equation with quadratic and cubic nonlinearities (mCH–Novikov–CH equation). It is a generalization of some classical equations, such as the Camassa–Holm (CH) equation, the modified Camassa–Holm (mCH) equation, and the Novikov equation. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that the peakons and periodic peakons are orbitally stable under small perturbations in the energy space.

Periodic motions are often desired in engineering applications. The traditional method of trajectory planning followed by feedback tracking may work well for maintaining the planned motion. However, it is desired for certain applications to achieve the motion as autonomous rather than forced oscillations so as to adapt the motion profile in response to disturbance and changing environment. This motivates us to consider feedback control design to embed a limit cycle oscillation in the closed-loop dynamics with the notion of orbital stability. Here we provide a tutorial review of basic concepts and analysis tools for limit cycles arising from autonomous nonlinear dynamics. The Floquet theory and contraction analysis are reviewed, simple planar oscillators are illustrated, and synchronization and coordination of coupled oscillators are discussed.

Abstract In present work, in order to avoid the spread of disease, the impulse control strategy is implemented to keep the density of infections at a low level. The SIR epidemic model with resource limitation including a nonlinear impulsive function and a state-dependent feedback control scheme is proposed and analyzed. Based on the qualitative properties of the corresponding continuous system, the existence and stability of positive order-k ( k ∈ Z + $k\in\mathbf{Z}^{+}$ ) periodic solution are investigated. By using the Poincaré map and the geometric method, some sufficient conditions for the existence and stability of positive order-1 or order-2 periodic solution are obtained. Moreover, the sufficient conditions which guarantee the nonexistence of order-k ( k ≥ 3 $k\geq3$ ) periodic solution are given. Some numerical simulations are carried out to illustrate the feasibility of our main results.

Abstract Considered here is the stability problem of solitary traveling waves with non-zero boundary of an equation describing the free surface waves of moderate amplitude in the shallow water regime. We employ a transform to convert the stability problem of solitary waves with non-zero boundary to the same one of solitary waves vanishing at infinity for a new equation, such that the abstract stability theorem proposed by Grillakis et al. can work on it. We then show that the solitary traveling waves with non-zero boundary are orbitally stable under certain parameter conditions.

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

In this article we prove of the orbital stability of the ground state for logarithmic Schrodinger equation in any dimension and under nonradial perturbations. This general stability result was announced by Cazenave and Lions [9, Remark II.3], but no details were given there.

Based on the concentration compactness principle, we shoe the existence of ground state solitary wave solutions for one-dimensional Schrodinger-Poisson systems with large L2-norm in the energy space. We also obtain orbital stability for ground state solitary waves.

In 1995 for a linear oscillator, Myshkis imposed a constant impulse to the velocity, each moment the energy reaches a certain level. The main feature of the resulting system is that it defines a nonlinear discontinuous semigroup. In this note we study the orbital stability of a one-parameter family of periodic solutions and state the existence of a period-doubling bifurcation of such solutions.

In this article, using a variational approach, we establish the nonlinear orbital stability of ground state solitary waves for a 2D Boussinesq-Benney-Luke system that models the evolution of three dimensional long water waves with small amplitude in the presence of surface tension.

Locomotion systems with variant constraints are familiar in real world applications, but the dynamics and control issues of variant constraint systems have not been sufficiently discussed to date. From the viewpoint of Lagrange−d’Alembert equations with additional variable constraints, this paper investigates the modeling approaches of a class of hybrid dynamical systems (HDS) with instantaneously variant constraints and the switching control techniques of stabilizing the HDS to given periodic orbits. It is shown that under certain conditions there possibly exist zero impact periodic orbits in the HDS, and the HDS can be stabilized to the period-one orbits by a linear controller with only partial state feedback, even though the HDS are generally underactuated nonholonomic systems. As an example, a one-legged planar hopping robot is employed to demonstrate the main results of modeling and control of a class of HDS.

This article addresses orbital stability of periodic travelling-wave solutions for coupled nonlinear Schrodinger equations. We prove the existence of smooth curves of periodic travelling-wave solutions depending on the dnoidal-type functions. Orbital stability analysis is developed in the context of Hamiltonian systems. We consider both the stability problem by periodic perturbations which have the same fundamental period as the corresponding periodic wave and the stability problem by periodic perturbations having two or more times the minimal period as the corresponding periodic wave.

The generalized regularized long-wave equation admits a family of negative solitary waves. We show that there is a critical wave speed dividing the range of stable and unstable negative solitary waves. Our proofs of stability and instability are based on a variant of the general theory by Grillakis, Shatah and Strauss.

It is well known that the Kirchhoff equation admits infinitely many simple modes, i.e., time periodic solutions with only one Fourier component in the space variable(s). We prove that for some form of the nonlinear term these simple modes are stable provided that their energy is large enough. Here stable means orbitally stable as solutions of the two-modes system obtained considering initial data with two Fourier components.

In this paper we review recent results on the existence of non-topological solitons in classical relativistic nonlinear field theories. We follow the Coleman approach, which is based on the existence of two conservation laws, energy and charge. In particular we show that under mild assumptions on the nonlinear term it is possible to prove the existence of solitons for a set of admissible charges. This set has been studied for the nonlinear Klein–Gordon equation, and in this paper we state new results in this direction for the Klein–Gordon–Maxwell system.

A distributed delay model of a class of three-neuron network has been investigated. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. A closed interval for a parameter of the system is identified in which Hopf-bifurcating periodic solution occurs for each point of such interval. The orbital stability of such bifurcating periodic solution at the extreme points of the interval is ascertained. Lastly global bifurcation aspect of such periodic solutions is studied. The results are illustrated by numerical simulations.