In this work, we propose a new method to verify the orbital stability criterion of traveling wave solutions. Here the trajectories corresponding to smooth solitary waves satisfy a certain form, i.e., 12y¯2+G(x¯)=h¯, where G(x¯) is analytic and h¯ is a real parameter related to the wave speed c. This is a successful application of ordinary differential equations to partial differential equations, where the idea of monotonicity of the period function of the planar systems is used to verify the orbital stability of traveling wave solutions of nonlinear evolution equations. And the orbital stability of smooth solitary waves of the Kudryashov-Sinelshchikov equation and the cubic Camassa-Holm-type equation is obtained using this method. In addition, the details of the proof for these two examples exhibit slight variations due to the more complex trajectories corresponding to the smooth solitary waves of the cubic Camassa-Holm-type equation.
Living systems exhibit behaviors in which they actively explore actions suited to changing constraints or environmental conditions in order to maintain their functionality. A major challenge in understanding such active behaviors lies in the fact that the relationship between internal system dynamics and external constraints cannot be predetermined. In other words, it is essential to elucidate the dynamical principles through which these relationships are formed. The development of decentralized autonomous control methods is expected to contribute to the discovery of such principles. A central issue is how internal states should be modulated in response to changes in the external environment. To achieve environmental adaptation under unknown environmental changes, the internal state must be described solely in terms of internal variables that encode the system’s internal condition. That is, environmental adaptability must be represented as a self-organizing dynamical process. Furthermore, environmental adaptability is observed even in unicellular organisms that lack brain-like information-processing organs, suggesting the existence of decentralized autonomous control mechanisms. In this paper, we propose a continuum model of a phase-control module for locomotion systems that autonomously adjusts the relationship between functional states and internal system parameters according to situational demands, within the framework of self-organization theory. In the proposed model, body segments are connected by actively oscillating springs, and modules composed of interactions between two populations of oscillators are incorporated. One population controls the oscillation period of the springs, whereas the other regulates the timing of ground contact. By appropriately configuring the phase difference between these oscillator populations, unidirectional crawling locomotion is generated. Through mathematical analysis of the continuum model, we clarify the conditions under which locomotion behavior emerges.
This paper investigates the nonlinear Rayleigh–Taylor (RT) instability of a three-dimensional incompressible viscous quantum magnetohydrodynamic (QMHD) model for dense magnetized plasmas, where quantum effects become dynamically relevant. The flow is posed in a horizontally periodic and vertically unbounded domain, and the equilibrium consists of a vertically stratified density profile under gravity together with a constant horizontal magnetic field. Within this viscous QMHD framework, we establish nonlinear Hadamard instability of the Rayleigh–Taylor equilibrium. For the linearized problem, we derive an explicit direction-dependent instability criterion σhz(Q,M,r*)>1, where Q and M denote the dimensionless quantum and magnetic parameters, respectively; here r*=|k1*|/|k*| describes the orientation of the horizontal wave vector relative to the imposed field. The criterion separates gravitational destabilization from two stabilizing mechanisms: quantum dispersion penalizes density-gradient variations, whereas horizontal magnetic tension acts only on perturbations with a component along the magnetic field. Consequently, transverse modes with k1*=0 are not directly stabilized by the horizontal field, and the magnetic contribution to the instability threshold vanishes for such modes.
This paper is devoted to the study of dynamics of a stochastic memristive Hindmarsh-Rose lattice system with additive noise. Based on the discussion of global existence of solutions, the pullback absorbing property and asymptotic compactness of the random dynamical system are first proved in the infinite lattice space through Ornstein-Uhlenbeck transformation and uniform tail estimates. In this way, the existence of a unique global random attractor for this stochastic lattice system is successfully obtained. In addition, the existence of invariant measures for the transition semigroup of the system is also established via the Krylov-Bogolyubov method. Moreover, the limiting behavior of these invariant measures is further investigated to show that any limit of a sequence of invariant measures of the perturbed systems must be an invariant measure of the limiting deterministic system.
We construct pointwise bounded controls that steer a finite, periodic, fully actuated active harmonic lattice with Rayleigh–Helmholtz velocity forcing to flock states in finite time. In our setting, an initial Lyapunov-feedback phase brings the velocities and elastic forces into a prescribed small neighborhood of zero. Then, subsequent trajectory planning and cancellation of the known drift achieve exact arrival at a moving flock. For prescribed stationary or moving profiles satisfying a strict control-admissibility condition, a two-stage interpolation reaches the target. We provide explicit sufficient time intervals for the stages following the initial feedback phase. Finally, we formulate the associated minimum-time problem and its formal Hamilton–Jacobi equation, leaving the continuity of the value function open.
