Nonautonomous Case Research Articles

Phase Difference in Lateral Synchronization of Pedestrian Floors Using a Modified Hybrid Van der pol/Rayleigh Oscillator

The modified hybrid Van der Pol/Rayleigh (MHVR) oscillator was originally proposed by the authors to model the lateral oscillations of a pedestrian walking on a rigid floor and it was shown that for the autonomous case, the MHVR oscillator can correctly fit the experimental data. The case of a pedestrian walking on a laterally moving floor is modeled by a nonautonomous oscillator. The case of a floor subjected to a harmonic lateral motion has been then studied by the authors, with focus on the amplitude and stability of the entrained response, i.e. the response having the same frequency as that of the given periodic excitation. For the nonautonomous (moving floor) case, the main focus of this paper is on the analysis of the phase difference between the oscillator entrained response and the external excitation. Both analytical and numerical calculations have been performed. The approximate analytical method is the harmonic balance method. Then, the model is used to represent the experimental results for the pedestrian lateral oscillations during walking. Comparison is made for the examples along with discussions.

Geometric approach to dynamics obtained by deformation of time-dependent Lagrangians

The relationship of equations of motion of a Lagrangian \(\phi (L)\) to those of L is studied in the non-autonomous case, and the question of the existence of a function \(\phi \) such that \(\phi (L)\) is dynamically equivalent to L is answered.

Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.

Extinction, periodicity and multistability in a Ricker model of stage-structured populations

We study the dynamics of a second-order difference equation that is derived from a planar Ricker model of two-stage (adult-juvenile) biological populations. We obtain sufficient conditions for global convergence to zero in the non-autonomous case yielding general conditions for extinction in the biological context. We also study the dynamics of an autonomous special case of the equation that generates multistable periodic and non-periodic orbits in the positive quadrant of the plane.

Directional Point-Contact Josephson Junctions on Ba0.4K0.6(FeAs)2 Single Crystals

We investigated the Josephson effect between the conventional superconductor Pb0.7In0.3 and Ba0.4K0.6 (FeAs)2 single crystals by performing directional point-contact experiments. The junctions were irradiated with microwaves and the behavior of the Shapiro steps as a function of the square root of the microwave power was studied by means of the RSJ model extended to the nonautonomous case with an rf current-source term. The analysis reveals that, for current injection along the ab plane, the current-phase relation deviates from being simply described by the standard sin(ϕ) (as we indeed observed along the c axis) and a dominant sin(2ϕ) component is observed. The results are compared with a theory of the dc Josephson effect, formulated by using a method based on the calculation of temperature Green’s function in the junction within the tight-binding model (Burmistrova et al., Phys. Rev. B 91, 214501, 2015). The comparison shows that the experimental results here presented strongly favor an s ±-wave symmetry of the order parameter in this compound.

Attractors for autonomous and nonautonomous 3D Benjamin–Bona–Mahony equations

In this paper we consider the long time behavior of the three dimensional Benjamin–Bona–Mahony equations for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Benjamin–Bona–Mahony equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to V′. For the nonautonomous case, we only assume that f(x, t) is translation bounded instead of translation compact. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Benjamin–Bona–Mahony equations and also obtain the existence of the uniform attractor.

Differentiability of Palmer's linearization theorem and converse result for density functions

We study differentiability properties in a particular case of the Palmer's linearization theorem, which states the existence of a homeomorphism H between the solutions of a linear ODE system having exponential dichotomy and a quasilinear system. Indeed, if the linear system is uniformly asymptotically stable, sufficient conditions ensuring that H is a C2 preserving orientation diffeomorphism are given. As an application, we generalize a converse result of density functions for a nonlinear system in the nonautonomous case.

Spectral invariants of periodic nonautonomous discrete dynamical systems

For an interval map, the poles of the Artin–Mazur zeta function provide topological invariants which are closely connected to topological entropy. It is known that for a time-periodic nonautonomous dynamical system F with period p, the p-th power [ζF(z)]p of its zeta function is meromorphic in the unit disk. Unlike in the autonomous case, where the zeta function ζf(z) only has poles in the unit disk, in the p-periodic nonautonomous case [ζF(z)]p may have zeros. In this paper we introduce the concept of spectral invariants of p-periodic nonautonomous discrete dynamical systems and study the role played by the zeros of [ζF(z)]p in this context. As we will see, these zeros play an important role in the spectral classification of these systems.

Pullback attractors for generalized evolutionary systems

We give an abstract framework for studying nonautonomous PDEs,called a generalized evolutionary system. In this setting, we define the notionof a pullback attractor. Moreover, we show that the pullback attractor, inthe weak sense, must always exist. We then study the structure of these attractors and the existence of a strong pullback attractor. We then apply ourframework to both autonomous and nonautonomous evolutionary systems asthey first appeared in earlier works by Cheskidov, Foias, and Lu. In this con-text, we compare the pullback attractor to both the global attractor (in theautonomous case) and the uniform attractor (in the nonautonomous case). Finally, we apply our results to the nonautonomous 3D Navier-Stokes equationson a periodic domain with a translationally bounded force. We show that theLeray-Hopf weak solutions form a generalized evolutionary system and mustthen have a weak pullback attractor.

On the Use of Nonlinear Boundary-Value Problems to Estimate the Cloud-Formation Potential of Aerosol Particles

This paper investigates the transient growth of aerosol particles in a humid environment. It seeks to explore the dependence of the fraction of droplet-forming particles on statistical properties of the distribution of dry particle diameters, as well as on the rate of temperature decay associated with vertical motion through the atmosphere. Low-dimensional, autonomous models are investigated using basic tools of dynamical systems analysis that establish the parameter-dependent existence and stability of families of equilibrium distributions of wet particle diameters. In the fully nonautonomous case, an original heuristic parameterization of the fraction of droplet-forming particles is derived in terms of a scalar, nonlinear boundary-value problem. To address the failure of the heuristic parameterization to account for the potential of a dynamic reversal of growth following an initial increase in particle diameter, a finely resolved, high-dimensional boundary-value formulation for the aerosol dynamics is investigated using methods of numerical continuation. In order to reduce the computational complexity of the numerical scheme, an adaptive asynchronous discretization algorithm is developed in which the state variables are partitioned across distinct temporal meshes. Results obtained from the proposed numerical scheme are compared with the estimated fractions of droplet-forming particles based on available approximate criteria in the literature and demonstrate relative errors as large as 20The analysis further uncovers a hitherto-unknown linear relationship between the arithmetic expectation and variance of a log-normal size distribution of dry diameters for fixed fractions of droplet-forming particles.

Isotropic submanifolds and the inverse problem for mechanical constrained systems

We give a new characterization of the inverse problem of the calculus of variations that is easily extended to constrained systems, both in the autonomous and non-autonomous cases. The transition from unconstrained to constrained systems is given by passing from Lagrangian submanifolds to isotropic ones. If the constrained system is variational we use symplectic techniques to extend these isotropic submanifolds to Lagrangian ones and describe the solutions of the constrained system as solutions of a variational problem without constraints. Mechanical examples such as the rolling disk are provided to illustrate the main results.

On the linearization theorem for nonautonomous differential equations

This paper presents an improvement of Palmer's linearization theorem in [10]. Palmer's linearization theorem extended the Hartman–Grobman theorem to the nonautonomous case. It requires two essential conditions: (i) the nonlinear term is bounded and Lipschitzian; (ii) the linear system as a whole possesses an exponential dichotomy. The main purpose of this paper is to weaken assumptions (i) and (ii). Also in this paper we prove that the topologically equivalent function H(t,x) in the linearization theorem is always Hölder continuous (and has a Hölder continuous inverse), so as a result we generalize and improve Palmer's linearization theorem.

Attractors of equations of non-Newtonian fluid dynamics

This survey describes a version of the trajectory-attractor[4] method, which is applied to study the limit asymptotic behaviour of solutions of equations of non-Newtonian fluid dynamics. The trajectory-[4]attractor method emerged in papers of the Russian mathematicians Vishik and Chepyzhov and the American mathematician Sell under the condition that the corresponding trajectory spaces be invariant under the translation semigroup. The need for such an approach was caused by the fact that for many equations of mathematical physics for which the Cauchy initial-value problem has a global (weak) solution with respect to the time, the uniqueness of such a solution has either not been established or does not hold. In particular, this is the case for equations of fluid dynamics. At the same time, trajectory spaces invariant under the translation semigroup could not be constructed for many equations of non-Newtonian fluid dynamics. In this connection, a different approach to the construction of trajectory attractors for dissipative systems was proposed in papers of Zvyagin and Vorotnikov without using invariance of trajectory spaces under the translation semigroup and is based on the topological lemma of Shura– Bura. This paper presents examples of equations of non-Newtonian fluid dynamics (the Jeffreys system describing movement of the Earth's crust, the model of motion of weak aqueous solutions of polymers, a system with memory) for which the aforementioned construction is used to prove the existence of attractors in both the autonomous and the non-autonomous cases. At the beginning of the paper there is also a brief exposition of the results of Ladyzhenskaya on the existence of attractors of the two-dimensional Navier–Stokes system and the result of Vishik and Chepyzhov for the case of attractors of the three-dimensional Navier–Stokes system. Bibliography: 34 titles.

Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays

In this paper, we deal with a discrete Lotka-Volterra predator-prey model with time-varying delays. For the general non-autonomous case, sufficient conditions which ensure the permanence and global stability of the system are obtained by using differential inequality theory. For the periodic case, sufficient conditions which guarantee the existence of a unique globally stable positive periodic solution are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.MSC:34K20, 34C25, 92D25.

Dynamic behavior of positive solutions for a leslie predator-prey system with mutual interference and feedback controls.

We consider a Leslie predator-prey system with mutual interference and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain some sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain some sufficient conditions which guarantee the existence, uniqueness, and stability of a positive periodic solution.

Entropy of scalar reaction-diffusion equations

We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms.

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lu system, and fractional-order Duffing system.

Existence and construction of a function with sign semidefinite derivative for dynamic systems

The problem of finding a function with sign semidefinite derivative along the solutions to the system of ordinary differential equations is considered. It is shown that a sufficient existence condition is the existence of trajectories that leave a neighborhood of the equilibrium point both in forward or backward time. Two approaches to the construction of the desired function in the autonomous and nonautonomous cases are proposed. The results are illustrated by the example of a nonlinear autonomous system (Artstein's circle).

A modified hybrid Van der Pol/Rayleigh model for the lateral pedestrian force on a periodically moving floor

A single degree of freedom self-sustained oscillator is proposed in order to model the lateral oscillations of a pedestrian walking on a periodically moving floor and particularly on a shaking table. In a previous work, the authors have shown that a suitable form for the restoring force of such an oscillator corresponds with a modified hybrid Van der Pol/Rayleigh (MHVR) model, whose associated parameters have been identified in the autonomous (rigid floor) case for a group of twelve pedestrians.The MHVR oscillator is analyzed here for the non-autonomous case, where the moving floor is subjected to a harmonic excitation. It has been experimentally proven that in this case the pedestrian may change his (her) natural walking frequency and synchronize with the floor oscillation frequency: one says that the so-called “frequency entrainment” occurs. This means that, under certain conditions, the response frequency switches from the natural value to that of the external excitation. This paper discusses the steady “entrained” response of the MHVR model subjected to a harmonic excitation, in terms of response amplitude curves obtained using the Harmonic Balance Method. Experimental results available in the literature and involving pedestrians walking on a shaking table are compared with the model predictions for illustrative purposes.

Birth of the shape of a reachable set

We address a linear control system under geometric constraints on control and study its reachable sets starting at zero time from the origin. The main result is the existence of a limit shape of the reachable sets as the terminal time tends to zero. Here, a shape of a set stands for the set regarded up to an invertible linear transformation. Both autonomous and non-autonomous cases are considered.