Nonautonomous Case Research Articles

Averaging principles for multiscale stochastic Cahn–Hilliard system

In this paper, we will establish averaging principles for the multiscale stochastic Cahn–Hilliard system. The stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. Under suitable conditions, two kinds of averaging principle (the autonomous case and the nonautonomous case) are proved, and as a consequence, the multiscale system can be reduced to a single stochastic Cahn–Hilliard equation (averaged equation) with a modified coefficient, the slow component of multiscale stochastic Cahn–Hilliard system towards to the solution of the averaged equation in moment (the autonomous case) and in probability (the nonautonomous case).

Long-Time Dynamics of the Wave Equation with Nonlocal Weak Damping and Super-Cubic Nonlinearity in 3-D Domains, Part II: Nonautonomous Case

Long-Time Dynamics of the Wave Equation with Nonlocal Weak Damping and Super-Cubic Nonlinearity in 3-D Domains, Part II: Nonautonomous Case

Boundedness and stability of global solutions for some superlinear and nonautonomous heat equations

We consider the initial-boundary and boundary only problems for some superlinear and nonautonomous heat equations with subcritical nonlinear terms, and extend some of the well known results for autonomous heat equations to the corresponding nonautonomous case. More precisely, we first establish a priori estimates for global positive solutions of the initial-boundary problem for some superlinear and nonautonomous heat equations with subcritical nonlinear terms by introducing two different kinds of rescaling techniques. Next we get the existence of global bounded positive solutions which do not vanish in infinite time for the corresponding boundary only problem, and then we obtain the stability of such global bounded positive solutions by using a technical auxiliary function. This implies that such global bounded solutions are just the threshold solutions lying on the borderline between blow-up and global existence. At last we devote our attention to the more general threshold solutions for the initial-boundary problem of the superlinear and nonautonomous heat equations.

Dynamical analysis of an inverted pendulum with positive position feedback controller approximate uniform solution

The inverted pendulum is controlled in this article by using the nonlinear control theory. From classical analytical mechanics, its substructure equation of motion is derived. Because of the inclusion of the restoring forces, the Taylor expansion is employed to facilitate the analysis. An estimated satisfactory periodic solution is obtained with the aid of the modified Homotopy perturbation method. A numerical technique based on the fourth-order Runge–Kutta method is employed to justify the previous solution. On the other hand, a positive position feedback control is developed to dampen the vibrations of an IP system subjected to multi-excitation forces. The multiple time scale perturbation technique of the second order is introduced as a mathematical method to solve a two-degree-of-freedom system that simulates the IP with the PPF at primary and 1:1 internal resonance. The stability of these solutions is checked with the aid of the Routh–Hurwitz criterion. A set of graphs, based on the frequency response equations resulting from the MSPT method, is incorporated. Additionally, a numerical simulation is set up with RK-4 to confirm the overall controlled performance of the studied model. The quality of the solution is confirmed by the match between the approximate solution and the numerical simulation. Numerous other nonlinear systems can be controlled using the provided control method. Illustrations are offered that pertain to implications in design and pedagogy. The linearized stability of IP near the fixed points as well as the phase portraits is depicted for the autonomous and non-autonomous cases. Because of the static stability of the IP, it is found that its instability can be suppressed by the increase of both the generalized force as well as the torsional constant stiffness of the spring. Additionally, the presence of the magnetic field enhances the stability of IP.

Self-oscillating systems with controlled phase of external force

The purpose of this work is to study self-oscillatory systems under adaptive external action. This refers to the situation when the phase of the external action additionally depends on the dynamical variable of the oscillator. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with three-dimensional phase space. Methods. Methods of charts of dynamical regimes and charts of Lyapunov exponents are used, as well as the construction of phase portraits and stroboscopic sections. Results. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with a three-dimensional phase space. The pictures of characteristic dynamical regimes are described. Scenarios for the development of multidimensional chaos are described. Illustrations are given of the influence of the control parameter, which is responsible for the degree of dependence of the phase on the oscillator variable, on the dynamics of the system at different frequencies of action. Conclusion. The taling into account of the dependence of the phase on a dynamical variable leads to an extension of the tongues of subharmonic resonances, which are weakly expressed in the classical van der Pol oscillator. This is especially noticeable for even resonances of periods 2 and 4. For the generator of quasi-periodic oscillations in the non-autonomous case, three-frequency tori are observed, their regions begin to dominate with an increase in the adaptivity parameter, displacing the tongues of resonant two-frequency tori. A variety of multidimensional chaos characterized by an additional Lyapunov exponent close to zero is discovered, the possibility of developing hyperchaos as a result of destruction is shown.

Global dynamics of a mosquito population suppression model with stage and sex structure

<abstract><p>In this paper, we study a kind of mosquito population suppression model incorporating the growth stage as well as the sex structure of mosquitoes. For the general non-autonomous case, a threshold $ m^* $ for the number of sexually active sterile mosquitoes in the field is defined, and sufficient conditions for successful suppression and partial suppression of wild mosquito population are obtained. For the first special case when the release period of the sterile mosquito is equal to its sexual lifespan, we determined the initial population size under which the mosquito population can be suppressed finally for different release intensities. For the latter special case when sterile mosquitoes are released with a constant rate, we give a threshold $ u^* $ for the release rate, and investigate the dynamic behavior of the system based on this threshold. Finally, some numerical examples are presented to confirm the theoretical results.</p></abstract>

Non-occurrence of gap for one-dimensional non-autonomous functionals

Let \(F(y){{:}{=}}\int \nolimits _t^TL(s, y(s), y'(s))\,ds\) be a positive functional defined on the space \(W^{1,p}([t,T]; {{\mathbb {R}}}^n)\) (\(p\ge 1\)) of Sobolev functions with, possibly, one or both end point conditions. It is important, especially for the applications, to be able to approximate the infimum of F with the values of F along a sequence of Lipschitz functions satisfying the same boundary condition(s). Sometimes this is not possible, i.e., the so called Lavrentiev phenomenon occurs. This is the case of the seemingly innocent Manià’s Lagrangian \(L(s,y,y')=(y^3-s)^2(y')^6\) and boundary data \(y(0)=0, y(1)=1\); nevertheless in this situation the phenomenon does not occur with just the end point condition \(y(1)=1\). The paper focuses about the different set of conditions that are needed to avoid the Lavrentiev phenomenon for problems depending on the number of end point conditions that are considered. Under minimal assumptions on the (possibly) extended value, Lagrangian, we ensure the non-occurrence of the Lavrentiev phenomenon with just one end point condition, thus extending a milestone result by Alberti and Serra Cassano to non-autonomous case. We then introduce an additional hypothesis, satisfied when the Lagrangian is bounded on bounded sets, in order to ensure the non-occurrence of the phenomenon when dealing with both end point conditions; the result gives some new light even in the autonomous case.

Nonlinear Dynamics of Turbine Generator Shaft Trains: Evaluation of Bifurcation Sets Applying Numerical Continuation

Abstract The nonlinear dynamics of turbine generator shaft trains for power generation are investigated in this paper. Realistic models of rotors, pedestals, and nonlinear bearings of partial arc and lemon bore configuration are implemented to compose a nonlinear set of differential equations for autonomous (balanced) and nonautonomous (unbalanced—per ISO) cases. The solution branches of the dynamic system are evaluated with the pseudo-arc length continuation programed by the authors, and the respective limit cycles are evaluated by an orthogonal collocation method and investigated on their stability properties and quality of motion for the respective key design parameters for the rotor dynamic design of such systems, namely, bearing profile and respective pad length, preload and offset, pedestal stiffness and elevation (misalignment), and rotor slenderness. Model order reduction is applied to the finite element rotor model and the reduced system is validated in terms of unbalanced response and stability characteristics. The main conclusion of the current investigation is that the system has the potential to develop instabilities at rotating speeds lower than the threshold speed of instability (evaluated by the linear approach) for specific unbalance magnitude and design properties. Unbalance response (with stable and unstable branches) is evaluated in severely reduced time compared to this applying time integration methods, enabling nonlinear rotor dynamic design of such systems as a standard procedure, and revealing the complete potential of motions (not only local).

Dynamic Behaviors of an Obligate Commensal Symbiosis Model with Crowley–Martin Functional Responses

A two species obligate commensal symbiosis model with Crowley–Martin functional response was proposed and studied in this paper. For an autonomous case, local and global dynamic behaviors of the system were investigated, respectively. The conditions that ensure the existence of the positive equilibrium is coincidentla to the conditions of global stability of a positive equilibrium. For nonautonomous cases, persistent and extinction properties of the system are investigated.

Planar Schrödinger-Poisson system with critical exponential growth in the zero mass case

In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero mass{−Δu+ϕu=f(u),x∈R2,Δϕ=2πu2,x∈R2, where f∈C(R,R) has the critical exponential growth at infinity and there is no monotonicity restriction on f(u)/u3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ=liminf|t|→∞t2F(t)eα0t2>0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t2F(t)eα0t2 or tf(t)eα0t2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) [21] dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V(x)u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: −Δu=f(u) in R2.

Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property

The lattice sine-Gordon equation is an integrable partial difference equation on ${\mathbb Z}^2$, which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property.

Non-autonomous Linear Quadratic Non-cooperative Differential Games with Continuous Updating

The subject of this paper is a non-autonomous linear quadratic case of a differential game model with continuous updating. This class of differential games is essentially new where it is assumed that, at each time instant, players have or use information about the game structure defined on a closed time interval with a fixed duration. During the interval information about motion equations and payoff functions of players updates. It is non-autonomy that simulates this effect of updating information. A linear quadratic case for this class of games is particularly important for practical problems arising in the engineering of human-machine interaction. Here we define the Nash equilibrium as an optimality principle and present an explicit form of Nash equilibrium for the linear quadratic case. Also, the case of dynamic updating for the linear quadratic differential game is studied and uniform convergence of Nash equilibrium strategies and corresponding trajectory for a case of continuous updating and dynamic updating is demonstrated.

Inertial manifolds for parabolic differential equations: The fully nonautonomous case

<p style='text-indent:20px;'>We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t> s. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators <inline-formula><tex-math id="M1">\begin{document}$ (A(t))_{t\in { \mathbb {R}}} $\end{document}</tex-math></inline-formula> generates an evolution family <inline-formula><tex-math id="M2">\begin{document}$ (U(t,s))_{t\ge s} $\end{document}</tex-math></inline-formula> satisfying certain dichotomy estimates, and the nonlinear forcing term <inline-formula><tex-math id="M3">\begin{document}$ f(t,x) $\end{document}</tex-math></inline-formula> satisfies the Lipschitz condition such that certain dichotomy gap condition holds.</p>

A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

Abstract In this paper, we introduce measure-theoretic for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies in discrete type and the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we establish new variational principles for Bowen topological entropy for nonautonomous dynamical systems. JEL classification numbers: 37A35. Keywords: Nonautonomous, Measure-theoretical entropies, Variational principles.

Existence of Solutions for the Semilinear Abstract Cauchy Problem of Fractional Order

Abstract In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.

Mathematical modeling and assessment of barrier measures and temperature on the transmission and persistence of Novel coronavirus disease 2019 (COVID-19)

In December 2019, human cases of novel coronavirus infection were detected in Wuhan, China which have been named as COVID-19 by the World Health Organization (WHO). Since COVID-19 was first detected in China, the virus has reached more than 120 countries and was declared a global pandemic on March 11, 2020 by the WHO. In this paper, we have highlighted the influence of temperature on the spread of COVID-19. For this, the dynamic transmission of COVID-19 is modeled taking into account the influence of the temperature on the persistence of coronavirus in the environment. We also took into account the impact of proportion of people who respect the barrier measures published by the WHO on the scale of the COVID-19 pandemic. Taking into account the influence of the temperature on the persistence of the virus in the environment, the dynamic transmission has been described by a system of ordinary differential equations (ODEs). First, we analyzed the solutions of system in the case where the impact of the temperature on the virus is neglected. Second, we carried out the mathematical analysis of the solutions of the system in the non-autonomous case. Mathematical analyzes have enabled us to establish that the temperature and proportion of persons who respect the barrier rules can affect the spread of COVID-19. Some numerical simulations have been proposed to illustrate the behavior of the pandemic in some countries.

Nonlinear Compartment Models with Time-Dependent Parameters

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.

Jacobi multipliers and Hojman symmetry

The geometric intrinsic approach to Hojman symmetry is developed and use is made of the theory of the Jacobi last multipliers to find the corresponding conserved quantity for non divergence-free vector fields. The particular cases of autonomous Lagrangian and Hamiltonian systems are studied as well as the generalization of these results to normalizer vector fields of the dynamics. The nonautonomous cases, where normalizer vector fields play a relevant role, are also developed.

Bounded and almost periodic solvability of nonautonomous quasilinear hyperbolic systems

The paper concerns boundary value problems for general nonautonomous first-order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right-hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the “loss of smoothness” property of hyperbolic PDEs.

On the Second Lyapunov Method for Quaternionic Differential Equations

In this paper we mainly study the stability of quaternion-valued differential equations. We first establish the second Lyapunov method over quaternion field and then we use this theory to study the stability in both autonomous systems and periodic non-autonomous systems cases.