Relative asymptotic equivalence of dynamic equations on time scales

This paper aims to study the relative equivalence of the solutions of the following dynamic equations $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.

Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

On maintaining the stability of the equilibrium of nonlinear oscillators under conservative perturbations

The problem of Yu.N. Bibikov on maintaining the stability of the equilibrium position of two interconnected nonlinear oscillators under the action of small, in a certain sense, conservative perturbing forces is considered. With different methods of reducing the system to the Hamiltonian form, some features are revealed for the case when the perturbing forces of the interaction of two oscillators are potential. The conditions for preserving the stability and instability of the equilibrium of two oscillators for the case of sufficiently small disturbing forces are obtained. The problem of maintaining the stability of the equilibrium under conservative perturbations is also considered in the more general situation of an arbitrary number of oscillators with power potentials with rational exponents, which leads to the case of a generalized homogeneous potential of an unperturbed system. The example given shows the applicability of the proposed approach in the case when the order of smallness of the perturbing forces coincides with the order of smallness of the unperturbed Hamiltonian.

Number of Limit Cycles from a Class of Perturbed Piecewise Polynomial Systems

In this paper, we study Poincaré bifurcation of a class of piecewise polynomial systems, whose unperturbed system has a period annulus together with two invariant lines. The main concerns are the number of zeros of the first order Melnikov function and the estimation of the number of limit cycles which bifurcate from the period annulus under piecewise polynomial perturbations of degree [Formula: see text].

Damped Perturbations of Systems with Center-Saddle Bifurcation

An autonomous system of ordinary differential equations on the plane with a center-saddle bifurcation is considered. The influence of a class of time damped perturbations is investigated. The particular solutions tending to the fixed points of the limiting system are considered. The stability of these solutions is analyzed by Lyapunov function method when the bifurcation parameter of the unperturbed system takes critical and noncritical values. Conditions that ensure the persistence of the bifurcation in the perturbed system are described. When the bifurcation is broken, a pair of solutions tending to a degenerate fixed point of the limiting system appears in the critical case. It is shown that, depending on the structure and the parameters of the perturbations, one of these solutions can be stable, metastable or unstable, while the other solution is always unstable. The proposed theory is applied to the study of autoresonance capturing in systems with quadratically varying driving frequency.

The Admissible Control Correction Method in a Nonlinear Terminal Perturbed Problem

A solution of a nonlinear perturbed unconstrained point-to-point control problem, in which the unperturbed system is differentially flat, is considered in the paper. An admissible open-loop control in it is constructed using the covering method. The main part of the obtained admissible control correction in the limit problem is found by expanding the perturbed problem solution in series by the perturbation parameter. The first term of the expansion is determined by A.N. Tikhonov’s regularization of the Fredholm integral equation of the first kind. As shown by numerical experiments, the found structure of an admissible control allows one to find the final form of high precision point-to-point control based on the solution of an auxiliary variational problem in its neighborhood.

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

Inference of Granger-causal relations in molecular systems — a case study of the functional hierarchy among actin regulators in lamellipodia

Many cell regulatory systems implicate significant level of nonlinearity and redundancy among its components. The actin regulatory network driving the formation of a lamellipodium is prototypical of such system, containing tens of actin nucleating and modulating molecules with strong functional overlap. Due to instantaneous compensation, the standard paradigm of perturbing individual components followed by phenotyping provides limited information on the roles the targeted component plays in the unperturbed system. Accordingly, despite the very rich data on lamellipoidial actin assembly, we have an incomplete understanding of the actual contributions the individual modulators make towards the lamellipodial dynamics. Here, we present a case study of the implementation of Granger-causal inference of the functional cause-effect relations among actin regulators, using the constitutive image fluctuations reporting regulator recruitment/activation as the input. Our analytical pipeline defines specific active regions of actin regulators within the lamellipodia and lamella and establishes actin-dependent and actin-independent causal relations of actin regulators with F-actin and edge motion. We demonstrate the specificity and sensitivity of the pipeline and suggest the existence of two discrete yet independently operating F-actin networks that drive edge motion.

Geometric speed limit of neutrino oscillation

We investigate geometric quantum speed limit of neutrino oscillations in a presence of matter and CP-violation. We show that periodicity in the speed limit present in an unperturbed system becomes damped by interaction with a normal matter and decoherence. We also show that (hypothetical) CP-violation causes enhancement of periodicity and increases amplitude of an oscillating quantum speed limit and can quantify CP-violation.