Linear Hamiltonian Differential Systems Research Articles

Crossing Limit Cycles Bifurcating from Two or Three Period Annuli in Discontinuous Planar Piecewise Linear Hamiltonian Differential Systems with Three Zones

The main topic studied in this article is the number of crossing limit cycles bifurcating from two or three period annuli in discontinuous planar piecewise linear Hamiltonian differential systems with three zones. With regard to the studies already published in the literature on this subject, we highlight the following five aspects of our work: (1) the expressions of the first order Melnikov functions for suitable perturbations of a piecewise Hamiltonian system with three zones separated by two parallel lines are obtained explicitly; (2) the way the Melnikov functions are obtained is different from what has already been done for similar studies; (3) the expressions of the Melnikov functions are used to estimate the number of crossing limit cycles that bifurcate simultaneously from period annuli under suitable polynomial perturbations; (4) since the piecewise Hamiltonian system studied here has no symmetry, the number of crossing limit cycles bifurcating from the period annuli is greater than or equal to those obtained in systems already studied; (5) unlike other similar studies, we present a concrete example of a piecewise linear near-Hamiltonian differential system in which the lower bound of the number of limit cycles that bifurcate from the period annuli is reached.

Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

AbstractIn this paper, we consider two linear Hamiltonian differential systems that depend in general nonlinearly on the spectral parameter λ and with Dirichlet boundary conditions. For the Hamiltonian problems, we do not assume any controllability and strict normality assumptions and also omit the classical Legendre condition for their Hamiltonians. The main result of the paper, the relative oscillation theorem, relates the difference of the numbers of finite eigenvalues of the two problems in the intervals and , respectively, with the so‐called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for and . As a corollary to the main result, we prove the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in . The consideration is based on the comparative index theory applied to the continuous case.

Non–limit‐circle and limit‐point criteria for symplectic and linear Hamiltonian systems

AbstractSeveral necessary and/or sufficient conditions for the existence of a non–square‐integrable solution of symplectic dynamic systems with general linear dependence on the spectral parameter on time scales are established and a sufficient condition for the limit‐point case is derived. Almost all presented results are new even in the continuous and discrete cases, that is, for the linear Hamiltonian differential systems and for the discrete symplectic systems, respectively.

Persistence of periodic solutions from discontinuous planar piecewise linear Hamiltonian differential systems with three zones

Persistence of periodic solutions from discontinuous planar piecewise linear Hamiltonian differential systems with three zones

Limit cycles of planar piecewise linear Hamiltonian differential systems with two or three zones

In this paper, we study the existence of limit cycles in continuous and discontinuous planar piecewise linear Hamiltonian differential system with two or three zones separated by straight lines and such that the linear systems that define the piecewise one have isolated singular points, i.e. centers or saddles. In this case, we show that if the planar piecewise linear Hamiltonian differential system is either continuous or discontinuous with two zones, then it has no limit cycles. Now, if the planar piecewise linear Hamiltonian differential system is discontinuous with three zones, then it has at most one limit cycle, and there are examples with one limit cycle. More precisely, without taking into account the position of the singular points in the zones, we present examples with the unique limit cycle for all possible combinations of saddles and centers.

The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems

In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.

Distribution and number of focal points for linear Hamiltonian systems

In this paper we consider the question of distribution and number of left and right focal points for conjoined bases of linear Hamiltonian differential systems. We do not assume any complete controllability (identical normality) condition. Recently we obtained the Sturmian separation theorem for this case which provides optimal lower and upper bounds for the numbers of left and right focal points of every conjoined basis in terms of the principal solutions at the endpoints of the interval. In this paper we show that for any two given integers within these bounds there exists a conjoined basis with these prescribed numbers of left and right focal points. We determine such conjoined bases by their initial conditions. Our approach is to transfer the problem through the comparative index into matrix analysis. The main results are new even for completely controllable linear Hamiltonian systems. As an application we extend a classical result for controllable systems by Reid (1971) about the existence of conjoined bases with an invertible first component.

Sturmian comparison theorems for completely controllable linear Hamiltonian systems in singular case

In this paper we consider two linear Hamiltonian differential systems on an open unbounded interval. We assume that the systems are completely controllable and satisfy the Sturmian majorant condition and the Legendre condition. We derive a singular Sturmian comparison theorem for the case when the minorant system has a solution, which is principal at both endpoints of the considered interval. The main result is new even for the second order differential equations and it generalizes the singular comparison theorem obtained by Aharonov and Elias (2010). We illustrate our new theory by several examples.

Comparison theorems for conjoined bases of linear Hamiltonian systems without monotonicity

In this paper we generalize comparison results for conjoined bases $$Y(t),{{\hat{Y}}}(t)$$ of two linear Hamiltonian differential systems proved by Elyseeva (J Math Anal Appl 444:1260–1273, 2016). In our consideration we do not impose classical monotonicity assumptions such that the majorant condition $${\mathcal {H}}(t)-\hat{\mathcal {H}}(t)\ge 0$$ for their Hamiltonians $${\mathcal {H}}(t), \hat{\mathcal {H}}(t)$$ and the Legendre conditions for $${\mathcal {H}}(t),\hat{\mathcal {H}}(t)$$. Our new comparison theorems are presented in terms of the so-called oscillation numbers associated with $$Y(t), {{\hat{Y}}}(t),$$ and the transformed conjoined basis $${\hat{Z}}^{-1}(t)Y(t)$$, where $${\hat{Z}}(t)$$ is a symplectic fundamental solution matrix of the Hamiltonian system with the Hamiltonian $$\hat{\mathcal {H}}(t)$$. The consideration is based on the comparative index theory applied to the continuous case.

Antiprincipal solutions at infinity for symplectic systems on time scales

In this paper we introduce a new concept of antiprincipal solutions at infinity for symplectic systems on time scales. This concept complements the earlier notion of principal solutions at infinity for these systems by the second author and Sepitka (2016). We derive main properties of antiprincipal solutions at infinity, including their existence for all ranks in a given range and a construction from a certain minimal antiprincipal solution at infinity. We apply our new theory of antiprincipal solutions at infinity in the study of principal solutions, and in particular in the Reid construction of the minimal principal solution at infinity. In this work we do not assume any normality condition on the system, and we unify and extend to arbitrary time scales the theory of antiprincipal solutions at infinity of linear Hamiltonian differential systems and the theory of dominant solutions at infinity of symplectic difference systems.

Relative oscillation of linear Hamiltonian differential systems without monotonicity

We generalize the concept of relative oscillation introduced by O. Došlý (2017) for the pair of linear differential Hamiltonian systems under the majorant condition H(t)−Hˆ(t)≥0,t∈[a,∞) for their Hamiltonians H(t),Hˆ(t). In this paper we omit the majorant condition presenting a generalization of the relative oscillation theorem for the case when the matrix H(t)−Hˆ(t) is nonnegative or nonpositive definite in a right (left) neighborhood of any t0∈(a,∞). We also omit controllability assumptions for the Hamiltonian systems.

Oscillation theorems for linear Hamiltonian systems with nonlinear dependence on the spectral parameter and the comparative index

We consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. We adapt the theory developed by A.A. Abramov for the Hamiltonian spectral problems based on piecewise constant transformations of their conjoined bases using the notion of the comparative index. We introduce the concept of oscillation numbers which generalize the notion of proper focal points and prove the oscillation theorem relating the number of finite eigenvalues in the given interval with the values of the oscillation numbers at the end points of this interval.

The comparative index and transformations of linear Hamiltonian differential systems

In this paper we investigate mutual oscillatory behaviour of two linear differential Hamiltonian systems related via symplectic transformations. The main result extends our previous results in [30], where we presented new explicit relations connecting the multiplicities of proper focal points of conjoined bases Y(t) of the Hamiltonian system and the transformed conjoined bases Y˜(t)=R−1(t)Y(t). In the present paper we omit restrictions on the symplectic transformation matrix R(t) concerning the constant rank of its components. As consequences of the main result we prove generalized reciprocity principles which formulate new sufficient conditions for R(t) concerning preservation of (non)oscillation of the abnormal Hamiltonian systems as t → ∞. The main tool of the paper is the comparative index theory for discrete symplectic systems implemented into the continuous case.

On square integrable solutions and principal and antiprincipal solutions for linear Hamiltonian systems

New results in the Weyl-Titchmarsh theory for linear Hamiltonian differential systems are derived by using principal and antiprincipal solutions at infinity. In particular, a non-limit circle case criterion is established and a close connection between the Weyl solution and the minimal principal solution at infinity is shown in the limit point case. In addition, the square integrability of the columns of the minimal principal solution at infinity is investigated. All results are obtained without any controllability assumption. Several illustrative examples are also provided.

Relative oscillation of linear Hamiltonian differential systems

We discuss the concept of relative oscillation for linear Hamiltonian differential systems. We prove a statement which enables to study this oscillation using the criteria of the classical oscillation theory. We also discuss some other aspects of the problem.

On symplectic transformations of linear Hamiltonian differential systems without normality

In this paper we investigate oscillations of conjoined bases of linear Hamiltonian differential systems related via symplectic transformations. Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for their Hamiltonians. The main result of the paper presents new explicit relations connecting the multiplicities of proper focal points of Y ( t ) and the transformed conjoined basis Y ̃ ( t ) = R − 1 ( t ) Y ( t ) , where the symplectic transformation matrix R ( t ) obeys some additional assumptions on the rank of its components. As consequences of the main result we formulate the generalized reciprocity principle for the Hamiltonian systems without normality. The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.

On some aspects of the Bohl transformation for Hamiltonian and symplectic systems

The classical Bohl transformation [4] from 1906 concerns the second order linear differential equations and states, roughly speaking, that a pair of linearly independent solutions of a second order differential equation can be expressed via the sine and cosine functions. Since that time, this transformation has been extended in various directions and became e.g. the theoretical basis for the deeply developed transformation theory of second order linear differential equations [8]. In our paper we discuss this transformation for linear Hamiltonian differential systems and discrete symplectic systems. We provide an alternative proofs to some know results and these new proofs enable to give a new insight into the topics. We also formulate some open problems associated with the discrete Bohl transformation.

Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index

In this paper we present comparison results for focal points of conjoined bases Y(t),Yˆ(t) of two linear Hamiltonian differential systems under the majorant condition H(t)−Hˆ(t)≥0 for their Hamiltonians H(t),Hˆ(t). Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for Hˆ(t). The main result of the paper connects the difference between the number of proper focal points of Y(t),Yˆ(t) with the number of proper focal points of the transformed conjoined basis Zˆ−1(t)Y(t), where Zˆ(t) is a symplectic fundamental solution matrix of the Hamiltonian system associated with Hˆ(t). Focal points of this transformed basis coincide with focal points of the matrix Wronskian of Y(t),Yˆ(t). The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.

Principal solutions at infinity for time scale symplectic systems without controllability condition

In this paper we introduce a new concept of a principal solution at infinity for nonoscillatory symplectic dynamic systems on time scales. The main ingredient is that we avoid the controllability (or normality) condition, which is traditionally assumed in this theory in the current literature. We show that the principal solutions at infinity can be classified according to the eventual rank of their first component and that the principal solutions exist for all values of the rank between explicitly given minimal and maximal values. The minimal value of the rank is connected with the eventual order of abnormality of the system and it gives rise to the so-called minimal principal solution at infinity. We show that the uniqueness property of the principal solutions at infinity is satisfied only by the minimal principal solution. In this study we unify and extend to arbitrary time scales the recently introduced theory of principal and recessive solutions at infinity for possibly abnormal (continuous time) linear Hamiltonian differential systems and (discrete time) symplectic systems. Moreover, the new theory on time scales also shows that in some results from the continuous time theory the needed assumptions can be simplified.

Recessive solutions for nonoscillatory discrete symplectic systems

In this paper we introduce a new concept of a recessive solution for discrete symplectic systems, which does not require any eventual controllability assumption. We prove that the existence of a recessive solution is equivalent to the nonoscillation of the system and that recessive solutions can have any rank between explicitly given lower and upper bounds. The smallest rank corresponds to the minimal recessive solution, which is unique up to a right nonsingular multiple, while the largest rank yields the traditional maximal recessive solution. We also present a method for constructing some (but not all) recessive solutions having a block diagonal structure from systems in lower dimension. Our results are new even for special discrete symplectic systems, such as for even order Sturm–Liouville difference equations and linear Hamiltonian difference systems.