The notion of persistent center has recently been extended to a linearizable persistent centers. We define the linearizability of persistent p:−q resonant center and explain the relation between p:−q ...resonant system and the corresponding persistent p:−q resonant system in terms of integrability and linearizability problems. We demonstrate the obtained results for persistent p:−q resonant systems with quadratic nonlinearities for p=1, q=2 and p=2, q=3.
In 7, Herman and Yoccoz prove that for any given locally analytic (at z=0) power series f(z)=z(λ+∑i=1∞aizi) over a complete non-Archimedean field of characteristic 0 if |λ|=1 and λ is not a root of ...unity, then f is locally linearizable at z=0. They ask the same question for power series over fields of positive characteristic.
In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable. More precisely, given a complete non-Archimedean field K of positive characteristic and a power series f(z)=z(λ+∑i=1∞aizi)∈K〚z〛 with λ not a root of unity and |1−λ|<1, we prove a sufficient condition (Criterion ★) for f to be non-linearizable. This phenomenon of prevalence for power series over fields of positive characteristic being non-linearizable was initially conjectured in 6, p 147 by Herman, and formulated into a concrete question by Lindahl as 9, Conjecture 2.2.
As applications of Criterion ★, we prove the non-linearizability of three families of polynomials.
Tasks and objects are two predominant ways of specifying distributed problems where processes should compute outputs based on their inputs. Roughly speaking, a
task
specifies, for each set of ...processes and each possible assignment of input values, their valid outputs. In contrast, an
object
is defined by a sequential specification. Also, an object can be invoked multiple times by each process, while a task is a one-shot problem. Each one requires its own
implementation
notion, stating when an execution satisfies the specification. For objects,
linearizability
is commonly used, while tasks implementation notions are less explored.
The article introduces the notion of
interval-sequential
object, and the corresponding implementation notion of
interval-linearizability
, to encompass many problems that have no sequential specification as objects. It is shown that interval-sequential specifications are
local
, namely, one can consider interval-linearizable object implementations in isolation and compose them for free, without sacrificing interval-linearizability of the whole system. The article also introduces the notion of
refined tasks
and its corresponding satisfiability notion. In contrast to a task, a refined task can be invoked multiple times by each process. Also, objects that cannot be defined using tasks can be defined using refined tasks. In fact, a main result of the article is that interval-sequential objects and refined tasks have the same expressive power and both are complete in the sense that they are able to specify any prefix-closed set of well-formed executions.
Interval-linearizability and refined tasks go beyond unifying objects and tasks; they shed new light on both of them. On the one hand, interval-linearizability brings to task the following benefits: an explicit operational semantics, a more precise implementation notion, a notion of state, and a locality property. On the other hand, refined tasks open new possibilities of applying topological techniques to objects.
In this paper, complex integrability and linearizability of cubic Z2-equivariant systems with two 1:q resonant singular points are investigated, and the necessary and sufficient conditions on complex ...integrability and linearizability of the systems with two 1:(−q) resonant saddles are obtained for q=1,2,3,4. Moreover, for general positive integer q, the complex integrability and linearizability conditions are classified, and the sufficiency of the conditions is proved. Further, the linearizability conditions of cubic Z2-equivariant systems with two 1:q resonant node points are also classified.
In this paper, complete integrability and linearizability of cubic Z2 systems with two non-resonant and elementary singular points are investigated. First of all, four simple normal forms are ...obtained based on the coefficients and eigenvalues of cubic Z2 systems. Then, the integrable and linearizable conditions are classified according to the four different cases respectively, and the problem is solved thoroughly for cubic Z2 systems with two non-resonant singular points.
In this paper, we study the problems of simultaneous integrability and non-linearizability of arbitrary double weak saddles and sole weak focus for a planar cubic Liénard system with cubic damping ...and restoring force. First of all, we discriminate the equilibria type: two of saddle type and one of focus-center type. Next, via computing saddle values and focus values, we deduce the integrability condition under which the system is simultaneously integrable at double weak saddles and sole weak focus. The sufficiency of the condition is proved as well. Lastly, we come to the conclusion that the system is not linearizable at the three equilibria by means of computing linearizability constants.
MSC: 34C07; 34C23; 34D10; 37G10
•The studied system is a planar cubic Li\'{e}nard system with cubic damping and restoring force.•The simultaneous necessary and sufficient integrability conditions at arbitrary double weak saddles and sole weak focus are given.•The non-linearizability is characterized.
In a family of real quadratic three dimensional systems symmetric with respect to a plane we look for subfamilies having center manifolds filled with isochronous periodic orbits. Eleven such ...subfamilies are detected and it is shown that for ten of them there are Darboux type substitutions transforming the subfamilies to systems which are linear on center manifolds. We also give an example of a 3-dim quadratic system with a compact isochronous periodic annulus.
In this work we consider a family of cubic, with respect to the first derivative, second-order ordinary differential equations. We study linearizability conditions for this family of equations via ...generalized nonlocal transformations. We construct linearizability criteria in the explicit form for some particular cases of these transformations. We show that each linearizable equation admits a quadratic rational first integral. Moreover, we demonstrate that generalized nonlocal transformations under certain conditions preserve Lax integrability for equations from the considered family. Consequently, we find that any linearizable equation from the considered family possesses a Lax representation. This provides a connection between two different approaches for studying integrability and demonstrates that the Lax technique can be effectively applied to finding and classifying dissipative integrable autonomous and non-autonomous nonlinear oscillators. We illustrate our results by several examples of linearizable equations, including a non-autonomous generalization of the Rayleigh–Duffing oscillator, a family of chemical oscillators and a generalized Duffing–Van der Pol oscillator.
•Linearizabitiy of cubic autonomous and non-autonomous nonlinear oscillators.•Lax representation for the dissipative systems.•Explicit linearizability criteria for cubic nonlinear oscillators.
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