Narrow Results

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family ${\{{\mathrm{\Omega }}^t\}}_{t\in D}$ of integrable one-forms ${\mathrm{\Omega }}^t$ defined in a neighborhood $U\subset {ℂ}^n$ of the initial singular point, and parametrized by the disc $D\subset ℂ$. The initial foliation is defined by ${\mathrm{\Omega }}^0$. The second type, more restrictive, is given by an integrable holomorphic one-form $\mathrm{\Omega }(x,t)$ defined in the product $U\times D\subset {ℂ}^n\times ℂ$. Then, the initial foliation is defined by the slice restriction $\mathrm{\Omega }(x,0)$. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by $df=0$ for some germ of holomorphic function $f\in {{𝒪}}_n$ at the origin $0\in {ℂ}^n,n\ge 3$. We assume that the germ $f$ is irreducible and that the typical fiber of $f$ is simply-connected. This is the case if outside of a dimension $\le n-3$ analytic subset $Y\subset {ℂ}^n$, the analytic hypersurface $X_f:(f=0)$ has only normal crossings singularities. We then prove that, if cod sing $\mathrm{\Omega }(x,0)\ge 2$ then the (germ of the) developing foliation given by $\mathrm{\Omega }(x,t)=0$ also exhibits a holomorphic first integral. For the general case, i.e. cod sing $\mathrm{\Omega }(x,0)\ge 1$, we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations ${\{{{ℱ}}_t\}}_{t\in ℂ,0}$, of a local pencil ${{ℱ}}_0:\frac{f}{g}=\mathrm{\text{constant}}$, for $f,g\in {{𝒪}}_n$. For dimension $n=2,$ we consider $f=x,g=y$. For dimension $n\ge 3,$ we assume some generic geometric conditions on $f$ and $g$. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type $\widehat{F}=\frac{f^{1+\widehat{\lambda }(t)}}{g^{1+\widehat{\mu }(t)}}{e}^{Ĥ(x,y,t)}$ with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form $M=\frac{f}{g}{e}^{P(t)+H(x,y,t)}$ with some additional properties, provided that for $n=2$ the axes remain invariant for the foliations ${{ℱ}}_t$.

First integrals admitted by second-order nonlinear ordinary differential equations modeling the temperature distribution in a straight fin are obtained. After imposing the boundary conditions these first integrals give a relationship between temperature at the fin tip and the temperature gradient at the base of the fin in terms of the fin parameters. These first integrals are plotted and analyzed. The results obtained show how the temperature at the fin tip can be controlled by the temperature gradient at the base for fixed fin parameters.

We consider several discrete dynamical systems for which some invariants can be found. Our study includes complex Möbius transformations as well as the third-order Lyness recurrence.

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

In this work, it is proved that any degenerate center is limit of a linear type center and when the degenerate center has an analytic first integral then it is limit of an analytic linear type center. A new method to detect integrability developed in [Giné & Santallusia, 2011] is applied to the degenerate center problem. Moreover, a review of the most important recent contributions to the degenerate center problem is given.

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

We obtain necessary and sufficient conditions for the existence of a center on a local center manifold for three 4-parameter families of quadratic systems on ℝ³. We also give a positive answer to an open question posed in [Dias & Mello, 2010] related to similar systems.

In this paper, we study the Lotka–Volterra complex cubic systems. We obtain necessary conditions of integrability for these systems with some restriction on the parameters. The sufficiency is proved for all conditions, except one which remains open, using different methods.

In this paper, the unique normal form for a class of three-dimensional nilpotent vector fields with symmetry is investigated. The key technique used is a combination of multiple Lie brackets, linear grading function, new notations of block matrices and first integral of the linear part of the given vector field which avoids complicated calculations. The first and the second order normal forms are computed successively, and the uniqueness of the second order normal form is proved under certain conditions.

In this paper, we consider a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and an active memristive device. The circuit is described by a four-parameter system of ordinary differential equations. We study in detail the role of parameters in the dynamics of the system. Using the existence of first integrals, we show that the circuit may present a continuum of stable periodic orbits, which arise due to the occurrence of infinitely many simultaneous zero-Hopf bifurcations on a line of equilibria located in the region where the memristance is negative and, consequently, the memristive device is locally-active. These bifurcations lead to multistability, which is a difficult and interesting problem in applied models, since the final state of a solution depends crucially on its initial condition. We also study the control of multistability by varying a parameter related to the state variable of the memristive device. All analytical results obtained were corroborated by numerical simulations.

The Maxwell–Bloch system arises naturally from laser models, which describes the dynamics of a two-state quantum system coupled to the basic electromagnetic resonator model. In this paper, we study its integrability and give a complete classification of the irreducible Darboux polynomials, which can help us find all polynomial and rational first integrals of the Maxwell–Bloch system. To this end, in the case that the loss rate of field is zero, we use the characteristic curve method and weight homogeneous polynomials to investigate the integrability of the Maxwell–Bloch system, and in the case that the loss rate of field is nonzero, we provide a linear scaling of variables to transform the Maxwell–Bloch system into the Lorenz system and obtain its integrability from some corresponding results of the Lorenz system. Our results show that although the Maxwell–Bloch system has chaotic behaviors for a large range of its parameters, it is nonchaotic when its parameters satisfy one of seven conditions, which may help us understand the complex and rich dynamics of nonlinear laser models.

We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differential systems separated by the straight-line $x=0$ having in $x>0$ the general quadratic isochronous center $ẋ=-y+x^2$, $ẏ=x(1+y)$ after an affine transformation, and in $x<0$ an arbitrary quadratic isochronous center except for the quadratic isochronous center $ẋ=-y+x^2-y^2$, $ẏ=x(1+2y)$ which has been studied in [Ghermoul et al., 2021]. For these piecewise-continuous differential systems the upper bound of crossing limit cycles is 2, and there are specific examples having one crossing limit cycle.

In this paper, we study the limit cycles in the discontinuous piecewise linear planar systems separated by a nonregular line and formed by linear Hamiltonian vector fields without equilibria. Motivated by [Llibre & Teixeira, 2017], where an open problem was posed: Can piecewise linear differential systems without equilibria produce limit cycles? We prove that such systems have at most two limit cycles, and the limit cycles must intersect the nonregular separation line in two or four points. More precisely, the exact upper bound of crossing limit cycles is two, and this upper bound can indeed be reached: either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points. Based on Poincaré map, the stability of various limit cycles is also proved. In addition, we give some concrete examples to illustrate our main results.

Although Lyapunov exponents have been widely used to characterize the dynamics of nonlinear systems, few methods are available so far to obtain a priori bounds on their magnitudes. Recently, sufficient conditions to rule out the existence of attractors with positive Lyapunov exponents have been derived via a Lyapunov approach based on the second additive compound matrices of the system Jacobian. This paper first provides some insights into this approach by showing how the several available techniques for computing Lyapunov functions can be fruitfully applied to Lorenz and Thomas systems to derive explicit conditions on their system parameters, which ensure that there are no attractors with positive Lyapunov exponents. Then, the approach is extended to the case of nonlinear systems with a first integral of motion and its application to the memristor Chua’s circuit is discussed.

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua’s circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds.

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems.

In this paper, we investigate the existence of limit cycles in piecewise linear systems separated by two parallel straight lines, where in one of these straight lines, the piecewise differential system is continuous, and in the other, it is discontinuous. Based on the methods of Poincaré map and first integral, we prove that these systems can have at most two limit cycles. Additionally, taking into account the type of singular points in the zones, we present examples with one or two limit cycles for all possible combinations.

We consider the Sprott cubic conservative jerk differential equation $\stackrel{\mathrm{...}}{x}-a(1-x^2)x+x^2ẋ=0,$ with $a\in ℝ$. It is known that this differential equation exhibits chaotic motion for some values of the parameter a. Here, we study when this differential equation has no chaotic motion, i.e. when it has first integrals, and then we describe its dynamics.

This paper is dedicated to studying the maximum number of crossing limit cycles of discontinuous planar piecewise-linear differential systems with three regions separated by a nonregular line and formed by linear centers and linear Hamiltonian systems without equilibria. There are two types of crossing limit cycles: one that intersects the nonregular line at three points and the other that intersects the nonregular line at four points, named three-point crossing limit cycles and four-point crossing limit cycles, respectively. We prove that the maximum numbers of three-point and four-point crossing limit cycles are three and one, respectively, and both reach their upper bounds. Furthermore, we show by concrete examples that one four-point crossing limit cycle and two three-point crossing limit cycles can exist simultaneously.