Quadratic Hamiltonian Systems Research Articles

2D-Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II

In earlier research, we developed two techniques designed to expand the construction of a periodic orbit dividing surface for Hamiltonian systems with three or more degrees of freedom. Our methodology involved transforming a periodic orbit into a torus or cylinder, thereby elevating it to a higher-dimensional structure within the energy surface (refer to [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). Recently, we introduced two new methods for creating dividing surfaces, which do not rely on periodic orbits. Instead, we used 2D surfaces (geometric entities) or 3D surfaces in a Hamiltonian system with three degrees of freedom (see [Katsanikas & Wiggins, 2024a, 2024b, 2024c]). In these studies, we applied these surfaces within a quadratic normal-form Hamiltonian system with three degrees of freedom. This series of two papers (this paper and [Katsanikas et al., 2024]) extends our results to 2D-generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper focuses on presenting the second method of constructing 2D-generating surfaces.

2D Generating Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – I

In previous studies, we developed two techniques aimed at expanding the scope of constructing a periodic orbit dividing surface within a Hamiltonian system with three or more degrees of freedom. Our approach involved extending a periodic orbit into a torus or cylinder, thereby elevating it into a higher-dimensional entity within the energy surface (see [Katsanikas & Wiggins, 2021a, 2021b, 2023a, 2023b]). Recently, we introduced two alternative methods for creating dividing surfaces, distinct from the utilization of periodic orbits, by employing 2D surfaces (geometric entities) or 3D surfaces within a Hamiltonian system with three degrees of freedom (refer to [Katsanikas & Wiggins, 2024a, 2024b, 2024c]). In these studies, we applied these surfaces in a quadratic normal form Hamiltonian system with three degrees of freedom. In this series of two papers, we extend our results to 2D generating surfaces for quartic Hamiltonian systems with three degrees of freedom. This paper presents the first method of constructing 2D generating surfaces.

Periodic Orbit-Dividing Surfaces in Rotating Hamiltonian Systems with Three Degrees of Freedom – I

In this paper, we extend the notion of periodic orbit-dividing surfaces (PODSs) to rotating Hamiltonian systems with three degrees of freedom. First, we present how to apply our first method for the construction of PODSs [ Katsanikas & Wiggins , 2021a , 2023a ] to rotating Hamiltonian systems with three degrees of freedom. Then, we study the structure of these surfaces in a rotating quadratic normal-form Hamiltonian system with three degrees of freedom.

Periodic Orbit-Dividing Surfaces in Rotating Hamiltonian Systems with Two Degrees of Freedom

In this paper, we extend the notion of periodic orbit-dividing surfaces (PODS) to rotating Hamiltonian systems with two degrees of freedom. First, we present a method that enables us to apply the classical algorithm for the construction of PODS [ Pechukas & McLafferty , 1973 ; Pechukas , 1981 ; Pollak & Pechukas , 1978 ; Pollak , 1985 ] in rotating Hamiltonian systems with two degrees of freedom. Then we study the structure of these surfaces in a rotating quadratic normal-form Hamiltonian system with two degrees of freedom.

Periodic Orbit Dividing Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – II

In prior studies [ Katsanikas & Wiggins , 2021a , 2021b , 2023a , 2023b ], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored specifically for Hamiltonian systems with three or more degrees of freedom. These approaches, as described in the aforementioned papers, were applied to a quadratic Hamiltonian system in its normal form with three degrees of freedom. Within this framework, we provide a more intricate geometric characterization of this entity within the family of 4D toratopes which elucidates the structure of the dividing surfaces discussed in these works. Our analysis affirmed the nature of this construction as a dividing surface with the property of no-recrossing. These insights were derived from analytical findings tailored to the Hamiltonian system discussed in these publications. In this series of papers, we extend our previous findings to quartic Hamiltonian systems with three degrees of freedom. We establish the no-recrossing property of the PODS for this class of Hamiltonian systems and explore their structural aspects. Additionally, we undertake the computation and examination of the PODS in a coupled scenario of quartic Hamiltonian systems with three degrees of freedom. In the initial paper [ Gonzalez Montoya et al. , 2024 ], we employed the first methodology for constructing PODS, while in this paper, we utilize the second methodology for the same purpose.

Periodic Orbit Dividing Surfaces in a Quartic Hamiltonian System with Three Degrees of Freedom – I

In prior work [Katsanikas & Wiggins, 2021a, 2021b, 2023c, 2023d], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored for Hamiltonian systems possessing three or more degrees of freedom. The initial approach, outlined in [Katsanikas & Wiggins, 2021a, 2023c], was applied to a quadratic Hamiltonian system in normal form having three degrees of freedom. Within this context, we provided a more intricate geometric characterization of this object within the family of 4D toratopes that describe the structure of the dividing surfaces discussed in these papers. Our analysis confirmed the nature of this construction as a dividing surface with the no-recrossing property. All these findings were derived from analytical results specific to the case of the Hamiltonian system discussed in these papers. In this paper, we extend our results for quartic Hamiltonian systems with three degrees of freedom. We prove for this class of Hamiltonian systems the no-recrossing property of the PODS and we investigate the structure of these surfaces. In addition, we compute and study the PODS in a coupled case of quartic Hamiltonian systems with three degrees of freedom.

Existence of solutions for superquadratic or asymptotically quadratic fractional Hamiltonian systems

Existence of solutions for superquadratic or asymptotically quadratic fractional Hamiltonian systems

3D Generating Surfaces in Hamiltonian Systems with Three Degrees of Freedom – I

In our earlier research (see [Katsanikas & Wiggins, 2021a , 2021b , 2023a , 2023b , 2023c ]), we developed two methods for creating dividing surfaces, either based on periodic orbits or two-dimensional generating surfaces. These methods were specifically designed for Hamiltonian systems with three or more degrees of freedom. Our prior work extended these dividing surfaces to more complex structures such as tori or cylinders, all within the energy surface of the Hamiltonian system. In this paper, we introduce a new method for constructing dividing surfaces. This method differs from our previous work in that it is based on 3D surfaces or geometrical objects, rather than periodic orbits or 2D generating surfaces (see [Katsanikas & Wiggins, 2023a ]). To explain and showcase the new method and to present the structure of these 3D surfaces, the paper provides examples involving Hamiltonian systems with three degrees of freedom. These examples cover both uncoupled and coupled cases of a quadratic normal form Hamiltonian system. Our current paper is the first in a series of two papers on this subject. This research is likely to be of interest to scholars and researchers in the field of Hamiltonian systems and dynamical systems, as it introduces innovative approaches to constructing dividing surfaces and exploring their applications.

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop

Heteroclinic loop bifurcations by perturbing a class of Z2-equivariant quadratic switching Hamiltonian systems with nilpotent singular points

This paper investigates the problem of heteroclinic loop bifurcation by perturbing a class of Z2-equivariant quadratic switching systems with nilpotent singular points. Firstly, it provides sufficient and necessary conditions for the occurrence of a heteroclinic loop. Next, the system is perturbed by piecewise polynomial systems of degree n≥1. The paper then considers the lower bound for the maximum number of limit cycles that bifurcate from the generalized heteroclinic loop, finding at least n+[n2] limit cycles.

Symmetric Phase Portraits of Homogeneous Polynomial Hamiltonian Systems of Degree 1, 2, 3, 4, and 5 with Finitely Many Equilibria

Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree.

The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom – IV

Recently, we presented two methods of constructing periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom [Katsanikas & Wiggins, 2021a, 2021b]. These methods were illustrated with an application to a quadratic normal form Hamiltonian system with three degrees of freedom. More precisely, in these papers we constructed a section of the dividing surfaces that intersect with the hypersurface [Formula: see text]. This was motivated by studies in reaction dynamics since in this model reaction occurs when the sign of the [Formula: see text] coordinate changes. In this paper, we continue the work of the third paper [Katsanikas & Wiggins, 2023] of this series of papers to construct the full dividing surfaces that are obtained by our algorithms and to prove the no-recrossing property. In the third paper we did this for the dividing surfaces of the first method [Katsanikas & Wiggins, 2021a]. Now we are doing the same for the dividing surfaces of the second method [Katsanikas & Wiggins, 2021b]. In addition, we computed the dividing surfaces of the second method for a coupled case of the quadratic normal form Hamiltonian system and we compared our results with those of the uncoupled case. This paper completes this series of papers about the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom.

The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom-III

In two previous papers [Katsanikas & Wiggins, 2021a, 2021b], we developed two methods for the construction of periodic orbit dividing surfaces for Hamiltonian systems with three or more degrees of freedom. We applied the first method (see [Katsanikas & Wiggins, 2021a]) in the case of a quadratic Hamiltonian system in normal form with three degrees of freedom, constructing a geometrical object that is the section of a 4D toroidal structure in the 5D energy surface with the space [Formula: see text]. We provide a more detailed geometrical description of this object within the family of 4D toratopes. We proved that this object is a dividing surface and it has the no-recrossing property. In this paper, we extend the results for the case of the full 4D toroidal object in the 5D energy surface. Then we compute this toroidal object in the 5D energy surface of a coupled quadratic normal form Hamiltonian system with three degrees of freedom.

Limit cycles near a homoclinic loop connecting a tangent saddle in a perturbed quadratic Hamiltonian system

In this paper, we study bifurcation of limit cycles from a homoclinic loop connecting a saddle of tangent type for a quadratic Hamiltonian system perturbed by nth degree polynomials, n=1,2,…,13. The main tool is the asymptotic expansion of the related Abelian integral near the homoclinic loop, and the maximal number of independent coefficients gives exact number of limit cycles. Our aim is to obtain more limit cycles by exploring more coefficients in the asymptotic expansion. However, it is usually very difficult to obtain the coefficients of the terms with degree greater than or equal to 2 in the asymptotic expansion. To overcome the difficulty, we derive two auxiliary systems and investigate the expansions for the related Abelian integral. The coefficients of lower degree terms in the new asymptotic expansions are equivalent to those of higher degree terms in the original asymptotic expansion. We obtain n−1−n−24 limit cycles near the non-regular homoclinic loop and n−24 limit cycles near the center, and it totally has at least n−1 limit cycles, when n∈{1,2,…,13}. The cyclicity of period annulus is also estimated by the first order Melnikov functions for n=3.

Nilpotent Center in a Continuous Piecewise Quadratic Polynomial Hamiltonian Vector Field

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line [Formula: see text], where these kinds of systems have a nilpotent center at [Formula: see text], which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with nonelementary singular points as we did here.

The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian Systems with Three or More Degrees of Freedom – I

We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a ([Formula: see text])-dimensional object in the ([Formula: see text])-dimensional energy surface. We present our methods using benchmark examples for two and three degrees of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end, we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.

Complex planar Hamiltonian systems: Linearization and dynamics

<p style='text-indent:20px;'>Global dynamics of complex planar Hamiltonian polynomial systems is difficult to be characterized. In this paper, for general complex quadratic Hamiltonian systems of one degree of freedom, we obtain some sufficient conditions on the existence of family of invariant tori. We also complete characterization on locally analytic linearizability of complex planar Hamiltonian systems with homogeneous nonlinearity of degrees either 2 or 3 at a nondegenerate singularity, and present their global dynamics. For these classes of systems we also prove existence of families of invariant tori, together with isochronous periodic orbits.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Entanglement and Squeezing in a Quadratic Hamiltonian System

We investigate the generation of entanglement and squeezing in a quadratic Hamiltonian system. We first analytically solve the time development of the field operators. By exploiting the analytical solutions of the time development of the field operators, we calculate the entanglement parameter defined by Pezze and Smerzi and quadrature squeezing. We show that squeezing and entanglement can be periodically generated. We also show that the more entanglement and stronger squeezing can be obtained by increasing the nonlinear interaction.

Stable and unstableperiodic solutions for quadratic contact Hamiltonians with a small parameter

In this paper, we study the existence of periodic solutions for a class of quadratic contact Hamiltonian system with a small parameter by averaging theory.We have proved that the quadratic contact Hamiltonian system has a stable periodic solution and an unstable periodic solution when the corresponding parameter is small enough.