Rotated Vector Fields Research Articles

Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

Rotational vector-based analysis of turbulent structures in channel flow using large eddy simulation simulation

Even with modern measurement techniques and data from direct numerical simulation (DNS), it is very difficult to identify the individual attached eddies and understand their dynamical behavior due to the multi-scale nature of the eddies in wall-bounded flows, which puts these issues at the center of the current debate. However, Liutex vector ( L for short), a rotational vector field with information on both the rotation axis and swirling strength, has recently been developed by Prof. Liu’s group as a more accurate and clear definition of a vortex. Combining conventional and L methods may provide more detailed information about the complex flow structures as well as insights into the flow’s mixing and transport features. We simulated the channel flow in large eddy simulation by implementing an inflow condition based on the box turbulence. After validating the results with DNS data, we used L isosurfaces and their vector profiles to track ordered flow structures in wall-bounded turbulence. Based on the data, we observe numerous turbulent phenomena that have been described in other works with different visualization techniques. Moreover, the shear contamination on the wall is the most severe while all the root-mean-square L component variations are negligible. Due to the presence of background shear, the peak location of vorticity fluctuation is closer to the wall than the corresponding L fluctuation, and the displacement of peak location brought on by shear contamination is greatest for the spanwise component (z-component) of the vorticity fluctuation. According to the two-point correlation of L components, the streamwise size of turbulent structures does not vary considerably with , however, the spanwise size of turbulent structures increases gradually as increases.

Normal forms of double Hopf oscillators with radial nonlinearities

We are concerned with the simplest normal forms of non-resonant double Hopf oscillators with radial nonlinearities. Due to the required Lie algebraic structure for preserving the radial structure in the normal form process, we appeal to the Lie algebra of radial and rotational transformation generators. The structure constants of the Lie algebra are presented, where we notice a Lie ideal structure for the rotational vector fields. This gives rise to an effective reduction technique through the use of its corresponding quotient Lie algebra. Yet, computations of kernel spaces of the generalized homological maps seem a formidable project and thus, we employ the multiple Lie brackets method to skip explicit kernel space derivations. The simplest normal forms and orbital normal forms of the nonlinear family are presented. We also address their parametric normal forms as multiple parametric perturbations of the corresponding vector fields. Transformation symmetry groups associated with truncated normal forms are also derived. An efficient algorithm is provided for the normal form computations using computer algebra systems. Finally, we derive state-feedback controllers for a two-arm vibrator using normal forms of radial double Hopf singularity. This is to illustrate how our normal forms can be applied on classical control engineering problems as a new alternative nonlinear controller technique to the existing methods in nonlinear control theory.

Relationship between threshold and bifurcations for paradoxical firing responses along with seizure induced by inhibitory stimulation

Inhibition-induced enhancement or paradoxical response of firing was related to Hopf bifurcation instead of saddle-node bifurcation on an invariant cycle (SNIC), due to the “negative” threshold and rotated vector fields. In the present letter, by changing multiple parameters to adjust the vector fields, the condition for the paradoxical response and “negative” threshold is extended to SNIC near a codimension-2 bifurcation appearing prior to the Hopf bifurcation, which presents a comprehensive relationship between bifurcations and threshold. Especially, the result for a special current can well explain the enhanced firing along with seizure induced by inhibitory interneuron, implying that SNIC far from the codimension-2 bifurcation of pyramidal neuron is a potential candidate to avoid seizure.

Mathematical justification of the point vortex dynamics in background fields on surfaces as an Euler–Arnold flow

The point vortex dynamics in background fields on surfaces is justified as an Euler–Arnold flow in the sense of de Rham currents. We formulate a current-valued solution of the Euler–Arnold equation with a regular-singular decomposition. For the solution, we first prove that, if the singular part of the vorticity is given by a linear combination of delta functions centered at \(q_n(t)\) for \(n=1,\ldots ,N\), \(q_n(t)\) is a solution of the point vortex equation. Conversely, we next prove that, if \(q_n(t)\) is a solution of the point vortex equation for \(n=1,\ldots ,N\), there exists a current-valued solution of the Euler–Arnold equation with a regular-singular decomposition such that the singular part of the vorticity is given by a linear combination of delta functions centered at \(q_n(t)\). As a corollary, we generalize the Bernoulli law to the case where the flow field is a curved surface and where the presence of point vortices is taken into account. From the viewpoint of the application, the mathematical justification is of significance since the point vortex dynamics in the rotational vector field on the unit sphere is regarded as a mathematical model of geophysical flow in order to take effect of the Coriolis force on inviscid flows into consideration.

Cylindrical spikes

The Geroch/Stephani transformation is a solution-generating transformation, and may generate spiky solutions. The spikes in solutions generated so far are either early-time permanent spikes or transient spikes. We want to generate a solution with a late-time permanent spike. We achieve this by applying Stephani’s transformation with the rotational Killing vector field (KVF) of the locally rotationally symmetric Jacobs solution. The late-time permanent spike occurs along the cylindrical axis. Using a mixed KVF, the generated solution also features a rich variety of transient structures. We introduce a new technique to analyse these structures. Our findings lead us to discover a transient behaviour, which we call the overshoot transition. These discoveries compel us to revise the description of transient spikes.

Time-Extremal Navigation in Arbitrary Winds on Conformally Flat Riemannian Manifolds

This paper aims at solving Zermelo’s navigation problem on conformally flat Riemannian manifolds admitting a ship’s variable self-speed, under the action of arbitrary winds including space and time dependence for both perturbation and ship’s speed. Our approach is a variational one under application of the Euler–Lagrange equations with reference to the initial studies of this problem. First of all, we distinguish the navigation cases in non-critical, i.e. weak or strong, and critical winds, which are then unified into an arbitrary wind. After having considered the second variation of a given functional, we obtain the conditions for both time-minimal, i.e. the typical solutions to Zermelo’s problem, and time-maximal extremals. The anomalous paths are also emphasized. Moreover, some classification results are presented with respect to the kinds of perturbation considered separately and under an arbitrary wind. This study is illustrated at its end by a two-dimensional example including a prolate ellipsoid in the presence of a rotational vector field, wherein the solution types are being compared.

A cyclic pursuit framework for networked mobile agents based on vector field approach

This paper proposes a pursuit formation control scheme for a network of double-integrator mobile agents based on a vector field approach. In a leaderless architecture, each agent pursues another one via a cyclic topology to achieve a regular polygon formation. On the other hand, the agents are exposed to a rotational vector field such that they rotate around the vector field centroid, while they keep the regular polygon formation. The main problem of existing approaches in the literature for cyclic pursuit of double-integrator multiagent systems is that under those approaches, the swarm angular velocity and centroid are not controllable based on missions and agents capabilities. However, by employing the proposed vector field approach in this paper, while keeping a regular polygon formation, the swarm angular velocity and centroid can be determined arbitrary. The obtained results can be extended to achieve elliptical formations with cyclic pursuit as well. Simulation results for a team of eight mobile agents verify the accuracy of the proposed control scheme.

Homoclinic, heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them comparing with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods. We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.

Rotation invariants of vector fields from orthogonal moments

Vector field images are a type of new multidimensional data that appear in many engineering areas. Although the vector fields can be visualized as images, they differ from graylevel and color images in several aspects. To analyze them, special methods and algorithms must be originally developed or substantially adapted from the traditional image processing area. In this paper, we propose a method for the description and matching of vector field patterns under an unknown rotation of the field. Rotation of a vector field is so-called total rotation, where the action is applied not only on the spatial coordinates but also on the field values. Invariants of vector fields with respect to total rotation constructed from orthogonal Gaussian–Hermite moments and Zernike moments are introduced. Their numerical stability is shown to be better than that of the invariants published so far. We demonstrate their usefulness in a real world template matching application of rotated vector fields.

Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.

New developments on constant angle property in $${\mathbb {S}}^2\times {\mathbb {R}}$$ S 2 × R

In the present paper, we classify curves and surfaces in \({\mathbb {S}}^2\times \mathbb {R}\), which make constant angle with a rotational Killing vector field. We obtain the explicit parametrizations of such curves and surfaces, and we find examples in some particular cases. Finally, we give the complete classification of minimal constant angle surfaces in \({\mathbb {S}}^2\times \mathbb {R}\).

Minimal surfaces in a Randers sphere with the rotational Killing vector field

The minimal surfaces in Finsler geometry with respect to the Busemann–Hausdorff measure and the Holmes–Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. Let (p1,p2,p3,p4) be the coordinates of R4 and (S3,F˜) be a Randers sphere of flag curvature K=1 with the navigation data (h˜,W˜), where h˜ is the standard sphere metric and W˜=ε(0,0,−p4,p3), 0<ε<1, is a Killing vector field. In this paper, we study the rotationally invariant minimal surface in (S3,F˜) generated by rotating the curve (x(s),y(s),z(s),0) in the upper half sphere of S2 around the p1p2-plane, s∈R. We first show that such a rotational BH-minimal surface in (S3,F˜) is either a great 2-sphere or the catenoid in (S3,h˜). Then we give a classification of the rotational HT-minimal surfaces, where we use the angle data to analyze the solutions of the system of ODE that characterizes the HT-minimality and prove that, such a rotational HT-minimal surface must be a great 2-sphere, an HT-minimal torus, or a rotational surface of unduloid type. As a special case, we obtain a distinguished embedded compact HT-minimal torus depending on ε. The completeness of these surfaces is also studied.

Analytic Integrability of Two Lopsided Systems

In this paper, we present two classes of lopsided systems and discuss their analytic integrability. The analytic integrable conditions are obtained by using the method of inverse integrating factor and theory of rotated vector field. For the first class of systems, we show that there are [Formula: see text] small-amplitude limit cycles enclosing the origin of the systems for [Formula: see text], and ten limit cycles for [Formula: see text]. For the second class of systems, we prove that there exist [Formula: see text] small-amplitude limit cycles around the origin of the systems for [Formula: see text], and nine limit cycles for [Formula: see text].

Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy.

In this paper, we analyze a general predator-prey model with state feedback impulsive harvesting strategies in which the prey species displays a strong Allee effect. We firstly show the existence of order-1 heteroclinic cycle and order-1 positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-1 positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.

Travelling waves in the Holling-Tanner model with weak diffusion.

For a wide range of parameters, we study travelling waves in a diffusive version of the Holling-Tanner predator-prey model from population dynamics. Fronts are constructed using geometric singular perturbation theory and the theory of rotated vector fields. We focus on the appearance of the fronts in various singular limits. In addition, periodic travelling waves of relaxation oscillation type are constructed using a recent generalization of the entry-exit function.

Transversal conics and the existence of limit cycles

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré–Bendixson regions by using transversal conics. We present several examples of known systems in the literature showing different features about limit cycles: hyperbolicity, Hopf bifurcation, sky-blue bifurcation, rotated vector fields, etc., for which the obtained Poincaré–Bendixson region allows to locate the limit cycles. Our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles.

Existence of limit cycles and homoclinic bifurcation in a plant–herbivore model with toxin-determined functional response

In this paper we study a two-dimensional toxin-determined functional response model (TDFRM). The toxin-determined functional response explicitly takes into consideration the reduction in the consumption of plants by herbivore due to chemical defense, which generates more complex dynamics of the plant–herbivore interactions. The purpose of the present paper is to analyze the existence of limit cycles and bifurcations of the model. By applying the theories of rotated vector fields and the extended planar termination principle, we establish the conditions for the existence of limit cycles and homoclinic loop. It is shown that a limit cycle is generated in a supercritical Hopf bifurcation and terminated in a homoclinic bifurcation, as the parameters vary. Analytic proofs are provided for all results, which generalize the results presented in [11].

Center conditions and limit cycles for a class of nilpotent-Poincarè systems

A class of nilpotent-Poincarè system is discussed in this paper. Center conditions are obtained by methods of inverse integrating factor and theory of rotated vector field. When n>2, we proved that there are 2n+4 small amplitude limit cycles enclosing the origin O(0,0). When n=2, there are 14 limit cycles enclosing the origin O(0,0).

Homoclinic bifurcation of prey–predator model with impulsive state feedback control

In this paper, homoclinic bifurcations of a prey–predator system with state impulse are investigated. Firstly, the existence of order-1 homoclinic cycle of system (1.2) with δ=0 is investigated; Secondly, choosing q as a control parameter, the sufficient conditions of existence and stability of order-1 periodic solution of system (1.2) with δ=0 are obtained by means of the geometry theory of semi-continuous dynamic systems; Thirdly, on the basis of the theory of rotated vector fields, homoclinic bifurcation of system (1.2) about parameter δ are also studied; Finally, some simulations are provided to prove the main results. The methods used in this paper are intuitive to prove the existence of order-1 homoclinic cycle and homoclinic bifurcations.