Simultaneous Bifurcation Research Articles

Cyclicity of period annulus for a class of quadratic reversible systems with a nonrational first integral

In this paper, we study the quadratic perturbations of a one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. By the criterion function for determining the lowest upper bound of the number of zeros of Abelian integrals, we obtain that the cyclicity of either period annulus is two. To the best of our knowledge, this is the first result for the cyclicity of period annulus of the one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. Moreover, the simultaneous bifurcation and distribution of limit cycles from two-period annuli are considered.

Zero-Hopf bifurcation in the general Van der Pol-Duffing equation

In this work we study the invariant sets which emerge from the zero-Hopf bifurcations that general Van der Pol-Duffing equations can exhibit. We provide sufficient conditions for the simultaneous bifurcation of three periodic solutions and two invariant torus from the origin of this system. We use recent results related to the averaging method in order to analytically obtain our results. We also provide numerical examples for all the analytical results that we provide.

Simultaneous Bifurcation of Limit Cycles and Critical Periods

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback.

We consider the Yamada model for an excitable or self-pulsating laser with saturable absorber and study the effects of delayed optical self-feedback in the excitable case. More specifically, we are concerned with the generation of stable periodic pulse trains via repeated self-excitation after passage through the delayed feedback loop and their bifurcations. We show that onset and termination of such pulse trains correspond to the simultaneous bifurcation of countably many fold periodic orbits with infinite period in this delay differential equation. We employ numerical continuation and the concept of reappearance of periodic solutions to show that these bifurcations coincide with codimension-two points along families of connecting orbits and fold periodic orbits in a related advanced differential equation. These points include heteroclinic connections between steady states and homoclinic bifurcations with non-hyperbolic equilibria. Tracking these codimension-two points in parameter space reveals the critical parameter values for the existence of periodic pulse trains. We use the recently developed theory of temporal dissipative solitons to infer necessary conditions for the stability of such pulse trains.

On the Detuned $$2\!{:}\!4$$ Resonance

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials this concerns the short axial orbits and in galactic dynamics the resulting stable periodic orbits are called "banana" orbits. Galactic potentials are symmetric with respect to the co-ordinate planes whence the potential -- and the normal form -- both have no cubic terms. This $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetry turns the 1:2 resonance into a higher order resonance and one therefore also speaks of the 2:4 resonance. In this paper we study the 2:4 resonance in its own right, not restricted to natural Hamiltonian systems where $H = T + V$ would consist of kinetic and (positional) potential energy. The short axial orbit then turns out to be dynamically stable everywhere except at a simultaneous bifurcation of banana and "anti-banana" orbits, while it is now the long axial orbit that loses and regains stability through two successive period-doubling bifurcations.

Bifurcation Features of Periodic Solutions of the Mackey–Glass Equation

Bifurcations of periodic solutions of the well-known Mackey–Glass equation from its unique equilibrium state under varying equation parameters are considered. The equation is used as a mathematical model of variations in the density of white blood cells (neutrophils). Written in dimensionless variables, the equation contains a small parameter multiplying the derivative, which makes this equation singular. It is shown that the behavior of solutions to the equation with initial data from a fixed neighborhood of the equilibrium state in the equation phase space is described by a countable system of nonlinear ordinary differential equations. This system has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. One fast variable and a countable number of slow variables can be extracted from this system of equations. As a result, the averaging method can be applied to the obtained system. It is shown that the equilibrium states of the averaged system of equations in slow variables are associated with periodic solutions of the same stability type in the original equation. The possibility of simultaneous bifurcation of a large number of periodic solutions (multistability bifurcation) is shown. It is also shown that, with a further increase in the bifurcation parameter, each of the periodic solutions exhibits the transition to a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Mackey–Glass equation is characterized by chaotic multistability.

Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli

We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system (x′,y′)=(−y((x−1)2+y2),x((x−1)2+y2)) has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli, first separately and second simultaneously. This problem is a generalization of [24] to the piecewise systems class. When the polynomial perturbation has degree n, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree as the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. When the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three types of configurations: (9,3), (6,6) and (4,8). In the non-piecewise scenario, only 5 limit cycles were found.

Synchrony and the Dynamic Dichotomy in a Class of Matrix Population Models

Temporal synchronization in population dynamics is a common ecological phenomenon that can have important consequences for conservation and management strategies. Though many environmental factors may cause cycling, it is also possible for cycles to occur due to intrinsic properties of the population, such as competition between individuals at different developmental stages. For discrete-time matrix models, competition between stages may result in synchronous cycles in which stages are temporally separated. In this paper we define a class of matrix models which exhibit a simultaneous bifurcation of positive equilibria and synchronous 2-cycles when the extinction equilibrium destabilizes. We also show that a dynamic dichotomy exists between the two steady states such that, if the steady states are bifurcating forward, then they have opposite stability conditions. Which of the two steady states is stable depends on measures of the within-class and between-class competition. These results extend the fundamental bifurcation theorem to a class of matrix models whose inherent projection matrix has index of imprimitivity two.

Limit Cycle Bifurcation of Infinity and Degenerate Singular Point in Three-Dimensional Vector Field

Our work focuses on investigating limit cycle bifurcation for infinity and a degenerate singular point of a fifth degree system in three-dimensional vector field. By using singular value method to compute focal values carefully, we give the expressions of the focal values (Lyapunov constants) at the origin and at infinity. Moreover, we obtain that four limit cycles at most can bifurcate from the origin and three limit cycles can bifurcate from infinity. At the same time, we show the structure of limit cycles from the origin and the infinity. It is interesting for this kind of nonlinear phenomenon that a string of large limit cycles encircle a string of small limit cycles by simultaneous Hopf bifurcation, which is hardly seen for similar published results in three-dimensional vector field, our result is new.

Bifurcation of Periodic Solutions of the Mackey– Glass Equation

We study the bifurcation of the equilibrium states of periodic solutions for the Mackey– Glass equation. This equation is considered as a mathematical model of changes in the density of white blood cells. The equation written in dimensionless variables contains a small parameter at the derivative, which makes it singular. We applied the method of uniform normalization, which allows us to reduce the study of the solutions behavior in the neighborhood of the equilibrium state to the analysis of the countable system of ordinary differential equations. We poot out the equations in ”fast” and ”slow” variables from this system. Equilibrium states of the ”slow” variables equations determine the periodic solutions. The analysis of equilibrium states allows us to study the bifurcation of periodic solutions depending on the parameters of the equation and their stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions is shown. This situation is called the multistability phenomenon.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

TCTAP C-223 Simultaneous Transcatheter Aortic Valve Implantation and Bifurcation Stenting of the Left Main Coronary Artery

### Patient initials or identifier number Vilk-013 ### Relevant clinical history and physical exam 77 y.o. woman Diagnosis: severe aortic stenosis. Stable angina. Arterial hypertension. Heart failure NYHA 2. CT-data: annulus = 2,2 x 2,8 cm; aorta - 3,7 cm; Sino-size tubular isthmus = 3,1 cm;

Abelian integrals and limit cycles for a class of cubic polynomial vector fields of Lotka–Volterra type with a rational first integral of degree two

In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka–Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree n. The analysis is carried out by estimating the number of zeros of the corresponding Abelian integrals. Moreover, using Chebyshev criterion, we show that the sharp upper bound for the number of zeros of the Abelian integrals defined on each period annulus is 3 for n=3. The simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered. All configurations (u,v) with 0≤u,v≤3, u+v≤5 are realizable.

Limit Cycles Bifurcations for a Class of 3-Dimensional Quadratic Systems

Our work is concerned with the Hopf bifurcation for a class of 3-dimensional quadratic symmetric vector field. By making use of singular values methods and computing carefully, we give the expressions of the first four focal values of the two singular point that system has. Both singular symmetric points can become fine foci of fourth order at the same time. Moreover, we obtain that from each one can bifurcate 4 small limit cycles under a certain coefficients' perturbed condition, consequently, at most eight limit cycles can appear by simultaneous Hopf bifurcation.

Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x′=−yF(x,y), y′=xF(x,y), where the set {F(x,y)=0} consists of k different isolated points, is defined by k+1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n. Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1, the provided upper bound is reached. Finally, the case k=2 is also treated.

Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry

Three-dimensional symmetric piecewise linear differential systems near the conditions corresponding to the fold-Hopf bifurcation for smooth systems are considered. By introducing one small parameter, we study the bifurcation of limit cycles in passing through its critical value, when the three eigenvalues of the linear part at the origin are at the imaginary axis of the complex plane.The simultaneous bifurcation of three limit cycles is proved. Conditions for stability of these limit cycles are provided, and analytical expressions for their period and amplitude are obtained.Finally, we apply the achieved theoretical results to a generalized version of Chua’s circuit, showing that the fold-Hopf bifurcation takes place for a certain range of parameters.

The supports of higher bifurcation currents

Let (f_\lambda) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold \Lambda. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on \Lambda, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).

Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits

In this paper, we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation z ̇ = i z + z 3 , after a small polynomial perturbation. We first show that, under small perturbations of the form ε P 2 m − 1 ( z , z ̄ ) , where P 2 m − 1 ( z , z ̄ ) is a polynomial of degree 2 m − 1 in which the power of z is odd and the power of z ̄ is even, the only possible distribution of limit cycles is ( u , u ) for all values of u = 0 , 1 , 2 , … , m − 3 . Hence, the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of z ̇ = i z + z 3 is m − 3 , for m ≥ 4 . Then we consider a perturbation of the form ε P m ( z , z ̄ ) , where P m ( z , z ̄ ) is a polynomial of degree m in which the power of z is odd and obtain the upper bound m − 5 , for m ≥ 6 . Moreover, we show that the distribution ( u , v ) of limit cycles is possible for 0 ≤ u ≤ m − 5 , 0 ≤ v ≤ m − 5 with u + v ≤ m − 2 and m ≥ 9 .

Defect-deformational surface layer instability as a universal mechanism for forming lattices and nanodot ensembles under the effect of ion and laser beams on solid bodies

The defect-deformational (DD) model of surface nanostructure formation under the effect of ion and laser beams is extended to an anisotropic case. The DD theory describes from a unified point of view simultaneous bifurcations in the dependences of grating orientation and period on the ion beam incidence angle, grating rotation, and transition to the formation of a nanodot structure, all of which have been recently observed during ion sputtering of Si. A nonlinear two-dimensional Kuramoto-Sivashinsky DD-equation (DDKS) that describes the formation of lamella-like structures and ordered hexagonal ensembles of nanodots upon normal incidence of ion and laser beams on solid bodies is obtained.

BIFURCATION ANALYSIS ON NONSMOOTH TORUS DESTRUCTION SCENARIO OF DELAYED-PWM SWITCHED BUCK CONVERTER

In this paper, we propose a qualitative and quantitative dynamical study about the evolution of nonsmooth torus in a Digital Delayed Pulse-Width Modulator (PWM) switched buck converter. We explain the birth and destruction of the torus by successive discontinuity induced bifurcations (DIBs). The Digital-PWM control is based on Zero Average Dynamics (ZAD) strategy and a one-period delay is included in the control law. The control parameter (ks) of the ZAD strategy can be varied in a large range, ideally (-∞, ∞). On these borders, the dynamical behavior is the same and thus an annulus-like parameter space is considered. Under variation of ks, the system gets closer to a codimension-two bifurcation point, where two simultaneous border-collision bifurcations (BC) meet. The system leaves the high-order periodic behavior and a high-order band torus appears. Wrinkles and nonsmoothness due to more and more successive border-collisions bifurcations cause the torus destruction in high-order band chaos with tent-map-like structures and Mandelbrot-like sets. Finally, the chaotic bands merge in one-band chaos. The switched converter is modeled as a piecewise linear system where an analytical expression of the Poincaré map is available. Characteristics as duality, symmetry and recurrent behavior are determined using additional numerical methods for the Poincaré map decomposition in periodic sequences.