Bifurcation Of Limit Cycles Research Articles

Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The piecewise smooth linear differential systems with two centers formed in two ways, one is that a center-fold point at the origin, the other is a center-fold at the origin and another unique center point exists. We first explore the expression of the first order Melnikov function. Then by using the Melnikov function method, we give estimations of the number of limit cycles bifurcating from the period annulus. For the latter case, the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.

Influence of Coil Current and Oil Film Thickness on Hopf Bifurcation of MLDSB

Magnetic-liquid double-suspension bearing (MLDSB) is composed of an electromagnetic supporting system and a hydrostatic supporting system. Due to its greater supporting capacity and stiffness, it is appropriate for middle-speed applications, overloading, and frequent starting. However, because it contains two sets of systems, its structure and rotor support system are more complex. It contains strong nonlinear links. When the parameters of the system change, the bearing rotor may feature Hopf bifurcation, resulting in system flutter and reducing the operational stability of the magnetic fluid double-suspension bearing rotor, which has become one of the key problems restricting its development and application. As key parameters of MLDSB, the coil current and oil film thickness exert a major impact on Hopf bifurcation. Therefore, the mathematical model of MLDSB is established in this paper, and the border and direction of Hopf bifurcation, period, and amplitude of limit cycle are analyzed. The calculation, simulation, and experimental results show that when the coil current and oil film thickness of the bearing system are greater than the boundary value of the Hopf bifurcation, Hopf bifurcation will occur, resulting in the vibration of the bearing rotor and affecting the stability of the system. In addition, when analyzing the combined effects of coil current and oil film thickness on the Hopf bifurcation of the system, it was found that the boundary value of Hopf bifurcation in the system is reduced compared with when it is are affected solely due to the coupling of the two parameters. The period, amplitude and vibration speed of limit cycle increase with increases in the coil current and oil film thickness. Hopf bifurcation experiment was conducted on MLDSB testing system. The results show that Hopf bifurcation does not occur when i0 < 0.5 A, the bearing rotor operates stably in the balanced position, i0 > 1.0 A, Hopf bifurcation occurs in the system, and the bearing rotor vibrates with equal amplitude, which reduces the stability of operation. The research in this paper can provide a theoretical reference for the Hopf bifurcation analysis of MLDSB.

Bifurcations of Limit Cycles in Rotating Shafts Mounted On Partial Arc and Lemon Bore Journal Bearings in Elastic Pedestals

Abstract The paper investigates the bifurcations encountered in a simple rotor dynamic system interacting with nonlinear impedance forces, generated by the supporting journal bearings of realistic profile geometry. Bearing configurations of finite arc length and of finite width, as implemented in standard design of turbomachinery have been selected, namely the cylindrical partial arc and the elliptical (lemon) bore profile. The way in which the key design parameters influence the stability of elastic or rigid Jeffcott rotor is discussed. In the scope of this study, the following bearing design parameters are considered: arc length, length to diameter ratio, geometric preload and offset, and properties of the supporting pedestal by codimension-two studies. The bearing model is coupled to a six degree of freedom shaft-disc-pedestal model with nonlinear forces calculated from the journal kinematics, bearing design and operating conditions by numerical evaluation of the Reynolds equation for laminar, isothermal flow on a two dimensional mesh. An autonomous system of differential equations is implemented. Stability of fixed points and of limit cycles for this system is evaluated applying numerical continuation. The results confirm that minor variations in journal bearing design and pedestal properties have the potential to render substantial changes in the quality of stability and the bifurcation set of the rotor dynamic system. Specific bearing profiles render significant increment of instability threshold speed while at the same time supercritical Hopf bifurcations can be shifted to subcritical with resulting instability envelopes to be generated at speeds lower than the threshold speed.

Melnikov functions of arbitrary order for piecewise smooth differential systems in [formula omitted] and applications

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in Rn with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.

Stability and bifurcations in fractional-order gene regulatory networks

A fractional-order mathematical model of a large-scale GRN is presented, focusing on the positivity of solutions and on the stability analysis of equilibria based on Matignon’s theorem. With special emphasis on cyclic GRNs, necessary and sufficient conditions are obtained in terms of inequalities involving the system’s parameters and the fractional order, which guarantee the stability of equilibrium points. Based on the stability analysis, the critical values of the fractional order for which Hopf-type bifurcations may occur are detected. In this context, it is shown that oscillatory behavior due to a Hopf-type bifurcation (considering the fractional order as bifurcation parameter) is expected only in the case of an odd number of repressive genes. Numerical examples are given in order to demonstrate the usefulness of the proposed results and to exemplify the effect of the fractional order on the dynamics of cyclic GRNs, additionally revealing that period-doubling bifurcations of limit cycles may also occur in sufficiently large cyclic GRNs.

Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation

In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin.

Complex dynamics and Bogdanov-Takens bifurcations in a retarded van der Pol-Duffing oscillator with positional delayed feedback.

In this article, we will investigate a retarded van der Pol-Duffing oscillator with multiple delays. At first, we will find conditions for which Bogdanov-Takens (B-T) bifurcation occurs around the trivial equilibrium of the proposed system. The center manifold theory has been used to extract second order normal form of the B-T bifurcation. After that, we derived third order normal form. We also provide a few bifurcation diagrams, including those for the Hopf, double limit cycle, homoclinic, saddle-node, and Bogdanov-Takens bifurcation. In order to meet the theoretical requirements, extensive numerical simulations have been presented in the conclusion.

Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis

&lt;p style='text-indent:20px;'&gt;In the present study, a nonlinear model is formulated to demonstrate crop - weed interactions, when they both grow together on agricultural land and compete with each other for the same resources like sunlight, water, nutrients etc., under the aegis of herbicides. The developed model is mathematically analyzed through qualitative theory of differential equations to demonstrate rich dynamical characteristics of the system, which are important to be known for maximizing crop yield. The qualitative results reveal that the system not only exhibits stability of more than one equilibrium states, but also undergoes saddle - node, transcritical and Hopf bifurcations, however, depending on parametric combinations. The results of saddle - node and transcritical bifurcations help to plan strategies for maximum crop yield by putting check over the parameters responsible for the depletion of crops due to their interaction with weeds and herbicides. Hopf - bifurcation shows bifurcation of limit cycle through Hopf - bifurcation threshold, which supports that crop - weed interactions are not always of regular type, but they can also be periodic.&lt;/p&gt;

Stability analysis for Schnakenberg reaction-diffusion model with gene expression time delay

This paper considers both analytical and numerical solutions for a diffusive Schnakenberg model with gene-expression time delays, presenting an analysis of the effects of delays and diffusion on stability regions and bifurcation maps. A one-domain system was considered. Systems of delay ODEs were obtained using the Galerkin method. Theoretical conditions for the existence of steady-state and Hopf bifurcation curves were determined. In addition, Hopf bifurcation points and bifurcation stability regions were plotted in detail. The gene-expression time delays and diffusion rates influenced the stability regions for both reactant concentration controls in the system. The results showed that, with increases in time delays, the rates of the Hopf bifurcation points for both chemical concentrations controls decreased, while both parameters of the diffusion coefficient grew as the chemical control values increased. Numerical simulations for bifurcation diagrams, limit cycles, and periodic routes to chaotic behaviour planes confirm the solutions achieved from theory.

Limit Cycles Appearing From Piecewise Smooth Perturbations to a Reversible Nonlinear Center

This paper deals with the limit cycle bifurcation from a reversible differential center of degree [Formula: see text] due to small piecewise smooth homogeneous polynomial perturbations. By using the averaging theory for discontinuous systems and the complex method based on the Argument Principle, we obtain lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus around the center of the unperturbed system.

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.

Fifteen Limit Cycles Bifurcating from a Perturbed Cubic Center

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

Bifurcations underlying different excitability transitions modulated by excitatory and inhibitory memristor and chemical autapses

Memristor is an emerging modulation factor to mimic neuronal synapse, and the switch or transition between neuronal excitability classes is an important topic in nonlinear dynamics and neurophysiology. In the present investigation on the Morris-Lecar model, the memristive autapse is identified to induce excitability transitions different from those of the chemical one. The inhibitory and excitatory chemical autapses induce excitability switch from class I to II and from class II to I, respectively. However, the inhibitory and excitatory memristive autapses induce the transition from class II to I and from class I to II, respectively. Furthermore, comprehensive bifurcation mechanisms underlying the transitions are acquired. The class I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC), and class II excitability to the saddle-node (SN) bifurcation and Hopf bifurcation. Type I spiking corresponds to the SNIC or big homoclinic (BHom) orbit bifurcations, and type II spiking to the Fold Limit Cycle (FLC) bifurcation or supercritical Hopf bifurcation. Among various codimension-2 bifurcations, the saddle-node homoclinic orbit bifurcation, which is related to the SNIC, BHom, and SN bifurcations, is responsible for the excitability transition. A nameless degenerate bifurcation associated with the BHom and FLC bifurcations, is responsible for the spiking transition. In addition, the nullcline shapes present the geometric mechanism for the transitions. The obtained results present different roles of memristive and chemical autapses on modulating neuronal electrical activities and the underlying dynamical mechanisms, which are helpful for the design and development of novel memristive synapses with excitatory or inhibitory effects.

The focus case of a nonsmooth Rayleigh–Duffing oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation $$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$ for the case $$d>0$$ , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case $$d<0$$ , i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

Two-parameter non-smooth bifurcations of period-one motions in a plastic impacting oscillator

A 2-DOF plastic impacting oscillator is considered. The discontinuity boundaries of the system phase space are defined as well as the local maps and Poincaré map. Four cases of sliding bifurcation of limit cycles in impact system are described schematically. The period-one motion modes and occurrence regions of the system are identified in the two-parameter space by numerical simulation. One-parameter bifurcation diagrams and trajectories are computed to explain the evolutions of adjacent periodic motions and the non-smooth bifurcation behaviors including grazing and four types of sliding bifurcation. Due to the discontinuous grazing bifurcation there appear a narrow hysteresis domain in the two-parameter evolution between neighboring non-impacting and impacting motions. Grazing, switching–sliding and multi-sliding bifurcations of periodic sticking motions are all manifested as rising phenomenon of the time trajectory, but the different locations of the rises in the sticking phase correspond to the different types of non-smooth bifurcations. The grazing induced dynamics and the characteristics of rising events are discussed. Notably, the codimension-2 bifurcation points are detected and investigated by numerical continuation algorithm.

Bifurcation of limit cycles and isochronous centers on center manifolds for four‐dimensional systems

This paper is concerned with the bifurcation of limit cycles and isochronous centers on center manifolds for four‐dimensional nonlinear dynamic systems, which have a pair of purely imaginary eigenvalues and a pair of eigenvalues with negative real part for the Jacobian matrix corresponding to the singularity situated at the origin. In order to investigate the isochronous centers for four‐dimensional systems, a new algorithm to calculate so‐called isochronous constants is given. Moreover, with the calculation of the isochronous constants, the conditions for the existence of an isochronous center are determined in a way which does not need first computing the center manifolds of the four‐dimensional system. Finally, as an application, we investigate a class of quadratic polynomial systems for a model of airfoil and prove that the system can have five small limit cycles around the origin on the center manifold. Moreover, we apply our new methodology to find the sufficient and necessary conditions for the origin to be an isochronous center restricted to the center manifold, illustrating the reliability and availability of the new algorithm.

Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

Bifurcations of clusters and collective oscillations in networks of bistable units.

We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a cluster subspace, where the electrodes take at most three distinct states. We systematically analyze the dynamics and bifurcations of the model equations: We consider the dynamics on cluster subspaces of successively increasing dimension and analyze the bifurcations occurring therein. Most importantly, the system exhibits an equivariant transcritical bifurcation of limit cycles. From this bifurcation, several limit cycles branch, one of which is stable for arbitrarily many bistable units.

Limit cycles bifurcated from a focus-fold singularity in general piecewise smooth planar systems

We consider codimension-two focus-fold bifurcation for general piecewise smooth (PWS) planar systems. Some changes of variables and parameters are employed to derive a normal form, which is topologically equivalent to the original system. Making use of the normal form, we verify that observations of focus-fold bifurcation in some special piecewise linear systems also hold for general PWS systems, including the existence of two types of sliding limit cycles and coexistence of crossing and sliding limit cycles. It is shown that two nested crossing limit cycles can be bifurcated from the focus-fold singularity. This result is the improvement of the work [Han and Zhang (2010) [29]], in which only one bifurcated limit cycle is obtained. In addition, the sliding limit cycles are proved to be convergent in finite time.

Homoclinic bifurcation of limit cycles in near-Hamiltonian systems on the cylinder

In this paper, we study a near-Hamiltonian system on the cylinder. First, we establish some general methods on the existence of limit cycles bifurcating from closed orbits of type II by the Melnikov function method, then we derive the expansions of the first order Melnikov function and consider the bifurcation problem of limit cycles near a double homoclinic loop. As an application, we discuss the number of limit cycles of a class of cylinder pendulum-like systems.