Bifurcation Set Research Articles

Bifurcation of Euler buckling problem, revisited

We present a precise discussion of Golubitsky and Schaeffer's treatment on bifurcation of Euler buckling problem. We discuss smoothness of the problem and derive equations describing the bifurcation set $B$ and the hysteresis set $H$ up to order 3, which enable us to draw their figures approximately under suitable assumptions.

Dynamics of a discontinuous coupled electro-mechanical system oscillator with strong irrational nonlinearities and with two outputs

The dynamics of the nonlinear electromechanical device, consisting of a mechanical part with two outputs and an electrical part which acts as the server is strongly investigated in the present work. The mechanical part consists of two nonlinear oscillators with strong irrational nonlinearities having smooth or discontinuous characteristics, where nonlinearity is just due to the inclination of springs, the geometric configuration, which are both elastically coupled. While the electrical part is the Rayleigh equation. By using the Lagrangian formulation, the model equations are established and used to investigate the equilibrium points and their stabilities. Nest by using the multiple time scales method, the analytical solutions are found both for the case of large amplitude and the weak amplitude, leading to interesting bifurcation sets of the equilibria by varying the control parameters, the inclination angles and driven frequency. Finally, numerical investigations of the exact equation of the system are used to justify the validity of analytical results and to find new phenomena such as chaotic impulses, chaotic bursting and the train of kink signal generations.

BIFURCATION APPROACH TO ANALYSIS OF DIVERGENT LOSS STABILITY OF NONLINEAR DYNAMIC SYSTEM

A bifurcation approach to analysis of divergent loss of stability of a trivial solution of a nonlinear dynamical system is presented. It is shown that the bifurcation set in the critical case of one zero root locally coincides with the discriminant set of a third-degree polynomial, which defines the set of stationary states of the system in neighborhood of symmetric solution of the system. This approach makes it possible to obtain conditions for safe-dangerous loss of stability of the symmetric solution, which are equivalent to the conditions of M. M. Bautin with minimum possible computational costs.

Global analysis in Bazykin's model with Holling II functional response and predator competition

In this paper, we study the well-known Bazykin's model with Holling II functional response and predator competition. A detailed bifurcation analysis, depending on all four parameters, reveals a rich bifurcation structure, including supercritical and subcritical Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension at most 2, and a focus type degenerate Bogdanov-Takens bifurcation of codimension 3, originating from a nilpotent focus of codimension 3 which acts as the organizing center for the bifurcation set. Moreover, some sufficient conditions to guarantee the global asymptotical stability of the semi-trivial equilibrium or the unique positive equilibrium are also given. Our analysis indicates that we can classify the long-time dynamics of the model with a threshold value c0 for the natural mortality rate c of predators, in detail, the following are true. (i) When c≥c0, the prey will persist and predators will eventually go extinct for all positive initial populations. (ii) When c<c0, the prey and predators will coexist, for all positive initial populations, in the form of multiple positive equilibria or multiple periodic orbits. Our results can be seen as a complement to the work by Bazykin et al. [2–5], Hainzl [22,23], Kuznetsov [30].

TRAVELING WAVE SOLUTIONS OF TWO TYPES OF GENERALIZED BREAKING SOLITON EQUATIONS

<abstract> In this paper, the bifurcation theory of dynamical system is applied to study traveling waves of two types of generalized breaking soliton (GBS) equations which include many famous partial differential equation models. Without any parameter constraints, we investigate their traveling wave systems in detail from the geometric point of view. Due to the existence of a two dimensional invariant manifold, all bounded and unbounded orbits are identified and studied in different parameter bifurcation sets. Furthermore, by calculating complicated elliptic integrals along these orbits, we obtain exact expressions of all possible single wave solutions of two types of GBS equations. </abstract>

TRAVELING WAVE SOLUTIONS OF THE GENERALIZED (2+1)-DIMENSIONAL KUNDU-MUKHERJEE-NASKAR EQUATION

In this paper, we consider two types of traveling wave systems of the generalized Kundu-Mukherjee-Naskar equation. Firstly, due to the integrity, we obtain their energy functions. Then, the dynamical system method is applied to study bifurcation behaviours of the two types of traveling wave systems to obtain corresponding global phase portraits in different parameter bifurcation sets. According to them, every bounded and unbounded orbits can be identified clearly and investigated carefully which correspond to various traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation exactly. Finally, by integrating along these orbits and calculating some complicated elliptic integral, we obtain all type I and type II traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation without loss.

Duffing-type Equations: Singular Points of Amplitude Profiles and Bifurcations

We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.

Dynamics and control of loop reactors: a review

In the loop reactor (LR) the system is composed of several reactor units that are organized in a loop and the feeding takes place at one of several ports with switching of the feed port in a periodic way. In its simplest operation a pulse is formed and rotates around it, producing high temperatures which enable combustion of dilute streams. A limiting model with infinite number of units was derived. Rotating pulses, that are steady in a coordinate moving with the switch velocity, emerge in both asymptotic and discrete models when the ratio of switching to front propagation velocities is around unity. But this behavior exists over a narrow domain of this ratio. Simulations were conducted with generic first order Arrhenius kinetics. Experimental observations of simple frozen rotating pulses are reviewed. Outside the narrow frozen rotating patterns domain the system may exhibit multi- or quasi-periodic operation separated by domains of inactive reaction. The bifurcation set incorporates many ’finger’-like domains of complex frequency-locked solutions that allow to significantly extend the operation domain with higher feed temperature or concentration. Control is necessary to attain stable simple rotating frozen patterns within the narrow domains of active operation. Various control approaches that were suggested, or experimentally applied for this purpose, are reviewed. Actual implementation of combustion in LR will involve several reactants of different ignition temperatures and varying concentration. Design and control should be aimed at producing locked fronts and avoid extinction of the slower reaction.

Modal synchronization of coupled bistable van der Pol oscillators

The paper revisits recently revealed regimes of the "nonconventional synchronization" in systems of coupled bi-stable Van der Pol oscillators. These regimes are characterized by periodic (or quasiperiodic) almost complete energy exchanges between the coupled oscillators. In the paper it is demonstrated that such responses correspond to synchronization of the modulation amplitudes between symmetric and antisymmetric modes of the system, with persistent phase drift. This observation substantially simplifies the treatment, reduces the dynamics to simple phase cylinder and allows to reveal a rich set of global and local bifurcations of limit cycles and tori in the system. Among other findings, one encounters an unexpected regime of nonstationary asymmetric synchronization.

Development of a wheat aphid population dynamics model based on cusp catastrophe theory

Aphids are a major global wheat pest that can cause considerable loss of yield. Modeling of aphid population dynamics is an integral part of management strategies to manage or control aphid populations. In this paper, first, a wheat aphid population dynamics model was developed based on a logistic model and the Holling III functional response, which includes three factors: temperature, natural enemies and insecticide. Second, this model fitted with a cusp catastrophe model to describe how abrupt changes in the wheat aphid population were influenced by these factors. Finally, the system was validated with field data from 2016 to 2018. The bifurcation set of the cusp catastrophe model was deemed to be the quantified dynamic control threshold, so an outbreak of aphid’s population can be explained according to the variation of control variables. In short, this aphid population model was successfully validated on survey data, which can be used to guide the prevention and control of aphids.

Investigation of cusp catastrophe model of rock slope instability with general constitutive equations

The unsteadiness of rock slopes is often triggered by water-softening. Based on catastrophe theory and the general constitutive equations of the strain-softening section in the weak interlayer, the present study investigated the catastrophic failure mechanism of rock slopes. Using theoretical analysis, the cusp catastrophe model for the failure of a rock slope with a strain-softening section in the soft rock strata was established, and the balance equation, bifurcation set, and formulas of displacement jumping and energy jumping were deduced. The changes of displacement jumping and energy jumping following the variation of the systematic rigidity ratio are discussed with regard to the occurrence of water-softening. Regardless of which type of constitutive equation is adopted, the control variables and sudden jump values of the rock slope’s displacement and energy can be determined by the systematic rigidity ratio when the systematic rigidity ratio is constant. Whether a rock slope is stable or not depends only on the mechanical and geometrical parameters of the system itself, and is unrelated to the constitutive equation of the strain-softening section. The sudden jump values of the displacement and energy are both inversely proportional to the systematic rigidity ratio. By comparing the formula of the systematic rigidity ratio deduced under different constitutive equations with the results obtained by other studies, the results of the present study were verified through the consideration of actual engineering projects. Based on catastrophe theory, the present paper discusses the principle of correctly selecting a constitutive equation to assess the rock slope stability in practical engineering.

Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds

We construct a complete topological invariant of foliations of finite type defined by smooth functions on two-dimensional noncompact orientable manifolds. In particular, we describe a complete topological classification of noncompact bifurcations of such foliations. We establish a natural one-to-one correspondence between the set of all such bifurcations and the set of oriented coloured graphs of a special form. As a consequence, we obtain the Liouville and trajectory classifications of Hamiltonian systems of finite type on noncompact two-dimensional manifolds. Bibliography: 25 titles.

Periodic Orbit Bifurcations in Planar Hysteretic Systems without Equilibria

This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some [Formula: see text]D piecewise linear systems.

Risk state changes analysis of railway dangerous goods transportation system: Based on the cusp catastrophe model

In this paper, the cusp catastrophe model is applied to analyze the railway dangerous goods transportation system risk state changes. Firstly, a Risk-Accident Catastrophe Tree is proposed to formulate the process that how the risk factors cause the accident. Secondly, the whole risk factors are classified into five categories including failure of human behavior, machine failure, transported materials, environment problems and management problems. Next, the cusp catastrophe model of railway dangerous goods transportation system is established, the Split Coefficient is defined to evaluate bifurcation set. The results of the case study show that the risk state changes of railway dangerous goods transportation system satisfy bimodality, inaccessibility, sudden transitions and divergence, but not satisfy hysteresis. The larger the Split Coefficient is, the easier the trajectory of system control point crosses with bifurcation sets. The reason of the system state changes from a safe state to a risk state is: as long as the trajectory of system control point crosses with bifurcation sets, there must be a cusp catastrophe in the system, and the system control point will cross the fold surface, which makes the risk energy increase sharply, the structure, information and energy of the system will also be destroyed.

Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations.

Relative bifurcation sets and the local dimension of univoque bases

AbstractFix an alphabet$A=\{0,1,\ldots ,M\}$with$M\in \mathbb{N}$. The univoque set$\mathscr{U}$of bases$q\in (1,M+1)$in which the number$1$has a unique expansion over the alphabet$A$has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set$\mathscr{U}$are distributed over the interval$(1,M+1)$by determining the limit$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$for all$q\in (1,M+1)$. We show in particular that$f(q)&gt;0$if and only if$q\in \overline{\mathscr{U}}\backslash \mathscr{C}$, where$\mathscr{C}$is an uncountable set of Hausdorff dimension zero, and$f$is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of$\mathscr{U}$called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of$\mathscr{U}$with any interval, answering a question of Kalleet al [On the bifurcation set of unique expansions. Acta Arith.188(2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets.Adv. Math.308(2017), 575–598] on strongly univoque sets.

Effective equation for two coupled oscillators: Towards a global view of metamorphoses of the amplitude profiles

Dynamics of nonlinear coupled driven oscillators is investigated. Recently, we have demonstrated that the amplitude profiles – dependence of the amplitude A on frequency Ω of the driving force, computed by asymptotic methods in implicit form as FA,Ω=0, permit prediction of metamorphoses of dynamics which occur at singular points of the implicit curve FA,Ω=0. In the present study we strive at a global view of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point.

Memristor Synapse-Based Morris–Lecar Model: Bifurcation Analyses and FPGA-Based Validations for Periodic and Chaotic Bursting/Spiking Firings

Electromagnetic induction current sensed by the membrane potential in biological neurons can be characterized with a memristor synapse, which can be employed to demonstrate the real oscillating voltage patterns of Barnacle muscle fibers. This paper presents a 3D autonomous memristor synapse-based Morris–Lecar (abbreviated as m-ML) model, which is implemented through introducing a memristor synapse-based induction current to substitute the externally applied current in an existing 2D nonautonomous Morris–Lecar model. Making use of one- and two-parameter bifurcation plots and time-domain representations, diverse period-adding bifurcations as well as abundant periodic and chaotic burst firings are demonstrated. Through constructing the fold and Hopf bifurcation sets of fast spiking subsystem, bifurcation analyses of these chaotic and periodic burst firings are carried out. Moreover, the periodic and chaotic spiking firings and coexisting firing behaviors are illustrated by using one- and two-parameter bifurcation plots and local attraction basins. Finally, based on a field programmable gate array (FPGA) board, a compact digital electronic neuron is fabricated for the 3D m-ML model, from which periodic and chaotic bursting/spiking firings are experimentally measured to verify the results of the numerical simulations.

Two bifurcation sets arising from the beta transformation with a hole at 0

Given β∈(1,2], the β-transformation Tβ:x↦βx(mod1) on the circle [0,1) with a hole [0,t) was investigated by Kalle et al. (2019). They described the set-valued bifurcation set Eβ≔{t∈[0,1):Kβ(t′)≠Kβ(t)∀t′>t},where Kβ(t)≔{x∈[0,1):Tβn(x)≥t∀n≥0} is the survivor set. In this paper we investigate the dimension bifurcation set ℬβ≔{t∈[0,1):dimHKβ(t′)≠dimHKβ(t)∀t′>t},where dimH denotes the Hausdorff dimension. We show that if β∈(1,2] is a multinacci number then the two bifurcation sets ℬβ and Eβ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for β a multinacci number we have dimH(Eβ∩[t,1])=dimHKβ(t) for any t∈[0,1). This confirms a conjecture of Kalle et al. for β a multinacci number.

Dynamic Analysis of Influence of Soil Damping Nonlinearity on Foundation Stability

Aiming at the sliding instability problem of foundation-based system under the action of earthquake and other dynamics, nonlinear cusp catastrophe analysis model which conforms the actual working behavior of foundation-based system is established by using the catastrophe theory. It describes the whole process of catastrophic instability of the system and analyze the basic stability. Based on the research of many nonlinear factors contained in the system that has been considered so far, the inherent influence of the nonlinear factors of soil damping in the actual working environment is considered and analyzed for the first time. Under the linear, quadratic and cubic nonlinear conditions, the most general expression of soil damping reflecting the system’s instability and catastrophe of the system bifurcation set is derived. The results show that the nonlinear characteristics of damping directly affect the correct evaluation of the ultimate bearing capacity of the foundation. Although ignoring the nonlinear influence of damping will not change its catastrophic properties, but will underestimate the ultimate bearing capacity of the foundation.