Global Bifurcation Analysis Research Articles

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.

Dynamic and chaotic behavior of periodically forced fermentors for bioethanol production

In this preliminary paper, the dynamic responses of bioethanol fermentors to sinusoidal periodic perturbations of the feed concentration have been investigated. It is shown that the steady state autonomous model of the unforced fermentor can be reduced analytically to a cubic polynomial. This analytical form enables to implement the global bifurcation analysis to divide the parameter space into regions with different number of steady state solutions; moreover, the discriminant of the cubic polynomial is used for the analysis of the nature of the number of the steady state solutions. The bifurcation analysis shows that the locus of the fold points is equivalent to the locus of the discriminant of the cubic polynomial in the ( D, C so ) plane. It has been shown that the periodic operation of the autonomous fermentor gives higher average ethanol concentration than the corresponding unstable steady state. The investigation reveals that the centre of forcing has significant effect on the dynamic response of the periodically forced fermentors. Chaotic behavior is developed when the centre of forcing is relatively close to a homoclinic infinite period orbit, while quasi-periodicity is developed when the centre of forcing is in the neighborhood of a Hopf bifurcation point. The system shows interesting phase planes at certain forcing amplitudes. It has been shown that within the parameters range studied the best policy for production of bioethanol is to operate the forced fermentors in the frequency locking regions at small forcing amplitudes.

Predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator-prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.

Lobe transport analysis of the Kelvin–Stuart cat’s eyes driven flow

Mixing and transport in the driven Kelvin-Stuart cat's eyes dynamical system is studied using lobe transport theory and the topological approximation method (TAM). The application of the TAM also provides a global bifurcation analysis. Lobe areas are calculated using the Melnikov amplitude function, which has been derived for the Kelvin-Stuart system. Results indicate that regions, originally in the exterior above the vortex chain, can be transported to the exterior below the vortex chain (and vice versa) by passing through the interior, and that a region within the interior of a given vortex can be transported to the interior of a neighboring vortex, or the interior of a vortex several vortices distant from the given vortex. Cumulative transport is shown to decrease with increasing perturbation frequency for a fixed value of perturbation strength. Cumulative transport increases with increasing perturbation strength for a fixed value of the structure index L. Cumulative transport approaches a characteristic maximum value for each set of parameter values. Results demonstrate a linear dependence of the maximum cumulative transport upon a universal flux function of the form proposed by Rom-Kedar and Poje, suggesting a possible scaling in the transport dependent on the structure index L.

Global Hopf Bifurcation Analysis for a Time‐Delayed Model of Asset Prices

A time‐delayed model of speculative asset markets is investigated to discuss the effect of time delay and market fraction of the fundamentalists on the dynamics of asset prices. It proves that a sequence of Hopf bifurcations occurs at the positive equilibrium v, the fundamental price of the asset, as the parameters vary. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined using normal form method and center manifold theory. Global existence of periodic solutions is established combining a global Hopf bifurcation theorem with a Bendixson′s criterion for higher‐dimensional ordinary differential equations.

Global bifurcation analysis and chaos of an arch structure with parametric and forced excitation

Abstract The global bifurcations and chaotic motions are investigated analytically for an arch structure with parametric and forced excitation. The critical curves separating the chaotic and non-chaotic regions are drawn, which show that the system in the case of 1:1 resonance is more easily chaotically excited than the case of 1:2 resonance. There exist “uncontrollable regions” or “chaotic bands” for the system as the natural frequency varies. There also exists a “controllable frequency” for the system with linear and cubic parametric excitation. The system can be chaotically excited through infinite subharmonic bifurcations of odd/even orders. Numerical results agree with the analytical ones.

Radially symmetric critical points of non-convex functionals

We investigate critical points of the functionalover a ball in ℝn. Here, W is radially symmetric but not convex. We embed the functional into a family of functionalswhere E0,0(u) = E(u). A global bifurcation analysis yields a branch of non-trivial critical points depending on λ and positive ε, where we can set λ = 0. The geometric properties preserved on that branch, due to the maximum principle, prove compactness such that the singular limit as ε ↘ 0 exists. Under natural conditions on W and G the critical point obtained in this way is a minimizer of the original functional. That plan can be carried out only under the restriction of radial symmetry, since the maximum principle applies only to special elliptic equations of fourth order. That restriction, however, is not essential since every minimizer of the functional is radially symmetric.

Characteristic point sequences in local and global bifurcation analysis of Filippov systems

We explain the set of rules behind of the LabView toolbox for bifurcation analysis of Filippov systems denominated SPTCont 1.0. This software can detect nonsmooth bifurcations in n-dimensional systems using integration-free algorithms based on the evaluation of the vector fields on the discontinuity boundary (DB). In this paper, we present the characteristic point sequences that the software detects to guarantee the existence of local and global nonsmooth bifurcations in planar Filippov systems (n = 2). These sequences can be extended to three-dimensional or higher dimension Filippov systems. Boolean-valued functions are used to formulate the conditions of existence for each point defined in the sequences. Dynamics on DB and cycles are defined in function of the set of points. The full catalog of codim 1 local and global bifurcations is used to define the characteristic point sequence when the bifurcation parameter is varied. Finally, an illustrative example is analyzed using step-by-step routines of SPTCont 1.0.

Bifurcation analysis and limit cycle oscillation amplitude prediction methods applied to the aeroelastic galloping problem

A global stability and bifurcation analysis of the transverse galloping of a square section beam in a normal steady flow has been implemented. The model is an ordinary differential equation with polynomial damping nonlinearity. Six methods are used to predict bifurcation, the amplitudes and periods of the ensuing Limit Cycle Oscillations: (i) Cell Mapping, (ii) Harmonic Balance, (iii) Higher Order Harmonic Balance, (iv) Centre Manifold linearization, (v) Normal Form and (vi) numerical continuation. The resulting stability predictions are compared with each other and with results obtained from numerical integration. The advantages and disadvantages of each technique are discussed. It is shown that, despite the simplicity of the system, only two of the methods succeed in predicting its full response spectrum. These are Higher Order Harmonic Balance and numerical continuation.

Global Hopf bifurcation analysis on a BAM neural network with delays

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large.

Global bifurcations and chaos for a rotor-active magnetic bearing system with time-varying stiffness

In this paper, we investigate the global bifurcations and chaotic dynamics in the rotor-active magnetic bearings (AMB) system with 8-pole legs and time-varying stiffness. From the averaged equation obtained in another paper, the theory of normal form is applied in this paper to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Since the normal form obtained here is not the simplest one, the methods of choosing other complementary space and utilizing the inner product are presented to further reduce the normal form and obtain a simpler normal form. Based on simpler the normal form obtained above, a global perturbation method is utilized for the analysis of global bifurcations and chaotic dynamics of the rotor-AMB system. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation of the rotor-AMB system with the time-varying stiffness. These results indicate that the chaotic motions can occur in the rotor-AMB system with time-varying stiffness. Numerical simulations verify the analytical predictions. The jumping and catastrophic phenomena of the amplitude for the chaotic oscillations in the system are also found by using numerical simulations.

Global Hopf bifurcation analysis on a BAM neural network with delays

A BAM neural network with three neurons is considered. Sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numerical simulations are presented to support the theoretical results found.

On nonsmooth bifurcations in a simple friction oscillator

AbstractWe consider a simple friction oscillator with one degree of freedom using a non‐smooth, three‐parameter fricion model accounting for both, a decreasing and increasing dynamic friction force depending on the relative velocity. An analysis of global non‐smooth bifurcations is performed numerically and parameter domains are computed for the existence of stick‐slip oscillations. Special emphasis is placed upon the occurrence of stick‐slip oscillations with negative relative velocity. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt

Abstract In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equations of motion for viscoelastic moving belt with the external damping and parametric excitation are given. The four-dimensional averaged equation under the case of primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving belt. From the averaged equations obtained here, the theory of normal form is used to give the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on normal form, the energy-phrase method is employed to analyze the global bifurcations and chaotic dynamics in parametrically excited viscoelastic moving belt. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type multi-pulse homoclinic orbits in the averaged equation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense in parametrically excited viscoelastic moving belt. The chaotic motions of viscoelastic moving belts are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits is observed from three-dimensional phase space.

A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences

In this work we propose and investigate a family of models, which admits as particular cases some well known mathematical models of tumor-immune system interaction, with the additional assumption that the influx of immune system cells may be a function of the number of cancer cells. Constant, periodic and impulsive therapies (as well as the non-perturbed system) are investigated both analytically for the general family and, by using the model by Kuznetsov et al. [V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Bull. Math. Biol. (1994) 56(2) 295–321), via numerical simulations. Simulations seem to show that the shape of the function modeling the therapy is a crucial factor only for very high values of the therapy period T, whereas for realistic values of T, the eradication of the cancer cells depends on the mean values of the therapy term. Finally, some medical inferences are proposed.

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.

Global Bifurcation Analysis of a Controlled Underactuated Mechanical System

In this paper, bifurcation theory is employed to classify different dynamical behaviors arising in an underactuated mechanical system subject to bounded controls. The methodology is applied to an inertia wheel pendulum consisting of a simple pendulum with a rotating disk at the end. Restricting the magnitude of the control action places an important obstacle to the design of a continuous controller capable of swinging-up and stabilize the pendulum at the inverted position: the arm only can reach that position by means of oscillations of increasing amplitude. The controller is derived from a simple nonlinear state-feedback law, followed by a saturating device that limits the maximum amplitude of the control action applied to the system. This bound gives birth to a rich dynamical behavior, including pitchfork and Hopf bifurcations of equilibria, saddle-node bifurcations of periodic orbits, homoclinic and heteroclinic bifurcations. The global dynamics is analyzed in terms of certain control gains and a two-parameter bifurcation diagram is derived. It is shown that the dynamics on this bifurcation diagram is organized in a pair of codimension-two rotationally symmetric bifurcation points. Finally, it is found out that when the control gains lie on a certain region in the parameter space simultaneous stabilization of the upright position together with a large basin of attraction is obtained. Simulation results show that almost global stabilization of the system can be achieved.

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system.First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

Nonlinear vibrations of suspended cables—Part I: Modeling and analysis

This review article is the first of three parts of a Special Issue dealing with finite-amplitude oscillations of elastic suspended cables. This part is concerned with system modeling and methods of analysis. After shortly reporting on cable historical literature and identifying the topic and scope of the review, the article begins with a presentation of the mechanical system and of the ensuing mathematical models. Continuum equations of cable finite motion are formulated, their linearized version is reported, and nonlinear discretized models for the analysis of 2D or 3D vibration problems are discussed. Approximate methods for asymptotic analysis of either single or multi-degree-of-freedom models of small-sag cables are addressed, as well as asymptotic models operating directly on the original partial differential equations. Numerical tools and geometrical techniques from dynamical systems theory are illustrated with reference to the single-degree-of-freedom model of cable, reporting on measures for diagnosis of nonlinear and chaotic response, as well as on techniques for local and global bifurcation analysis. The paper ends with a discussion on the main features and problems encountered in nonlinear experimental analysis of vibrating suspended cables. This review article cites 226 references.

Global dynamics of the cable under combined parametrical and external excitations

The chaotic dynamics and global bifurcations of the suspended elastic cable under combined parametric and external excitations are investigated. The non-linear equations of motion of the elastic cable to small vibration of one support are derived. The averaged equations are obtained by using the method of multiple scales. Based on the averaged equations, the theory of normal form and Maple program are used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. On the basis of the normal form, global bifurcation analysis of the parametrically and externally excited suspended elastic cable is given by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of the elastic cable is also found by numerical simulation.