Codimension-two Bifurcation Research Articles

Nonintegrability of the unfolding of the fold-Hopf bifurcation

We consider the unfolding of the codimension-two fold-Hopf bifurcation and prove its meromorphic nonintegrability in the meaning of Bogoyavlenskij for almost all parameter values. Our proof is based on a generalized version of the Morales–Ramis–Simó theory for non-Hamiltonian systems and related variational equations up to second order are used.

Cascades of Global Bifurcations and Chaos near a Homoclinic Flip Bifurcation: A Case Study

We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward in understanding the generic cases of homoclinic flip bifurcations, which started with the study of the simpler cases \textbf{A} and \textbf{B}. In a three-dimensional vector field due to Sandstede, we compute relevant bifurcation curves in the two-parameter bifurcation diagram near the central codimension-two bifurcation in unprecedented detail. We present representative images of invariant manifolds, computed with a boundary value problem setup, both in phase space and as intersection sets with a suitable sphere. In this way, we are able to identify infinitely many cascades of homoclinic bifurcations that accumulate on specific codimension-one heteroclinic bifurcations between an equilibrium and various saddle periodic orbits. Our computations confirm what is known from theory but also show the existence of bifurcation phenomena that were not considered before. Specifically, we identify the boundaries of the Smale--horseshoe region in the parameter plane, one of which creates a strange attractor that resembles the R\"{o}ssler attractor. The computation of a winding number reveals a complicated overall bifurcation structure in the wider parameter plane that involves infinitely many further homoclinic flip bifurcations associated with so-called homoclinic bubbles.

Two codimension-two bifurcations of a second-order difference equation from macroeconomics

In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.

Stability, Bifurcation, and Quenching Chaos of a Vehicle Suspension System

This study investigated the dynamics and control of a nonlinear suspension system using a quarter-car model that is forced by the road profile. Bifurcation analysis used to characterize nonlinear dynamic behavior revealed codimension-two bifurcation and homoclinic orbits. The nonlinear dynamics were determined using bifurcation diagrams, phase portraits, Poincaré maps, frequency spectra, and Lyapunov exponents. The Lyapunov exponent was used to identify the onset of chaotic motion. Finally, state feedback control was used to prevent chaotic motion. The effectiveness of the proposed control method was determined via numerical simulations.

Bifurcation analysis of the rock–paper–scissors game with discrete-time logit dynamics

In this study, we investigate a discrete-time version of logit dynamics, as applied to the rock–paper–scissors (RPS) game. First, we show that around the Nash equilibrium point, an attracting closed invariant curve appears due to the Neimark–Sacker bifurcation. Next, near the resonance point, we find a period-three attracting cycle, which can be thought of as a counterpart to the cyclically stable set in the RPS game with best response dynamics. Moreover, we show that the cycle can coexist with an attracting closed invariant curve, a period-three saddle cycle, and the attracting or repelling Nash equilibrium point. Finally, we use the codimension-two bifurcation theory to specify the set of heteroclinic bifurcations that destroy the coexistence of the attractors.

Bifurcations and global dynamics in a toxin-dependent aquatic population model

The study of effects of environmental toxins on ecosystems is of great interest from both environmental and conservation points of view. In this paper, we present a global stability and bifurcation analysis of a toxin-dependent aquatic population model. Our analytical and numerical results show that both the environmental toxin level and the depuration capability of the population significantly affect the population persistence. The model exhibits a multifarious array of dynamics. While low levels of external toxin allow population persistence and high levels of toxin lead to an extirpation, intermediate toxin concentrations can produce very rich dynamics, such as transient oscillations, hysteresis, heteroclinic orbits, and a codimension-two bifurcation. In particular, a regime of bistability exists where the population is doomed to extinction or survival, depending on initial state of the system. As a practical implication of our study, the toxic effects of methylmercury on rainbow trout are scrutinized. The theory developed here provides a sound theoretical foundation for understanding the population effects of toxicity.

Codimension-two bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response

In this paper, we investigate dynamical behaviours of a discrete predator–prey model with nonmonotonic functional response. Codimension-2 bifurcations associated with 1:2, 1:3 and 1:4 resonances are analyzed by using bifurcation theory. Codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams and phase portraits, which not only illustrate the validity of the theoretical results but also display some interesting complex dynamical behaviours, are obtained by numerical simulations.

Effects of second diffusing component and cross diffusion on primary and secondary thermoconvective instabilities in couple stress liquids

Linear and weakly nonlinear analyses are made for the Rayleigh-Benard convection in two-component couple-stress liquids with the Soret effect. Conditions for pitchfork, Hopf, Takens-Bogdanov, and codimension-two bifurcations are presented. The Lorenz model is used to study the inverted bifurcation. Positive values of the Soret coefficient favor a pitchfork bifurcation, whereas negative values favor a Hopf bifurcation. Takens-Bogdanov and codimension-two bifurcations are not as much influenced by the Soret coefficient as pitchfork and Hopf bifurcations. The influence of the Soret coefficient on the inverted bifurcation is similar to the influence on the pitchfork bifurcation. The influence of other parameters on the aforementioned bifurcations is also similar as reported earlier in the literature. Using the Newell-Whitehead-Segel equation, the condition for occurrence of Eckhaus and zigzag secondary instabilities is obtained. The domain of appearance of Eckhaus and zigzag instabilities expands due to the presence of the Soret coefficient for positive values. The Soret coefficient with negative values enhances heat transport, while positive values diminish it in comparison with heat transport for the case without the Soret effect. The dual nature of other parameters in influencing heat and mass transport is shown by considering positive and negative values of the Soret coefficient.

Complex dynamics of a discrete predator–prey model with the prey subject to the Allee effect

In this paper, complex dynamics of the discrete predator–prey model with the prey subject to the Allee effect are investigated in detail. Firstly, when the prey intrinsic growth rate is not large, the basins of attraction of the equilibrium points of the single population model are given. Secondly, rigorous results on the existence and stability of the equilibrium points of the model are derived, especially, by analyzing the higher order terms, we obtain that the non-hyperbolic extinction equilibrium point is locally asymptotically stable. The existences and bifurcation directions for the flip bifurcation, the Neimark–Sacker bifurcation and codimension-two bifurcations with 1:2 resonance are derived by using the center manifold theorem and the bifurcation theory. We derive that the model only exhibits a supercritical flip bifurcation and it is possible for the model to exhibit a supercritical or subcritical Neimark–Sacker bifurcation at the larger positive equilibrium point. Chaos in the sense of Marotto is proved by analytical methods. Finally, numerical simulations including bifurcation diagrams, phase portraits, sensitivity dependence on the initial values, Lyapunov exponents display new and rich dynamical behaviour. The analytic results and numerical simulations demonstrate that the Allee effect plays a very important role for dynamical behaviour.

Extensional channel flow revisited: a dynamical systems perspective

Extensional self-similar flows in a channel are explored numerically for arbitrary stretching-shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching-shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier-Stokes solutions.

Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or nonorientable way, provided the corresponding genericity conditions hold. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or nonorientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $B$, which is characterized by the fact that the codimension-two point gives rise to an additional homoclinic bifurcation, namely, a two-homoclinic orbit. To explain how the global manifolds organize phase space, we consider Sandstede's three-dimensional vector field model, which features inclination and orbit flip bifurcations. We compute global invariant manifolds and their intersection sets with a suitable sphere, by means of continuation of suitable two-point boundary problems, to understand their role as separatrices of basins of attracting periodic orbits. We show representative images in phase space and on the sphere, such that we can identify topological properties of the manifolds in the different regions of parameter space and at the homoclinic bifurcations involved. We find heteroclinic orbits between saddle periodic orbits and equilibria, which give rise to regions of infinitely many heteroclinic orbits. Additional equilibria exist in Sandstede's model, and we compactify phase space to capture how equilibria may emerge from or escape to infinity. We present images of these bifurcation diagrams, where we outline different configurations of equilibria close to homoclinic flip bifurcations of case $B$; furthermore, we characterize the dynamics of Sandstede's model at infinity.

Non-linear dynamics and control of an automotive suspension system based on local and global bifurcation analysis

This paper details the non-linear dynamic behaviours and control of a non-linear semi-active suspension system using a quarter-car model under kinematic excitation by a road surface profile. The results of local and global bifurcation analysis indicate that the hysteretic non-linear characteristics of damping force cause the suspension system to exhibit codimension-two bifurcation, resulting in homoclinic orbits and a pitchfork bifurcation. The complex dynamic behaviour of automotive suspension systems was examined using a bifurcation diagram, phase portraits, a Poincare map, and frequency spectra. We also used Lyapunov exponent to identify the occurrence of chaotic motion and verify our analysis. Finally, a dither signal control was used to convert chaotic behaviours into periodic motion. Simulation results verify the effectiveness of the proposed control method.

Bursting Types and Bifurcation Analysis in the Pre-Bötzinger Complex Respiratory Rhythm Neuron

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast–slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast–slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov–Takens bifurcation point.

Non-linear dynamics and control of an automotive suspension system based on local and global bifurcation analysis

This paper details the non-linear dynamic behaviours and control of a non-linear semi-active suspension system using a quarter-car model under kinematic excitation by a road surface profile. The results of local and global bifurcation analysis indicate that the hysteretic non-linear characteristics of damping force cause the suspension system to exhibit codimension-two bifurcation, resulting in homoclinic orbits and a pitchfork bifurcation. The complex dynamic behaviour of automotive suspension systems was examined using a bifurcation diagram, phase portraits, a Poincaré map, and frequency spectra. We also used Lyapunov exponent to identify the occurrence of chaotic motion and verify our analysis. Finally, a dither signal control was used to convert chaotic behaviours into periodic motion. Simulation results verify the effectiveness of the proposed control method.

Chaos in the Takens–Bogdanov bifurcation with O(2) symmetry

ABSTRACTThe Takens–Bogdanov bifurcation is a codimension-two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and travelling and standing waves in the nonlinear regime and predicts the presence of modulated travelling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated travelling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension-two point.

Codimension-two bifurcations induce hysteresis behavior and multistabilities in delay-coupled Kuramoto oscillators

Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddle-node bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the Matlab Package DDE-Biftool, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, at most four coherent states (two of which are stable) and a stable incoherent state may coexist, and that the system undergoes Neimark-Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.

Robust chaos in a credit cycle model defined by a one-dimensional piecewise smooth map

We consider a family of one-dimensional continuous piecewise smooth maps with monotone increasing and monotone decreasing branches. It is associated with a credit cycle model introduced by Matsuyama, under the assumption of the Cobb-Douglas production function. We offer a detailed analysis of the dynamics of this family. In particular, using the skew tent map as a border collision normal form we obtain the conditions of abrupt transition from an attracting fixed point to an attracting cycle or a chaotic attractor (cyclic chaotic intervals). These conditions allow us to describe the bifurcation structure of the parameter space of the map in a neighborhood of the boundary related to the border collision bifurcation of the fixed point. Particular attention is devoted to codimension-two bifurcation points. Moreover, the described bifurcation structure confirms that the chaotic attractors of the considered map are robust, that is, persistent under parameter perturbations.

Codimension-two bifurcation and multistability coexistence in an inertial two-neuron system with multiple delays

Employing the codimension-two bifurcation analysis, multistability coexistence and the corresponding parameter regions can be exhibited in the nonlinear dynamical systems. In this paper, an inertial two-neuron system with multiple delays is established and the dynamic behaviors are studied. The neural system exhibits one/three equilibria by the pitchfork bifurcation. Analyzing the Hopf bifurcation, we found that multiple delays can induce system dynamics to exhibit the stability switching. Further, the stability regions with the delayed dependent/independent are illustrated, followed which one of the codimension-two bifurcations, i.e., the Hopf–Hopf bifurcation is obtained from the intersection points of the Hopf bifurcation curves. The dynamic behavior near the bifurcation singularity is analyzed. It follows that the original system exhibits the stability coexistence with two different periodic activities. Finally, the other codimension-two bifurcation, i.e., the Pitchfork–Hopf bifurcation is investigated, where the system characteristic equation has a zero and a pair of pure imaginary eigenvalues. Theoretical results and numerical simulations are illustrated to show neural system exhibits the stability coexistence with a pair of nontrivial equilibrium and a periodic activity.

Codimension-Two Bifurcation Analysis in DC Microgrids Under Droop Control

This paper addresses local and global bifurcations that may appear in electrical power systems, such as DC microgrids, which recently has attracted interest from the electrical engineering society. Most sources in these networks are voltage-type and operate in parallel. In such configuration, the basic technique for stabilizing the bus voltage is the so-called droop control. The main contribution of this work is a codimension-two bifurcation analysis of a small DC microgrid considering the droop control gain and the power processed by the load as bifurcation parameters. The codimension-two bifurcation set leads to practical rules for achieving a robust droop control design. Moreover, the bifurcation analysis also offers a better understanding of the dynamics involved in the problem and how to avoid possible instabilities. Simulation results are presented in order to illustrate the bifurcation analysis.

Border Collision and Smooth Bifurcations in a Family of Linear-Power Maps

In this work we describe some properties and bifurcations which occur in a family of linear-power maps typical in Nordmark’ systems. The continuous case has been investigated by many authors since a few years, while the discontinuous case has been considered only recently. In particular, having a vertical asymptote, it gives rise to new kinds of bifurcations. Organizing centers related to codimension-two bifurcation points, due to the intersection of a border collision bifurcation and a smooth fold bifurcation of cycles having a different symbolic sequence are evidenced. It is shown the relevant role played by a codimension-two point existing on any border collision bifurcation curve, and related to the smooth fold bifurcation of cycles with the same symbolic sequence. We recall some of the properties proved up to now, evidencing the rich structure which is still to be understood.