Codimension-two Bifurcation Points Research Articles

Bifurcation mechanisms underlying the nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Inaba et al, 2022) the bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered; these structures can be generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator. It is known that this oscillator can generate canards in the absence of perturbations. In this work, we investigate nested MMOs in a canard-generating driven BVP oscillator near a supercritical Hopf bifurcation point subjected to weak periodic perturbations. The analysis of this system is undertaken using the techniques presented by Kawakami, (1984). These techniques include the use of a shooting algorithm to derive the parameters of the forced oscillators at which bifurcations occur by solving the simultaneous equations that identify the conditions in which the solution of the forced oscillators is periodic and the characteristic multiplier(s) with the largest absolute value of the periodic solution on the Poincaré section crosses the unit circle. We also obtain several two-parameter bifurcation diagrams that describe the system investigated here. One motivation for this study is to demonstrate that the bifurcation structures present in nested MMOs are likely to be a widespread phenomenon. We hypothesize here that nested MMOs represent a stable phenomenon; this is likely to be the case because they appear and disappear consistently and are bound by sequential saddle–node and period-doubling bifurcation curves; this hypothesis is supported by the observation that codimension-two bifurcation points or cusps do not appear in the region close to these bifurcation boundaries.

Pulse-adding of temporal dissipative solitons: resonant homoclinic points and the orbit flip of case B with delay

We numerically investigate the branching of temporally localised, two-pulse solutions from one-pulse periodic solutions with non-oscillating tails in delay differential equations (DDEs) with large delay. Solutions of this type are commonly referred to as temporal dissipative solitons (TDSs) (Yanchuk et al 2019 Phys. Rev. Lett. 123 53901) in applications, and we adopt this term here. We show by means of a prototypical example that—analogous to travelling pulses in reaction–diffusion partial differential equations (Yanagida 1987 J. Differ. Equ. 66 243–62)—the branching of two-pulse TDSs from one-pulse TDSs with non-oscillating tails is organised by codimension-two homoclinic bifurcation points of a real saddle equilibrium (Homburg and Sandstede 2010 Handbook of Dynamical Systems Elsevier) in a corresponding profile equation. We consider a generalisation of Sandstede’s model (Sandstede 1997 J. Dyn. Differ. Equ. 9 269–88) (a prototypical model for studying codimension-two homoclinic bifurcation points in ordinary differential equations) with an additional time-shift parameter, and use Auto07p (Doedel 1981 Congr. Numer. 30 265–84; Doedel and Oldeman 2010 AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations Concordia University) and DDE-BIFTOOL (Sieber et al 2014 arXiv:1406.7144) to compute numerically the unfolding of these bifurcation points in the resulting DDE. We then interpret this model as the profile equation for TDSs in a DDE with large delay by exploiting the reappearance of periodic solutions in DDEs (Yanchuk and Perlikowski 2009 Phys. Rev. E 79 046221). In doing so, we identify both the non-orientable resonant homoclinic bifurcation and the orbit flip bifurcation of case B as organising centres for the existence of two-pulse TDSs in the DDE with large delay. We study the bifurcation curves emanating from these codimension-two points beyond a local neighbourhood in parameter space. In this way, we are able to discuss how folds of homoclinic bifurcations in an extended system bound the existence region of TDSs in the DDE with large delay. We also discuss the relation between a reduced multivalued-map (in the limit of infinite delay) and the existence of TDSs.

Global dynamics and bifurcation for a discontinuous oscillator with irrational nonlinearity

The objective is to carry out thorough analysis of global dynamics and bifurcation for a discontinuous oscillator, showing the transition of dynamics between the piecewise smooth system (with irrational nonlinearity) and the piecewise linear system. The existence of one or three equilibrium sets is proven by the approach of Filippov. Based on non-smooth Lyapunov functions and their set-valued derivatives, asymptotical stability especially finite-time asymptotical stability is shown for the equilibrium sets. In particular, some conditions are established guaranteeing the uniqueness of an equilibrium set and its global finite-time asymptotical stability. Making use of the parametric representation, codimension-one bifurcation curves and a codimension-two bifurcation point are obtained. We give numerical simulations and bifurcation diagram to illustrate the transition of dynamics and verify the results. We further clarify some previous misconceptions in Li et al. (2016).

Soret effect on the onset of viscous dissipation thermal instability for Poiseuille flows in binary mixtures

We investigate numerically the Soret effect on the linear instability properties in convection due to viscous dissipation in a horizontal channel filled with a binary fluid mixture. Two sets of boundary conditions of experimental interest are considered. Both have no-slip boundaries for the velocity and no mass flux through them. The lower boundary is considered adiabatic, while the upper boundary is isothermal for case A and inversely for case B. As no external temperature or concentration difference is imposed on the layer, the cause of thermal instability is the flow rate through the volumetric heating induced by the viscous dissipation and the Soret effect inherent to binary mixtures. It is found that longitudinal rolls (LR) represent the preferred mode for the onset of convection. For case A, both oscillatory and steady-state LR may develop depending on the value of the separation ratio ψ, which represents the ratio between the mass contribution and the temperature contribution to buoyancy forces. The dependence of the instability thresholds on the separation ratio is discussed near and far from the codimension-two bifurcation point. For case B, the basic state remains stable for positive separation ratios, while it loses its stability via a stationary bifurcation with zero wave number for negative values of the separation ratio. The relevance of the theoretical results for the observability of such instability in real systems is discussed. Finally, we suggest a protocol to determine Soret coefficients by using the stability diagrams obtained in the current paper.

Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints

The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

Turing-Hopf patterns in a morphochemical model for electrodeposition with cross-diffusion

This paper focuses on the impact of cross-diffusion for Turing-Hopf instability in a morphochemical model for electrodeposition (DIB) and completes the analysis on the role of cross-diffusion on pattern formation in electrodeposition we recently carried out in [Lacitignola et al. 2018]. We derive and discuss conditions for the destabilization a´la Turing of the Hopf limit cycle in the neighborhood of the codimension-two Turing-Hopf bifurcation point, resulting in oscillatory Turing patterns. Numerical validation of the above theoretical findings is provided both in the case of Weak and Strong Turing-Hopf instability. Experimental evidence of this kind of phenomenology is then discussed also in relation to the charging and discharging process of a battery.

Bifurcation scenario in the two-dimensional laminar flow past a rotating cylinder

Abstract

Codimension-two border collision bifurcation in a two-class growth model with optimal saving and switch in behavior

We consider a two-class growth model with optimal saving and switch in behavior. The dynamics of this model is described by a two-dimensional (2D) discontinuous map. We obtain stability conditions of the border and interior fixed points (known as Solow and Pasinetti equilibria, respectively) and investigate bifurcation structures observed in the parameter space of this map, associated with its attracting cycles and chaotic attractors. In particular, we show that on the x-axis, which is invariant, the map is reduced to a 1D piecewise increasing discontinuous map, and prove the existence of a corresponding period adding bifurcation structure issuing from a codimension-two border collision bifurcation point. Then, we describe how this structure evolves when the related attracting cycles on the x-axis lose their transverse stability via a transcritical bifurcation and the corresponding interior cycles appear. In particular, we show that the observed bifurcation structure, being associated with the 2D discontinuous map, is characterized by multistability, that is impossible in the case of a standard period adding bifurcation structure.

Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system

The Turing and Turing-Hopf bifurcations of a Leslie-Gower type predator-prey system with ratio-dependent Holling III functional response are investigated in this paper. Complex and interesting patterns induced by the bifurcations are identified theoretically and numerically. First the existence conditions of the Turing instability and the Turing-Hopf bifurcation are established from the theoretical analysis, respectively. Then by employing the technique of weakly nonlinear analysis, amplitude equations generated near the Turing instability critical value are derived. Various spatiotemporal patterns, such as homogeneous stationary state patterns, hexagonal patterns, coexisting patterns, stripe patterns, and their stability are determined via analyzing the obtained amplitude equations. Numerical simulations are presented to illustrate the theoretical analysis. Especially, the analogous-spiral and symmetrical wave patterns can be found near the codimension-two Turing-Hopf bifurcation point. A security center of the prey species can be found as well. These spatiotemporal patterns are explained from the perspective of the predators and prey species.

Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

Two scenarios for the onset and suppression of collective oscillations in heterogeneous populations of active rotators.

We consider the macroscopic regimes and the scenarios for the onset and the suppression of collective oscillations in a heterogeneous population of active rotators composed of excitable or oscillatory elements. We analyze the system in the continuum limit within the framework of Ott-Antonsen reduction method, determining the states with a constant mean field and their stability boundaries in terms of the characteristics of the rotators' frequency distribution. The system is established to display three macroscopic regimes, namely the homogeneous stationary state, where all the units lie at the resting state, the global oscillatory state, characterized by the partially synchronized local oscillations, and the heterogeneous stationary state, which includes a mixture of resting and asynchronously oscillating units. The transitions between the characteristic domains are found to involve a complex bifurcation structure, organized around three codimension-two bifurcation points: a Bogdanov-Takens point, a cusp point, and a fold-homoclinic point. Apart from the monostable domains, our study also reveals two domains admitting bistable behavior, manifested as coexistence between the two stationary solutions or between a stationary and a periodic solution. It is shown that the collective mode may emerge via two generic scenarios, guided by a saddle-node of infinite period or the Hopf bifurcation, such that the transition from the homogeneous to the heterogeneous stationary state under increasing diversity may follow the classical paradigm, but may also be hysteretic. We demonstrate that the basic bifurcation structure holds qualitatively in the presence of small noise or small coupling delay, with the boundaries of the characteristic domains shifted compared to the noiseless and the delay-free case.

Memory effects on binary choices with impulsive agents: Bistability and a new BCB structure.

After the seminal works by Schelling, several authors have considered models representing binary choices by different kinds of agents or groups of people. The role of the memory in these models is still an open research argument, on which scholars are investigating. The dynamics of binary choices with impulsive agents has been represented, in the recent literature, by a one-dimensional piecewise smooth map. Following a similar way of modeling, we assume a memory effect which leads the next output to depend on the present and the last state. This results in a two-dimensional piecewise smooth map with a limiting case given by a piecewise linear discontinuous map, whose dynamics and bifurcations are investigated. The map has a particular structure, leading to trajectories belonging only to a pair of straight lines. The system can have, in general, only attracting cycles, but the related periods and periodicity regions are organized in a complex structure of the parameter space. We show that the period adding structure, characteristic for the one-dimensional case, also persists in the two-dimensional one. The considered cycles have a symbolic sequence which is obtained by the concatenation of the symbolic sequences of cycles, which play the role of basic cycles in the bifurcation structure. Moreover, differently from the one-dimensional case, the coexistence of two attracting cycles is now possible. The bistability regions in the parameter space are investigated, evidencing the role of different kinds of codimension-two bifurcation points, as well as in the phase space and the related basins of attraction are described.

Spike-adding structure in fold/hom bursters

Square-wave or fold/hom bursting is typical of many excitable dynamical systems, such as pancreatic or other endocrine cells. Besides, it is also found in a great variety of fast-slow systems coming from other neural models, chemical reactions, laser dynamics, and so on. We focus on the spike-adding process and its connection with the homoclinic structure of the system. The creation of new fast spikes on a bursting neuron is an important phenomenon as it increases the duty cycle of the neuron. Here we mainly work with the Hindmarsh-Rose neuron model, a prototype of fold/hom bursting, but also with the pancreatic β-cell model, where, as already known from the literature, homoclinic bifurcations play an important role in bursting dynamics. Based on several numerical simulations, we present a theoretical scheme that provides a complete scenario of bifurcations involved in the spike-adding process and their connection with the homoclinic bifurcations on the parametric space. The global scheme explains the different phenomena of the spike-adding processes presented in literature (continuous and chaotic processes after Terman analysis) and moreover, it also indicates where each kind of spike-adding process occurs. Different elements are involved in the theoretical scheme, such as homoclinic isolas, canard orbits, inclination and orbit flip codimension-two bifurcation points and several pencils of period doubling and fold bifurcations, all of them illustrated with different numerical techniques. Some of these bifurcations needed in the process may be not visible on some numerical simulations because the organizing points are in different parametric planes due to the high dimension of the whole parameter space, but their effects are present. Therefore, we introduce a mechanism of the spike-adding process in fold/hom bursters in the whole space of parameters, even if apparently no role is played by the “far-away” homoclinic bifurcations. This fact is illustrated showing how the theoretical scheme provides a theoretical explanation to the different interspike-interval bifurcation diagrams (IBD) that have appeared in the literature for different models.

Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing–Hopf bifurcation point

Spatiotemporal dynamics of a ratio-dependent predator-prey model with time delay and the Neumann boundary condition are considered. The existence of the codimension-two Turing–Hopf bifurcation point is proved, so that both the Turing and Hopf bifurcations will occur simultaneously. Then, through the global Hopf bifurcation results, the global Hopf bifurcation of the ratio-dependent predator-prey model is investigated when diffusion is absent. Further, to identify the spatiotemporal dynamics during the Turing–Hopf bifurcation, the method of the multiple time scale analysis is employed to derive the amplitude equations near the codimension-two bifurcation point, instead of employing the center manifold theory and the normal form reduction. By analyzing the amplitude equations, it is found that the original ratio-dependent predator-prey model may exhibit the spatial, temporal or the spatiotemporal patterns, such as the nonconstant steady state, spatially homogeneous periodic solutions as well as the spatially inhomogeneous periodic solutions. Numerical simulations are carried out to validate the theoretical results.

Families of reversing and non-reversing Taylor vortex flows between two co-oscillating cylinders with different amplitudes

This paper deals with the centrifugal instability of time-modulated Taylor-Couette flow for the case in which the inner and outer cylinders are co-oscillating around zero mean with the angular velocities Ωin = Ω0 cos(ωt) and Ωout = εΩ0 cos(ωt), respectively (Ω0, ω, and ε denote, respectively, the amplitude, the frequency of the modulated rotation, and the amplitudes ratio). The small-gap equations for the stability of this flow with respect to axisymmetric disturbances are derived and solved on the basis of Floquet theory. We recover in the case ε = 0 where the outer cylinder is stationary while the inner is modulated the two well-known reversing and non-reversing Taylor vortex flows. Attention is focused on the evolution of these time-dependent flows when one allows the oscillation of the outer cylinder. It turns out that an increase in the parameter ε leads to the discovery of families of reversing and non-reversing flows and other interesting bifurcation phenomena including codimension-two bifurcation points. In addition, a proper tuning of this parameter ε provides a control of the onset of instability as well as the nature of the primary bifurcation. Moreover, it is shown that when ε > 1, the instability is suppressed in low frequencies and the flow is always stable in good agreement to what is obtained by a quasi-steady approach where transient instability is detected. This latter is attributed to the fluid inertia taking place when the cylinders are reversing their rotation’s direction. However, no effect of the parameter ε is observed in high frequencies where the instability develops in thin boundary Stokes layers near the oscillating cylinders.

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

A general method to calculate multi-parameter bifurcation diagram in the parameter space is designed based on top Lyapunov exponent and Floquet multiplier to study the effect of different combinations of system parameters on the system’s dynamics. Bifurcation and chaos of the forced and damped Duffing system in two-parameter plane are investigated by using the method designed in this work. The correlation and matching laws of the Duffing system between dynamic performance and system parameters are analyzed. The effect of different types of bifurcation curves on the bifurcating of coexisting attractors is investigated according to basins of attraction, bifurcation diagrams, top Lyapunov exponent spectrums, phase portraits, Poincare maps, and Floquet multipliers. The evolution of various bifurcation curves and codimension-two bifurcation in the parametric plane is studied as well. Coexisting attractors are found in the parameter plane. The results indicate that the different bifurcating curves are selective for the bifurcation of coexisting attractors. Both the pitchfork bifurcation curve and the period-doubling bifurcation curve just change the stability of some of the coexisting attractors, but have no effect on the stability of the other part of the attractors. The saddle-node bifurcation curve has an effect on the stability of all the coexisting attractors. A series of period-doubling bifurcation curves and codimension-two bifurcation points lead to chaos existence region in two-parameter plane. The special evolution of bifurcation points and bifurcation curves in two-parameter plane with the change of the system parameter is observed. The codimension-two bifurcation points and bifurcation curves play an important role in understanding nonlinear dynamics of the system in the parametric plane. The work in this study emphasizes the importance of the different combinations of system parameters on the system dynamics.

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.

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.

Border collision and fold bifurcations in a family of one-dimensional discontinuous piecewise smooth maps: unbounded chaotic sets

In this work we consider a class of generalized piecewise smooth maps, proposed in the study of engineering models. It is a class of one-dimensional discontinuous maps, with a linear branch and a nonlinear one, characterized by a power function with a term and a vertical asymptote. The bifurcation structures occurring in the family of maps are classified according to the invertibility or non-invertibility of the map, depending on the parameters characterizing the two branches. When the map is non-invertible we prove the persistence of chaos. In particular, the existence of robust unbounded chaotic attractors. The parameter space is characterized by intermingled regions of attracting cycles born by smooth fold bifurcations, issuing from codimension-two bifurcation points. The main result is related to the description of the relationship between two types of bifurcations, smooth fold bifurcations and border collision bifurcations (BCBs). We describe the particular role of codimension-two bifurcation points associated with these bifurcations related to cycles with the same symbolic sequences. We show that they exist related to the border collision of any admissible cycle. We prove that each BCB, each fold bifurcation and each homoclinic bifurcation is a limit set of infinite families of other BCBs. We prove that in the considered range all the unstable cycles are always homoclinic, and that an unbounded chaotic set always exists, either in an invariant set of zero measure or of full measure.

Nonlinear pattern selection and heat transfer in thermal convection of a viscoelastic fluid saturating a porous medium

The linear and nonlinear dynamics of convection in porous media heated from below and saturated by a viscoelastic fluid are examined. Two geometrical configurations are considered. For infinite layer, it is found that stationary structures (SS), traveling waves (TW) or standing waves (SW) can be stable at the onset of convection, depending on the viscoelastic properties. For weakly viscoelastic fluids, we find that SS are the preferred convective pattern. Otherwise, diluted polymers promote TW while concentrated ones favors SW. The heat transfer associated to the three types of convective patterns are evaluated and compared. For a square box, an amplitude equation is derived in the vicinity of the codimension-two bifurcation point where both the stationary and oscillatory instabilities occur simultaneously. The dynamics associated with the interaction between the two kind of the instability is studied. In particular when oscillatory convection appears first, we found that a secondary nonlinear transition to stationary convection may occur. This prediction is in accordance with the few existing simulation results. Possible connections between experiments and our findings are discussed.