Singularities and Groups in Bifurcation Theory II
Chapters (12)
The basic theme of this volume is that the symmetries of bifurcating systems impose strong restrictions on the form of their solutions and the way in which the bifurcation may take place. There are two major subthemes, which we might term “geometric” and “algebraic.” These lead us to introduce two pieces of mathematical machinery: group representation theory and equivariant singularity theory. The aim of this chapter is to describe, in a fairly concrete fashion, the requisite mathematical background. In this manner we hope to make the methods accessible to a wide audience.
In this chapter we begin to study the structure of bifurcations of steady-state solutions to systems of ODEs
$$ \frac{{dx}}{{dt}} + g(x,\lambda ) = 0 $$ (0.1) where g: ℝn × ℝ → ℝncommutes with the action of a compact Lie group Γ on V = ℝn. Steady-state solutions satisfy dx/dt = 0; that is, $$ g(x,\lambda ) = 0. $$ (0.2) We focus here on the symmetries that a solution x may possess and in particular define some simple “geometric” notions that will prove to be of central importance.
This Case Study focuses on Rayleigh-Bénard convection. The term convection refers to fluid motion caused by the interaction of temperature gradients with a gravitational field; motion occurs because hotter fluid is less dense and therefore tends to rise. In this Case Study we consider only carefully controlled laboratory experiments in which a horizontal layer of fluid is heated from below and the ensuing motion is observed. Of course, such experiments are intended to shed light on more dramatic geophysical occurrences of convection, such as in the atmosphere and in the interior of the earth (plate tectonics). See Koschmieder [1974], Schluter et al. [1965], Sattinger [1978].
From the geometry of equivariant bifurcation problems we move on to their algebra, that is, to singularity theory. Our aim in the next two chapters is to develop Γ-equivariant generalizations of the ideas introduced in Chapters II and III. In particular, in this chapter we develop machinery to solve the recognition problem for Γ-equivariant bifurcation problems. In the next chapter we adapt unfolding theory to the equivariant setting. We also give proofs of the main theorems. When specialized to Γ = 1 these will provide the promised proof of the Unfolding Theorem III, 2.3.
Unfolding theory is the study of parametrized families of perturbations of a given germ. In the symmetric setting, when a group Γis acting, we consider only Γ-equivalent perturbations. There is a general theory of Γ-unfoldings, analogous to unfoldings in the nonsymmetric case (Volume I, Chapter III). The heart of the singularity theory approach to bifurcations with symmetry is the equivariant unfolding theorem, which asserts that every Γ-equivariant mapping with finite Γ-codimension has a universal Γ-unfolding and gives a computable test for universality.
In this case study we analyze the Rivlin cube, a bifurcation problem introduced in Chapter XI. Our purpose there was to illustrate the phenomenon of spontaneous symmetry-breaking and to describe the kinds of results that can be obtained by a singularity-theoretic analysis. This case study has a different aim: to present complete calculations supporting the singularity theory analysis of a specific bifurcation problem. The Rivlin cube is an ideal example since the calculations are tractable. This case study has been written so as to be independent of Chapter XI.
Until now the theory developed has applied largely to steady-state bifurcation, the exception being the study of degenerate Hopf bifurcation without symmetry in Chapter VIII. There a dynamic phenomenon—the occurrence of periodic trajectories—was reduced to a problem in singularity theory by applying the Liapunov-Schmidt procedure. In the remainder of this volume we will show that this is a far-reaching idea and that dynamic phenomena in many different contexts can be studied by similar methods.
The object of this chapter is to study Hopf bifurcation with O(2) symmetry in some depth, including a formal analysis—that is, assuming Birkhoff normal form—of nonlinear degeneracies. The most important case, to which most others reduce, is the standard action of O(2) on ℝ2. Since this representation is absolutely irreducible the corresponding Hopf bifurcation occurs on ℝ2 ⊕ ℝ2. We repeat the calculations of XVI, §7(c), in a more convenient coordinate system and in greater detail. In §1 we find that there are two maximal isotropy subgroups, corresponding to standing and rotating waves, as in the example of a circular hosepipe. We also give a brief discussion of nonstandard actions of O(2), for which the standing and rotating waves acquire extra spatial symmetry. In §2 we derive the generators for the invariants and equivariants of O(2) × S1 acting on ℝ2 ⊕ ℝ2. In §3 we apply these results to analyze the branching directions of these solutions in terms of the Taylor expansion of the vector field.
In this chapter we give three illustrations of the general theory of symmetric Hopf bifurcation developed in Chapter XVI. We study systems with dihedral group symmetry Dn
, systems with O(3) symmetry (corresponding to any irreducible representation), and systems with the symmetry T2 +̇ D6 of the hexagonal lattice. For Dn
and T2 +̇ D6 we consider the stability of bifurcating branches. These examples illustrate several features of specific applications that have not yet appeared in our discussions, and they show the different levels at which the methods can be used.
In this chapter we introduce the idea of mode interaction. In order to do this, we review some by now elementary material concerning steady-state and Hopf bifurcation without symmetry.
In the final chapter of this volume we extend the ideas of the previous chapter to the O(2)-symmetric case by studying mode interactions in O(2)-symmetric systems.
The Taylor-Couette apparatus consists of a fluid contained between two coaxial, independently rotating cylinders. The experiments involve setting the speeds of rotation of the inner and the outer cylinders and observing the nature of the fluid flow between them. What astonishes observers of the experiment is the beautiful patterns that develop in the fluid flow. In this respect both theoreticians and experimentalists are united in an attempt to explain how these patterns form and why there are so many different types. Since the Taylor-Couette experiment is one of the simplest fluid flow experiments, it is a natural place to try to connect theory with experiment.
... In this work, we introduce the crease flow (in Eq. (2) below) in order to study the development of the caustics in the structure of the crease sets and the associated continuous change of these singularities upon parameter variation. This approach is from a bifurcation theory point of view using methods of the singularity theory of functions [6][7][8]. In our sense, fronts will be defined more generally than in previous references as bifurcation sets corresponding to fixed points of versal unfoldings (i.e., versal families or universal deformations), cf. ...
... Since a bifurcation problem g in n state variables, has codimension at least n 2 − 1 (cf. [6]), we find that the crease flow has codimension at least 3, instead of codimension at least 8 for the geodesic flow (where n ≥ 3). Therefore studying the crease submanifolds and their singularities using the geodesic flow is equivalent to expecting these singularities only in a geodesic flow problem with at least eight parameters. ...
... We note that these conditions for non-hyperbolicity, transversality, and nondegeneracy (together with condition (6) below) are very important in the study of singularities because they are responsible for putting any such system in its possible normal forms, and also ensure the typicality of the solutions cf. [1,6,7,[11][12][13]. They also imply, for instance, that the structures will in general be non-smooth because, even thought the variables in (3) are smooth and smoothly related, they guarantee that the variables x, y cannot be expressed as smooth functions of the λ when solving g(x, y, λ) = 0, with g given in (3) (the function g may in fact be infinitely differentiable). ...
The crease flow, replacing the Hamiltonian system used for the evolution of crease sets on black hole horizons , is introduced and its bifurcation properties for null hypersurfaces are discussed. We state the conditions of non-degeneracy and typicality for the crease submanifolds, and find their normal forms and versal unfoldings (codimension 3). The allowed boundary singularities are thus prescribed by the Arnold-Kazaryan-Shcherbak theorem for 3-parameter versal families, and hence identified as swallowtails and Whitney umbrellas of particular kinds. We further present the bifurcation diagrams describing crease evolution at the crossings of the bifurcation sets and elsewhere, and a typical example is studied. Some remarks on the connection of these results to the crease evolution on black hole horizons are also given.
... In the absence of symmetries, time-periodically forced systems, whose basic states are usually described by limit cycles, can lose their stability via a saddle-node bifurcation (synchronous) corresponding to a single real eigenvalue of the Floquet multiplier (based on Poincaré map) crossing the unit circle through +1. This bifurcation becomes a pitchfork in the presence of a spatial 2 symmetry while it becomes a pitchfork of revolution in the presence of the 2 symmetry group [1,2]. A period doubling bifurcation is another possibility by which this systems, common behaviors are observed notably the inhibition of the subharmonic responses due to the spatio-temporal symmetry, i.e. 2 × (2). ...
... This bifurcation becomes a pitchfork in the presence of a spatial 2 symmetry while it becomes a pitchfork of revolution in the presence of the 2 symmetry group [1,2]. A period doubling bifurcation is another possibility by which this systems, common behaviors are observed notably the inhibition of the subharmonic responses due to the spatio-temporal symmetry, i.e. 2 × (2). The third generic way in which time-periodically forced systems become unstable is via a Neimark-Sacker bifurcation where a pair of complex-conjugate Floquet multipliers cross the unit circle. ...
... The components of the two operators * and * are given in the Appendix A. The equations governing the dynamics of the considered flow configuration (10) and their boundary conditions (11) are invariant with respect to the symmetry group (2) × (2). This one represents the spatial symmetries of the system resulting from the arbitrary rotation , the reflection around any height acting as → − and the translation along the axis. ...
The effects of harmonically co-oscillating the inner and outer cylinders about zero mean rotation in a Taylor–Couette flow are examined numerically using Floquet theory, for the case where the fluid confined between the cylinders obeys the upper convected Maxwell model. Although stability diagrams and mode competition involved in the system were clearly elucidated recently by Hayani Choujaa et al. (2021) in weakly elastic fluids, attention is focused, in this paper, on the dynamic of the system at higher elasticity with emphasis on the nature of the primary bifurcation. In this framework, we are dealing with pure inertio-elastic parametric resonant instabilities where the elastic and inertial mechanisms are considered of the same order of magnitude. It turns out, on the one hand, that the fluid elasticity gives rise, at the onset of instability, to the appearance of a family of new harmonic modes having different axial wavelengths and breaking the spatio-temporal symmetry of the base flow: invariance in the axial direction generating the O(2)
symmetry group and a half-period-reflection symmetry in the azimuthal direction generating a spatio-temporal Z_2 symmetry group. On the other hand, new quasi-periodic flow emerging in the high frequency limit and other interesting bifurcation phenomena including bi and tricritical states are also among the features induced by the fluid elasticity. Lastly, and in comparison with the Newtonian configuration of this system, the fluid elasticity leads to a total suppression of the non-reversing flow besides emergence of instabilities with lower wavelengths. Such a comparison provides insights into the dynamics of elastic hoop stresses in altering the flow reversal in modulated Taylor–Couette flow.
... In this work, we introduce the crease flow (in Eq. (2) below) in order to study the development of the caustics in the structure of the crease sets and the associated continuous change of these singularities upon parameter variation. This approach is from a bifurcation theory point of view using methods of the singularity theory of functions [6,7,8]. In our sense, fronts will be defined more generally than in previous references as bifurcation sets corresponding to fixed points of versal unfoldings (i.e., versal families or universal deformations), cf. ...
... Since a bifurcation problem g in n state variables, has codimension at least n 2 − 1 (cf. [6]), we find that the crease flow has codimension at least 3, instead of codimension at least 8 for the geodesic flow (where n ≥ 3). Therefore studying the crease submanifolds and their singularities using the geodesic flow is equivalent to expecting these singularities only in a geodesic flow problem with at least eight parameters. ...
... We note that these conditions for non-hyperbolicity, transversality, and nondegeneracy (together with condition (6) below) are very important in the study of singularities because they are responsible for putting any such system in its possible normal forms, and also ensure the typicality of the solutions cf. [6,7,11,12,13,1]. They also imply, for instance, that the structures will in general be non-smooth because, even thought the variables in (3) are smooth and smoothly related, they guarantee that the variables ...
The crease flow, replacing the Hamiltonian system used for the evolution of crease sets on black hole horizons, is introduced and its bifurcation properties for null hypersurfaces are discussed. We state the conditions of nondegeneracy and typicality for the crease submanifolds, and find their normal forms and versal unfoldings (codimension 3). The allowed boundary singularities are thus prescribed by the Arnold-Kazaryan-Shcherbak theorem for 3-parameter versal families, and hence identified as swallowtails and Whitney umbrellas of particular kinds. We further present the bifurcation diagrams describing crease evolution at the crossings of the bifurcation sets and elsewhere, and a typical example is studied. Some remarks on the connection of these results to the crease evolution on black hole horizons are also given.
... Nonlinear dynamics are distinguished from linear dynamics in that they exhibit bifurcations and can be studied through bifurcation theory (52)(53)(54). A (local) bifurcation is a change in the number and/or stability of equilibrium solutions of a nonlinear dynamical process as a (bifurcation) parameter varies across a critical value. ...
... The results extend beyond the example of Figure 2 to an arbitrarily large number of agents and options, mixed-sign networks, and oscillations. Figure 3 illustrates the intimate connection between fast and flexible decision-making and bifurcation theory (52)(53)(54). The organizing bifurcation of fast and flexible decision-making between two options is the pitchfork bifurcation. ...
... Because the coupled opinion-attention dynamics are monotone dynamics on quadrants x > 0 and x < 0, there are no dynamical behaviors other than convergence to an equilibrium (67). For b ̸ = 0, the bifurcation behavior is predicted by unfolding theory (53). The input-output behavior is as in Figure 3c, characterized by hysteresis between neutrality and decision and the existence of opinion cascades for fast, strong opinion formation. ...
A multiagent system should be capable of fast and flexible decision-making to successfully manage the uncertainty, variability, and dynamic change encountered when operating in the real world. Decision-making is fast if it breaks indecision as quickly as indecision becomes costly. This requires fast divergence away from indecision in addition to fast convergence to a decision. Decision-making is flexible if it adapts to signals important to successful operation, even if they are weak or rare. This requires tunable sensitivity to input for modulating regimes in which the system is ultrasensitive and in which it is robust. Nonlinearity and feedback in the decision-making process are necessary to meeting these requirements. This article reviews theoretical principles, analytical results, related literature, and applications of decentralized nonlinear opinion dynamics that enable fast and flexible decision-making among multiple options for multiagent systems interconnected by communication and belief system networks. The theory and tools provide a principled and systematic means for designing and analyzing decision-making in systems ranging from robot teams to social networks.
Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 7 is May 2024. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
... Notations from group theory [51] define S M as a group with symmetry M, that is to say a set that is invariant under any permutation of its M elements. In the case of a CPVA, S M refers to a solution for which a subgroup of M absorbers have the exact same motion, so that any permutation of these M absorbers results in the same response. ...
... is the Schur complement of U in J [64]. det(U) can be easily computed using properties of block-circulant matrices [51], so the difficulty is to compute det(G), which involves U −1 . ...
... where G 1 and G 2 are 2 × 2 blocks. Finally, the determinant of the Jacobian can be computed using results from [51]. It is given by ...
Centrifugal pendulum vibration absorbers (CPVAs) are passive devices used to reduce torsional vibrations in rotating machines. Previous works showed that a CPVA configuration with two pendulums oscillating in phase opposition and at half the excitation frequency is efficient in reducing the rotor’s vibrations. This paper deals with a new generation of CPVAs, in which the pendulums admit a rotational motion relative to the rotor in addition to the traditional translational motion. The aim of this study is to assess the dynamic stability of a particular subharmonic solution of CPVAs composed of several pairs of pendulum. To do so, a new method based on an analytical perturbation technique is proposed. It leads to more general conclusions than previous studies as the results are derived for CPVAs with any even number of rocking pendulums. The validity of the analytical model is confirmed through a comparison with numerical resolutions of the system’s dynamics, and new design guidelines are proposed.
... Nodes that are part of the same orbit (i.e., in the same cluster) have synchronized dynamics x γ(i) ≡ x i for any γ ∈ Σ (see [61,Thm III.2]). Isotropy subgroups that are conjugate in Γ lead to cluster states with identical existence and stability criteria [8,59]. The remaining possible cluster states arise from the specific choice of Laplacian coupling. ...
... The presence of symmetry in a system imposes constraints on the form of the Jacobian matrix, which one can use to greatly simplify stability calculations. For periodic cluster states that one predicts from symmetry, there are well-established methods for stability calculations in symmetric systems to blockdiagonalize the Jacobian and generalize the MSF formalism [58,59]. Sorrentino et al. [146] extended these techniques to Laplacian cluster states. ...
... One can decompose the action of Σ on the phase space R N m into a collection of irreducible representations of Σ (i.e., the most trivial invariant subspaces under the action of Σ). Some of these subspaces are isomorphic to each other; we combine these subspaces to obtain "isotypic components" [58,59]. Each isotypic component is invariant under the variational equation (6.12), so one can determine the Floquet exponents by considering the restriction of this equation to each isotypic component. ...
There is enormous interest -- both mathematically and in diverse applications -- in understanding the dynamics of coupled oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled oscillator networks. We discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. We then focus in particular on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well-developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use \textit{saltation operators}, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electro-mechanical oscillators, and cooperation in cattle herds.
... Synchrony-breaking is an approach to classify the bifurcating steady-state solutions of dynamical systems on homogeneous networks. It is analogous to the successful approach of spontaneous symmetry-breaking [24,Golubitsky and Stewart], [27,Golubitsky et al.]. In its simplest form, symmetry-breaking addresses the following question. ...
... Given a symmetry group Γ acting on R n , a stable Γ-symmetric equilibrium x 0 , and a parameter λ, what are the possible symmetries of steady states that bifurcate from x 0 when x 0 (λ) loses stability at λ = λ 0 ? A technique that partially answers this question is the Equivariant Branching Lemma (see [27,XIII,Theorem 3.3]). This result was first observed by [51,Vanderbauwhede] and [6,Cicogna]. ...
... The undecided solution and its linearization. It is well known that equivariant maps leave fixed-point subspaces invariant [27,Golubitsky et al.]. Since admissible maps G are Γ-equivariant, it follows that the 1-dimensional undecided (fully synchronous) subspace V s = Fix(Γ) is flow-invariant; that is, invariant under the flow of any admissible ODE. ...
... 10,11 For such systems, group theory and equivariant dynamical system theory can provide a useful tool for exploring and understanding the dynamics. 12,13 For example, the symmetry properties of a network of oscillators can be used to facilitate model reductions. 14 Knowledge of the symmetries can also be used to discover possible cluster synchronization patterns. ...
... , N − 1 labels the specific primary cluster state. Following Ref. 13, these states may be characterized as discrete rotating waves when m is coprime to N, discrete standing waves when m = 0, and a discrete alternating wave when m = N/2. In our case of N = 4, m = 0 corresponds to the in-phase state [cf. ...
... Then, very shortly afterward, further pitchfork bifurcations take place, at which the ϕ j+2 − ϕ j = ϕ j − ϕ j−2 symmetry is lost, resulting in four branches with Z 1 symmetry. This is in agreement with the equivariant branching lemma, 13 which characterizes the isotropy subgroups of bifurcating solution branches. Only with both sets of symmetries gone, can all oscillators approach each other to form the compressed reverse splay state. ...
Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists.
... In [18], Golubitsky and Schaeffer also use the term "distinguished parameter" to refer to a specific parameter whose variation can be seen as typical or natural in a given model. They work with scalar equations of the form ...
... For the slow-fast models that we consider in this paper, the choice of distinguished parameter is indeed a "natural" one: the small parameter ε that controls the behavior of the slow variable. Note that the theory of singularities, as introduced in [18], allows to understand, for instance, the appearance of isola of equilibria in bifurcation diagrams (see [4]). The notion of geometric bifurcation incorporates a rather particular concept of codimension. ...
... These two cases are also discussed in [32,31]. The isola-type and saddle-type geometric bifurcations correspond to the codimension-one singularities named isola-center and simple bifurcation, respectively, in [18]. Nevertheless, as we already mention in the introduction, there exist singularities which do not match any geometric bifurcation, the hysteresis point, a codimension-one singularity in [18], is one example. ...
Inspecting a $p$-dimensional parameter space by means of $(p-1)$-dimensional slices, changes can be detected that are only determined by the geometry of the manifolds that compose the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, particularly, the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are discussed in the context of two models of neuron activity: the Hindmarsh-Rose and the FitzHugh-Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter.
... These two cases are also discussed by Wieczorek and Krauskopf 15,16 . The isola-type and saddle-type geometric bifurcations correspond to the codimension-one singularities named isola-center and simple bifurcation, respectively, by Golubitsky and Schaeffer 19 . Nevertheless, as we already mention in the introduction, there exist singularities which do not match any geometric bifurcation, the hysteresis point 19 , a codimension-one singularity, is one example. ...
... The isola-type and saddle-type geometric bifurcations correspond to the codimension-one singularities named isola-center and simple bifurcation, respectively, by Golubitsky and Schaeffer 19 . Nevertheless, as we already mention in the introduction, there exist singularities which do not match any geometric bifurcation, the hysteresis point 19 , a codimension-one singularity, is one example. Any two isola-type geometric bifurcations are equivalent in the sense that, using appropriate coordinates, their respective height functions can be written in a unique canonical form, either with a maximum or a minimum. ...
By studying a nonlinear model by inspecting a p-dimensional parameter space through (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\end{document}-dimensional cuts, one can detect changes that are only determined by the geometry of the manifolds that make up the bifurcation set. We refer to these changes as geometric bifurcations. They can be understood within the framework of the theory of singularities for differentiable mappings and, in particular, of the Morse Theory. Working with a three-dimensional parameter space, geometric bifurcations are illustrated in two models of neuron activity: the Hindmarsh–Rose and the FitzHugh–Nagumo systems. Both are fast-slow systems with a small parameter that controls the time scale of a slow variable. Geometric bifurcations are observed on slices corresponding to fixed values of this distinguished small parameter, but they should be of interest to anyone studying bifurcation diagrams in the context of nonlinear phenomena.
... Bifurcation theory soon emerged as a fruitful area of application of group theory in the context of structural mechanics. Sattinger [8] proposed a group-theoretic bifurcation theory, while specific bifurcation phenomena (such as singularities and symmetry-breaking) were studied in depth by a number of investigators [9][10][11]. ...
... Subspace S (5) -Based on operators of Equation (10) Subspace S (5,1) ...
... Moreover, breaking the symmetry can result in the unfolding of pitchfork bifurcation, so-called the imperfect pitchfork bifurcation (as shown in Fig. 3(b)). The extensions to encompass broader cases, as well as the results for non-symmetric cases, will depend on the sophisticated application of singular bifurcation theory [29], and we leave them to future study. ...
... The immediate limitation of this work is that the results are only theoretically valid for minimal models, and our findings are empirically instantiated for larger-size and structured networks. While formal analysis of larger networks is more challenging, several tools may be useful to consider in future work, such as the singularity theory and the equivariant bifurcation theory [29]. Another important step will be to connect this theory to the analysis and design of recurrent neural networks and learning rules in practical artificial intelligence tasks, especially ones such as meta-learning that involve adaptation of plasticity rules [32]. ...
In this paper, we study recurrent neural networks in the presence of pairwise learning rules. We are specifically interested in how the attractor landscapes of such networks become altered as a function of the strength and nature (Hebbian vs. anti-Hebbian) of learning, which may have a bearing on the ability of such rules to mediate large-scale optimization problems. Through formal analysis, we show that a transition from Hebbian to anti-Hebbian learning brings about a pitchfork bifurcation that destroys convexity in the network attractor landscape. In larger-scale settings, this implies that anti-Hebbian plasticity will bring about multiple stable equilibria, and such effects may be outsized at interconnection or 'choke' points. Furthermore, attractor landscapes are more sensitive to slower learning rates than faster ones. These results provide insight into the types of objective functions that can be encoded via different pairwise plasticity rules.
... In view of these observations, it seems plausible to expect that standard approaches to studying symmetry-induced bifurcations should apply directly in our situation, such as the ones described in [5,10]. Notice, however, that for these approaches to work one needs to study symmetry operators which are acting on the space containing the equilibrium solutions -and in our case this is the Hilbert space X. ...
... Thus, it is natural to find the pitchfork bifurcation point approximation in the form of a truncated series. If we denote the resulting discretization size by N ∈ N, then we consider the orthogonal projection P N : X → X defined via P N u(x) := N k=1 a k cos(kπx) for every u(x) = ∞ k=1 a k cos(kπx) in X , (39) see also (10). An analogous projection Q N can also be defined on the image space Y . ...
... According to [19] (pp. 296-297, Theorems 6.3 and 6.5), the bifurcations of the smallamplitude periodic solution of (15) are completely determined by the zero point of the ...
... We have the following theorem by the result of [19] (p. 383, Theorem 3.1). ...
In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multiple periodic solutions of the system are given. Secondly, the multiple periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric Hopf bifurcation theory. The normal form of the system on the central manifold and the bifurcation direction of the bifurcating periodic solutions are derived. Finally, numerical simulations are attached to demonstrate our theoretical results.
... Some standard methods of analytical and semi-analytical analysis exist that allow one to obtain first or even secondary bifurcations [1]. These methods include Lyapunov-Schmidt reduction [2,3], central manifold reduction [4,5], equivariant unfolding theory [6], etc. Some examples from the area of fluid dynamics are presented below. ...
... Lyapunov-Schmidt reduction was applied to 1D spatial Euler equations in [7] to obtain periodic solutions; the classical results of Youdovich with these approaches were obtained in [8][9][10], as well as other results in the field of fluid dynamics and Navier-Stokes equations [11][12][13]. Equivariant unfolding theory together with the Lyapunov-Schmidt reduction was applied to the Taylor-Couette flow in [6] by reducing O(2) × SO(2) group symmetries in the 1. ...
The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed.
... According to [17] (pp.296-297, Theorem 6.3 and 6.5 ), the bifurcations of small-amplitude periodic solution of (4.9) are comletely determined by the zero point of equation ...
... From the expressions of D and those above, we get the following expressions We have the following theorem by the result of [17] (pp.383, Theorem 3.1). ...
In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multi-periodic solutions of the system are given. Secondly, the multi-periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric bifurcation theory. The normal form of the system on the central manifold and the direction of the bifurcation periodic solution are given. Finally, numerical simulations are used to support our theoretical results.
MSC: 34K18, 35B32
... Lastly, the analytic results presented in the sequel make an explicit reference to the isotropy group of a point (measuring its symmetries relative to the action of S k × S d , Section 3), as well as to several irreducible representations of S d , Section 4; these may be safely ignored at first reading. By the Orbit-stabilizer theorem, the smaller the isotropy group, the larger the orbit, giving additional families of critical points with conjugate isotropy groups, see e.g., [14]. The length of the orbit shall be referred to below as the multiplicity of a given family of critical points. ...
... However, it may still give a local minimum, a local maximum or a saddle point. The isotropy group of C 1 , being large, allows us to use the representation theory of the symmetric group to determine the stability of C 1 (see [14] for a more complete account). The analysis proceeds by considering four irreducible representations (i.e., S d -invariant linear spaces containing no proper S d -invariant subspaces) occurring for the representation of S d : ...
We consider the non-convex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank one terms. Use is made of the rich symmetry structure to derive Puiseux series representations of families of critical points, and so obtain precise analytic estimates on the critical values and the Hessian spectrum. The sharp results make possible an analytic characterization of various geometric obstructions to local optimization methods, revealing in particular a complex array of saddles and local minima which differ by their symmetry, structure and analytic properties. A desirable phenomenon, occurring for all critical points considered, concerns the index of a point, i.e., the number of negative Hessian eigenvalues, increasing with the value of the objective function. Lastly, a Newton polytope argument is used to give a complete enumeration of all critical points of fixed symmetry, and it is shown that contrarily to the set of global minima which remains invariant under different choices of tensor norms, certain families of non-global minima emerge, others disappear.
... Equivariant bifurcation theory guarantees that there exists at least one symmetry-invariant solution branch that emerges out of such an SBP. 45 A further number of symmetry-broken solution branches also emerge out of SBPs depending on the normal form of the bifurcation. ...
Localized phenomena abound in nature and throughout the physical sciences. Some universal mechanisms for localization have been characterized, such as in the snaking bifurcations of localized steady states in pattern-forming partial differential equations. While much of this understanding has been targeted at steady states, recent studies have noted complex dynamical localization phenomena in systems of coupled oscillators. These localized states can come in the form of symmetry-breaking chimera patterns that exhibit coexistence of coherence and incoherence in symmetric networks of coupled oscillators and gap solitons emerging in the bandgap of parametrically driven networks of oscillators. Here, we report detailed numerical continuations of localized time-periodic states in systems of coupled oscillators, while also documenting the numerous bifurcations they give way to. We find novel routes to localization involving bifurcations of heteroclinic cycles in networks of Janus oscillators and strange bifurcation diagrams resembling chaotic tangles in a parametrically driven array of coupled pendula. We highlight the important role of discrete symmetries and the symmetric branch points that emerge in symmetric models.
... The scale-free chaos phase transition in the 3D HCVM is such that all critical curves tend to β = η = 0 for finite number of particles [25]. There are power laws in η that allow the calculation of critical exponents without having to increase N as we did in Section 4. The point β = η = 0 acts as an organizing center of codimension two for the phase transition in a sense that reminds of singularity theory [40]. The critical curves including η = 0 and β c (η; N) issue from the organizing center at finite N and the attractors in the regions between these lines have specific properties (non-chaotic, single-cluster chaos, multicluster chaos, deterministic chaos, etc). ...
Animal motion and flocking are ubiquitous nonequilibrium phenomena that are often studied within active matter. In examples such as insect swarms, macroscopic quantities exhibit power laws with measurable critical exponents and ideas from phase transitions and statistical mechanics have been explored to explain them. The widely used Vicsek model with periodic boundary conditions has an ordering phase transition but the corresponding homogeneous ordered or disordered phases are different from observations of natural swarms. If a harmonic potential (instead of a periodic box) is used to confine particles, then the numerical simulations of the Vicsek model display periodic, quasiperiodic, and chaotic attractors. The latter are scale-free on critical curves that produce power laws and critical exponents. Here, we investigate the scale-free chaos phase transition in two space dimensions. We show that the shape of the chaotic swarm on the critical curve reflects the split between the core and the vapor of insects observed in midge swarms and that the dynamic correlation function collapses only for a finite interval of small scaled times. We explain the algorithms used to calculate the largest Lyapunov exponents, the static and dynamic critical exponents, and compare them to those of the three-dimensional model.
... To gain deeper insight in the quasi-potential landscape changes in the presence of a transient signal, we calculated next the bifurcation diagrams during the subsequent increase/decrease in the signal amplitude. Even a low-amplitude spatial signal (step (i)) introduces an asymmetry to the system and thereby a universal unfolding of the PB [25], such that a marginally asymmetric steady state ( Fig 1E, gray solid lines) replaces the HSS (black solid lines in signal absence). Moreover, for the same parameter values, now also the IHSS (a remnant of the PB that disappeared) is also stable. ...
Author summary
During wound healing or embryonic development, cells in tissues or organs migrate over large distances by sensing local chemical cues. The migration response is based on cell polarization—the formation of a distinct front and back of the cell in the direction of the chemical cues. These cues are however disrupted and have a complex spatial-temporal profile. This suggests that cell polarity must be robustly established in signal direction, but also flexibly adapt when signals change. A large diversity of abstract and biochemically detailed models have been proposed to explain cell polarity, but they cannot fully describe the experimental observations. Here, we argue that cell polarization is a highly dynamic transient process, and must be studied via an explicit time-dependent form. We demonstrate that criticality organization uniquely enables formation of metastable polarized states that can be robustly maintained for a transient period even when the signals are disrupted, but also enable rapid adaptation to temporal or spatial signal changes. Using a combination of bifurcation and quasi-potential landscape analysis, we provide a framework to characterize non-asymptotic transients explicitly, and thereby further emphasize the necessity to change the mathematical formalism when describing biological systems that operate in changing environments.
... This disagrees with Parker and Dimitrov [2022], who conclude that bifurcations in the IB "are only of pitchfork type". To see the reason for this discrepancy, note that they employ the mathematical machinery in [Golubitsky et al., 1988] of bifurcations under symmetry. As pitchfork bifurcations are "common in physical problems that have a symmetry", [Strogatz, 2018, Section 3.4], then detecting only pitchforks by using the above machinery might not come as a surprise (their Theorem 5 and the comments below it imply that they are only able to detect bifurcations of pitchfork type). ...
The Information Bottleneck (IB) is a method of lossy compression. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory as the input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB's intimate relations with RD, sub-optimal solutions can be seen to collide or exchange optimality there. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB's first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, one needs not only to detect IB bifurcations but also to identify their type in order to handle them accordingly. Rather than approaching the optimal IB curve from sub-optimal directions, the latter allows us to follow a solution's trajectory along the optimal curve, under mild assumptions. Thereby, translating an understanding of IB bifurcations into a surprisingly accurate numerical algorithm.
... The decomposition of the domain of the function f into a direct sum of the kernel and its orthogonal complement is known as the Liapunov-Schmidt decomposition [11,38]. In our application, the (infinite dimensional) gradient, although invertible, has unbounded inverse because of the presence of small divisors δ k . ...
We prove the existence of ``pure tone'' nonlinear sound waves of all frequencies. These are smooth, space and time periodic, oscillatory solutions of the $3\times3$ compressible Euler equations in one space dimension. Being perturbations of solutions of a linear wave equation, they provide a rigorous justification for the centuries old theory of Acoustics. In particular, Riemann's celebrated 1860 proof that compressions always form shocks holds for isentropic and barotropic flows, but for generic entropy profiles, shock-free periodic solutions containing nontrivial compressions and rarefactions exist for every wavenumber $k$.
... Research in this area -which is called equivariant dynamics -has been highly active in the last decades and lots of remarkable results have been established. Background and more details on equivariant dynamics can be found, for example, in CHOSSAT and LAUTERBACH [23], FIELD [36], GOLUBITSKY and SCHAEFFER [49], and GOLUBITSKY, STEWART, and SCHAEF-FER [58] with no claim of this list being complete. The symmetries in question, throughout all these results, need to have an underlying structure themselves. ...
This thesis deals with the investigation of dynamical properties – in particular generic synchrony breaking bifurcations – that are inherent to the structure of a semigroup network as well the numerous algebraic structures that are related to these types of networks. Most notably we investigate the interplay between network dynamics and monoid representation theory as induced by the fundamental network construction in terms of hidden symmetry as introduced by RINK and SANDERS.
After providing a brief survey of the field of network dynamics in Part I, we thoroughly introduce the formalism of semigroup networks, the customized dynamical systems theory, and the necessary background from monoid representation theory in Chapters 3 and 4. The remainder of Part II investigates generic synchrony breaking bifurcations and contains three major results. The first is Theorem 5.11, which shows that generic symmetry breaking steady state bifurcations in monoid equivariant dynamics occur along absolutely indecomposable subrepresentations – a natural generalization of the corresponding statement for group equivariant dynamics. Then Theorem 7.12 relates the decomposition of a representation given by a network with high-dimensional internal phase spaces to that induced by the same network with one-dimensional internal phase spaces. This result is used to show that there is a smallest dimension of internal dynamics in which all generic l-parameter bifurcations of a fundamental network can be observed (Theorem 7.24).
In Part III, we employ the machinery that was summarized and further developed in Part II to feedforward networks. We propose a general definition of this structural feature of a network and show that it can equivalently be characterized in different algebraic notions in Theorem 8.35. These are then exploited to fully classify the corresponding monoid representation for any feedforward network and to classify generic synchrony breaking steady state bifurcations with one- or highdimensional internal dynamics.
... When f is the identity map, this equation defines an equivariant dynamical system [72,105,106]. In the general case, Eq. (47) defines dualities in general dynamical systems. ...
Dualities are hidden symmetries that map seemingly unrelated physical systems onto each other. The goal of this work is to systematically construct families of Hamiltonians endowed with a given duality and to provide a universal description of Hamiltonian families near self-dual points. We focus on tight-binding models (also known as coupled-mode theories), which provide an effective description of systems composed of coupled harmonic oscillators across physical domains. We start by considering the general case in which group-theoretical arguments suffice to construct families of Hamiltonians with dualities by combining irreducible representations of the duality operation in parameter space and in operator space. When additional constraints due to system-specific features are present, a purely group-theoretic approach is no longer sufficient. To overcome this complication, we reformulate the existence of a duality as a root-finding problem, which is amenable to standard optimization and numerical continuation algorithms. We illustrate the generality of our method by designing concrete toy models of photonic, mechanical, and thermal metamaterials with dualities.
We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of the system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, exhibits enhanced generalization, sample efficiency, and interpretability, with fewer trainable parameters and computational costs.
Bifurcation theory provides a very general means to classify the local changes in numbers of zeros of vector fields, but not a general means to find where a given bifurcation occurs, at least at higher codimensions. Instead, it turns out, these bifurcations can be found by looking for their underlying catastrophes. Here I show that the concept of underlying catastrophes can be extended to the umbilics. The umbilics are important in opening up qualitatively different forms of bifurcations beyond the ‘corank 1’ catastrophes of folds, cusps, swallowtails, etc. An example is given showing how four zeros of a vector field bifurcating from a single point, may do so either via a 3-parameter swallowtail catastrophe involving equilibria of similar stabilities, or via a 4-parameter umbilic catastrophe involving equilibria of opposing stabilities. This opens an avenue to studying spatiotemporal pattern formation around high codimension bifurcation points, and I conclude with some illustrative examples.
Disorder in parameters appears to influence the collective behavior of complex adaptive networks in ways that might seem unconventional. For instance, heterogeneities may, unexpectedly, lead to enhanced regions of existence of stable synchronization states. This behavior is unexpected because synchronization appears, generically, in symmetric networks with homogeneous components. Related works have, however, misidentified cases where disorder seems to play a critical role in enhancing synchronization, where it is actually not the case. Thus, in order to clarify the role of disorder in adaptive networks, we use normal forms to study, mathematically, when and how the presence of disorder can facilitate the emergence of collective patterns. We employ parameter symmetry breaking to study the interplay between disorder and the underlying bifurcations that determine the conditions for the existence and stability of collective behavior. This work provides a rigorous justification for a certain barycentric condition to be imposed on the heterogeneity of the parameters while studying the synchronization state. Theoretical results are accompanied by numerical simulations, which help clarify incorrect claims of disorder purportedly enhancing synchronization states.
The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.
The collective dynamics of interacting dynamical units on a network crucially depends on the properties of the network structure. Rather than considering large but finite graphs to capture the network, one often resorts to graph limits and the dynamics thereon. We elucidate the symmetry properties of dynamical systems on graph limits—including graphons and graphops—and analyze how the symmetry shapes the dynamics, for example through invariant subspaces. In addition to traditional symmetries, dynamics on graph limits can support generalized noninvertible symmetries. Moreover, as asymmetric networks can have symmetric limits, we note that one can expect to see ghosts of symmetries in the dynamics of large but finite asymmetric networks.
We reexamine the focusing effect crucial to the theorems that predict the emergence of spacetime singularities and various results in the general theory of black holes in general relativity. Our investigation incorporates the fully nonlinear and dispersive nature of the underlying equations. We introduce and thoroughly explore the concept of versal unfolding (topological normal form) within the framework of the Newman–Penrose–Raychaudhuri system, the convergence-vorticity equations (notably the first and third Sachs optical equations), and the Oppenheimer–Snyder equation governing exactly spherical collapse. The findings lead to a novel dynamical depiction of spacetime singularities and black holes, exposing their continuous transformations into new topological configurations guided by the bifurcation diagrams associated with these problems.
Let \(\textit{O}(3)\) denote the group of orthogonal \(3\times 3\) real matrices, and \({\mathcal {M}}\) the 5-dimensional real vector space of all \(3\times 3\) real symmetric matrices with trace zero. Let \(\lambda _1(A)\le \lambda _2(A)\le \lambda _3(A)\) be the eigenvalues of \(A\in {\mathcal {M}}\) and \(\Xi ^-=\{A\in {\mathcal {M}}\mid \lambda _1(A)=\lambda _2(A)\}\). \({\mathcal {M}}\) is an inner product space with the inner product \(\langle A,B\rangle ={\textit{trace}}(AB)\). Let \(G_3({\mathcal {M}})\) be the set of all 3-dimensional subspaces of \({\mathcal {M}}\), a 6-dimensional Grassman manifold. \(\textit{O}(3)\) acts on \({\mathcal {M}}\) on the left by conjugation via inner product preserving linear isomorphisms, which map any 3-dimensional subspace into another 3-dimensional subspace; thus \(G_3({\mathcal {M}})\) also has a left action of \(\textit{O}(3)\). \(G_3({\mathcal {M}})\) becomes a category, an action groupoid, with morphisms \((V,M,W)\in G_3({\mathcal {M}})\times \textit{O}(3)\times G_3({\mathcal {M}})\), where \(W=MVM^T\). Composition of morphisms is \((V_1,N,V_2)\circ (V_0,M,V_1)=(V_0,NM,V_2)\). Let \({\mathcal {C}}\) be a category whose objects (V, S) consist of a real inner product space V and \(S\subset V\), and whose arrows \((V,S)\rightarrow (W,T)\) consist of \(f:V\rightarrow W\), an inner product preserving real linear mapping such that \(f(S)\subset T\). We have the functor
where \(F^-(M):V\rightarrow W:A\mapsto MAM^T\). Suppose further that \(S_3\) denotes the group of permutations of \(\{1,2,3\}\), and \(\rho :S_3\rightarrow \textit{O}(3)\) denotes a group homomorphism which is isomorphic as a group representation to the natural representation of \(S_3\) on \({\mathbb {R}}^3\) (which permutes the coordinates). Let \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) denote the set of all \(V\in G_3({\mathcal {M}})\) whose isotropy subgroup contains \({\mathcal {S}}=\rho (S_3)\) as a subgroup. This paper completely describes the full subcategory \({\mathcal {L}}_{{\mathcal {S}}}\) of \(G_3({\mathcal {M}})\) with object set \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\), as well as the details of the above functor restricted to \({\mathcal {L}}_{{\mathcal {S}}}\). Thus all the members \(V\in {\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) are determined, as well as the smooth manifold structure on \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\); it is embedded as a one-dimensional submanifold of \(G_3({\mathcal {M}})\). The isotropy subgroups of all \(V\in {\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) are computed and all pairs \(V, W\in {\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) which are isomorphic via some \(M\in \textit{O}(3)\) are determined. The sets \(\Xi ^-\cap V\) are all determined, and the functorial mappings on morphism sets are computed. However, \({\mathcal {L}}_{{\mathcal {S}}}\) is not a Lie groupoid. The image of \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) under the functor \(\pi _1F^-\) is the collection of fibres of the smooth manifold \(\coprod _{V\in {\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})}V\), which is the total space of the canonical vector bundle over the base manifold \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\). The bifurcation points of the family of subsets \(\Xi ^-\cap V\) as V ranges over \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) (within this total space) are seen to be the points of \({\textit{Obj}}({\mathcal {L}}_{{\mathcal {S}}})\) with infinite isotropy subgroups. We also show how this mathematical problem arises naturally from a problem in mathematical chemistry. Hence certain features of numerical calculations of energy eigenvalue intersection patterns of the simple chemical system H3 are rationalized through linearization about the triple intersection point.
Normal Form Theory helps to simplify linear and nonlinear dynamical systems by reducing the number of terms in the equations and by introducing a symmetry in the system. It is demonstrated, how this Normal Form reduction is applied depending on the properties of the Jordan Normal Form for the linearized system.
In this paper, we prove the existence of non-radial solutions to the problem \(-\triangle u= f(x,u)\), \(u|_{\partial \Omega }=0\) on the unit ball \(\Omega :=\{x\in {\mathbb {R}}^3: \Vert x\Vert <1\}\) with \(u(x)\in {\mathbb {R}}^s\), where f is a sub-linear continuous function, differentiable with respect to u at zero and satisfying \(f(gx,u) = f(x,u)\) for all \(g\in O(3)\), \( f(x,-u)=- f(x,u)\). We investigate symmetric properties of the corresponding non-radial solutions. The abstract result is supported by a numerical example.
The universal unfolding of a normal form can be employed to reveal the general behaviors of a specific local bifurcation, while the computation of the normal form for high codimensional bifurcation still remains unsolved. This paper focuses on a vector field with codimension-3 triple Hopf bifurcation. Besides 1:1 internal resonance for two frequencies in semi-simple form, two cases are considered, corresponding to internal resonance and noninternal resonance between the first two frequencies and the third frequency, respectively. Based on a combination of center manifold and normal theory, all the coefficients in the normal form and the nonlinear transformation are derived explicitly in terms of the coefficients of the original vector field. Upon the recursive procedure established, a user friendly computer program can be easily developed using a symbolic computation language Maple to compute the coefficients up to an arbitrary order for a specific vector field with triple Hopf bifurcation. Furthermore, universal unfolding of the normal form is obtained, which can be used to display the topological structure in the neighborhood of bifurcation point. It is pointed out that different choices of the remaining terms in the nonlinear transformation may lead to different expressions of the normal form and the unfolding, which are qualitatively equivalent to each other.
The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB’s intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB’s first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB’s optimal tradeoff curve from sub-optimal directions, the latter allows us to follow a solution’s trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm.
Emergent behavior in complex networks can be predicted and analyzed via the mechanism of spontaneous symmetry-breaking bifurcation, in which solutions of related bifurcation problems lose symmetry as some parameters are varied, even though the equations that such solutions satisfy retain the full symmetry of the system. A less common mechanism is that of forced symmetry-breaking, in which either a bifurcation problem has symmetry on both the state variables and the parameters, or one where the equations have less symmetry when a certain parameter is varied. In this manuscript, it is shown that in certain networks with parameter mismatches the governing equations remain unchanged when the group of symmetries acts on both the state variables and the parameter space. Based on this observation we study the existence and stability of collective patterns in symmetric networks with parameters mismatches from the point of view of forced symmetry-breaking bifurcations. Treating the parameters as state variables, we perform center manifold reductions, which allow us to understand how the disorder in parameters affects the bifurcation points as well as the stability properties of the ensuing patterns. Theoretical results are validated with numerical simulations.
Electrical stimulation is an increasingly popular method to terminate epileptic seizures, yet it is not always successful. A potential reason for inconsistent efficacy is that stimuli are applied empirically without considering the underlying dynamical properties of a given seizure. We use a computational model of seizure dynamics to show that different bursting classes have disparate responses to aborting stimulation. This model was previously validated in a large set of human seizures and led to a description of the Taxonomy of Seizure Dynamics and the dynamotype, which is the clinical analog of the bursting class. In the model, the stimulation is realized as an applied input, which successfully aborts the burst when it forces the system from a bursting state to a quiescent state. This transition requires bistability, which is not present in all bursters. We examine how topological and geometric differences in the bistable state affect the probability of termination as the burster progresses from onset to offset. We find that the most significant determining factors are the burster class (dynamotype) and whether the burster has a DC (baseline) shift. Bursters with a baseline shift are far more likely to be terminated due to the necessary structure of their state space. Furthermore, we observe that the probability of termination varies throughout the burster’s duration, is often dependent on the phase when it was applied, and is highly correlated to dynamotype. Our model provides a method to predict the optimal method of termination for each dynamotype. These results lead to the prediction that optimization of ictal aborting stimulation should account for seizure dynamotype, the presence of a DC shift, and the timing of the stimulation.
A rigorous description of period doubling bifurcation of limit cycles in autonomous systems of first order differential equations based on tools of functional analysis and singularity theory is presented. It is an alternative approach which is independent of the theory of discrete-time dynamical systems, especially Poincaré sections. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. Also, stability of solutions is analysed and it is related to particular derivatives of the operator. Our approach is an adjustment of techniques used by Golubitsky and Schaeffer (Singularities and Groups in Bifurcation Theory: Volume 1. Springer, New York, 1985) in the study of Hopf bifurcation and it can be considered as a theoretical background for calculations presented in Kuznetsov et al. (SIAM J. Numer. Anal. 43:1407–1435, 2006). The normal form of a vector field derived in Iooss (J. Differ. Equ. 76:47–76, 1988) is not needed, since a given differential equation is considered as an algebraic equation. The theory used here concerns Fredholm operators, Lyapunov-Schmidt reduction and recognition problem for pitchfork bifurcation.KeywordsLimit cyclePeriod doublingFredholm operatorLyapunov-Schmidt reductionPitchfork bifurcation
The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.
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