Blue Sky Catastrophe Research Articles

Canard Mechanism and Rhythm Dynamics of Neuron Models

Canards are a type of transient dynamics that occur in singularly perturbed systems, and they are specific types of solutions with varied dynamic behaviours at the boundary region. This paper introduces the emergence and development of canard phenomena in a neuron model. The singular perturbation system of a general neuron model is investigated, and the link between the transient transition from a neuron model to a canard is summarised. First, the relationship between the folded saddle-type canard and the parabolic burster, as well as the firing-threshold manifold, is established. Moreover, the association between the mixed-mode oscillation and the folded node type is unique. Furthermore, the connection between the mixed-mode oscillation and the limit-cycle canard (singular Hopf bifurcation) is stated. In addition, the link between the torus canard and the transition from tonic spiking to bursting is illustrated. Finally, the specific manifestations of these canard phenomena in the neuron model are demonstrated, such as the singular Hopf bifurcation, the folded-node canard, the torus canard, and the “blue sky catastrophe”. The summary and outlook of this paper point to the realistic possibility of canards, which have not yet been discovered in the neuron model.

The contact geometry of the spatial circular restricted 3-body problem

We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range.

Noise-induced spiking-bursting transition in the neuron model with the blue sky catastrophe.

We study a special variant of the noise-induced transition between spiking and bursting regimes associated with the blue sky catastrophe bifurcation in the Hindmarsh-Rose neuron model. We show that in the parameter region close to the bifurcation value, where the only attractor of the system is the limit cycle of tonic spiking type, noise can transform the spiking oscillatory regime to the bursting one. This phenomenon is studied by means of power spectral density and interspike intervals statistics. We show that noise shifts the bifurcation value, so that bursting activity can be observed for a wider parameter range. Moreover, we reveal that the stochastic spiking-bursting transitions in this system are accompanied by the change in sign of the Lyapunov exponent. We perform a detailed quantitative analysis of these phenomena with an approach that uses a concept of the stochastic sensitivity function, the confidence domains method, and Mahalanobis metrics.

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

We prove the existence of chaotic motions in an equilateral planar circular restricted four body problem (CRFBP), establishing that the system is not integrable. The proof works by verifying the hypotheses of a topological forcing theorem for Hamiltonian vector fields on R4 which hypothesizes the existence of a transverse homoclinic orbit in the energy manifold of a saddle focus equilibrium. We develop mathematically rigorous computer assisted arguments for verifying these hypotheses, and provide an implementation for CRFBP. Due to the Hamiltonian structure, this also establishes the existence of a “blue sky catastrophe”, and hence an analytic family of periodic orbits of arbitrarily long period at nearby energy levels.Our method works far from any perturbative regime and requires no mass symmetry. Additionally, the method is constructive and yields additional byproducts such as the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and bounds on transport times.

Sensitivity of energy system investments to policy regulation changes: Too many, too fast?

In this paper we argue, that the interaction of energy policy regulations in the European electricity sector may be described by the slow-fast class of dynamical systems. Such systems may exhibit drastic changes in their dynamics known as bifurcations; one important being the so called blue sky catastrophe. Once reaching such a state, the slow system becomes unresponsive to the changes in the fast system. For the energy-policy nexus this translates into the energy system becoming unresponsive to policy interventions leading to a freeze in the system dynamics. Application of this result allows us to argue that caution is needed when updating economic policies to achieve a faster transition towards a low carbon energy supply structure. To avoid this risk the policy design should be aimed at long term incentive structures with less frequent but more consistent interventions.

Coexistence between silent and bursting states in a biophysical Hodgkin-Huxley-type of model.

Classification of the dynamical mechanisms that support bistability between bursting oscillations and silence has not yet been clarified in detail. The purpose of this paper is to demonstrate that the coexistence of a stable equilibrium point with a state of continuous bursting can occur in a slightly modified, biophysical model that describe the dynamics of pancreatic beta-cells. To realize this form of coexistence, we have introduced an additional voltage-dependent potassium current that is activated in the region around the original, unstable equilibrium point. It is interesting to note that this modification also leads the model to display a blue-sky catastrophe in the transition region between chaotic and bursting states.

Features of a blue-sky transition in an autonomous convective flow

The nonlinear dynamics of a buoyant air flow in the gap formed by a horizontal cylindrical heat source and a concentric isothermally cooled square enclosure have been recently studied, in the limits of a two-dimensional approximation, for different values of the system aspect ratio, defined as the ratio between the cavity side length and the minimum gap width. Accurate analyses have revealed that, for low values of the aspect ratio, the dominant pattern is the swaying motion of a buoyant plume, which undergoes a period-doubling cascade for increasing values of the Rayleigh number. On the other hand, a substantially different behaviour has been preliminarily observed for higher values of the aspect ratio, since, in a thinner gap, the flow exhibits Rayleigh-Bénard-like cellular patterns above the cylinder.The aim of this communication is to describe the peculiar route to chaos which has been detected for this second type of geometry. The scenario consists in the progressive winding of a periodic limit cycle on an attractor whose topology resembles that of a French horn, which is a typical signature of the codimension-2 Shilnikov bifurcation from periodic dynamics to chaos. A similar feature, known as the blue-sky catastrophe, has never been reported so far for confined convection of pure fluids.

Sufficient condition for the hyperbolicity of mappings of the torus

We consider mappings of the m-dimensional torus Tm (m ≥ 2) that are C 1-perturbations of linear hyperbolic automorphisms. We obtain sufficient conditions for such mappings to be one-to-one hyperbolic mappings (i.e., Anosov diffeomorphisms). These results are used to study the blue-sky catastrophe related to the vanishing of a saddle-node invariant torus with a quasiperiodic winding in a system of ordinary differential equations.

Sensitivity of Energy System Investments to Policy Regulation Changes: Application of the Blue Sky Catastrophe

In this paper we argue, that the interaction of technology and economic policy regulations in the energy sector may be described by the so-called slow-fast class of dynamical systems. It is known that such systems may exhibit the blue sky catastrophe, a special type of bifurcation. Application of this result allows us to argue that caution is needed when updating economic policies in the energy sector to avoid the onset of catastrophic developments in the system's transformation, when energy system dynamics becomes unresponsive to policy updates.

A Farey staircase from the two-extremum return map of a Josephson junction

We report the presence of a Farey staircase in the simulated current–voltage characteristics between the second and third harmonic steps of an underdamped Josephson junction under external electromagnetic radiation. The steps constituting the staircase are interrupted by chaotic intervals. The dynamics is due to a two-extremum return map and is not associated with phase locking on an invariant torus. On decreasing the current, the third harmonic step ends in the bifurcation known as blue-sky catastrophe.

Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations

Necessary and sufficient conditions of the existence of homoclinic trajectories in Lorenz-like systems are obtained. The Shimizu–Morioka, Lu, and Chen systems are considered. For special Lorenz-like system with zero saddle value, cascade of bifurcations is obtained. Birth of strange attractor, blue sky catastrophe and 9 homoclinic bifurcations are discovered.

Blue sky catastrophe in systems with nonclassical relaxation oscillations

The feasibility of the well-known blue sky bifurcation in a class of three-dimensional singularly perturbed systems of ordinary differential equations with one fast and two slow variables is studied. A characteristic property of the considered systems is that so-called nonclassical relaxation oscillations occur in them. The same name is used for oscillations with slow components, which are asymptotically close to some time-discontinuous functions and a δ-like fast component. Cases when the blue sky catastrophe results in a stable relaxation cycle or a stable two-dimensional invariant torus are analyzed. The problem of the appearance of homoclinic structures is also considered.

Cascade of bifurcations in Lorenz-like systems: Birth of a strange attractor, blue sky catastrophe bifurcation, and nine homoclinic bifurcations

A Lorenz-like system with a single parameter and a zero saddle value is constructed for which a cascade of bifurcations is obtained, namely, the birth of limit cycles and a strange attractor, nine homoclinic bifurcations, and a blue sky catastrophe bifurcation.

Blue sky catastrophe as applied to modeling of cardiac rhythms

A new mathematical model for the electrical activity of the heart is proposed. The model represents a special singularly perturbed three-dimensional system of ordinary differential equations with one fast and two slow variables. A characteristic feature of the system is that its solution performs nonclassical relaxation oscillations and simultaneously undergoes a blue sky catastrophe bifurcation. Both these factors make it possible to achieve a phenomenological proximity between the time dependence of the fast component in the model and an ECG of the human heart.

Blue Sky Catastrophe in Systems with Non-classical Relaxation Oscillations

The feasibility of the well-known blue sky bifurcation in a class of three-dimensional singularly perturbed systems of ordinary differential equations with one fast and two slow variables is studied. A characteristic property of the considered systems is that so-called nonclassical relaxation oscillations occur in them. The same name is used for oscillations with slow components, which are asymptotically close to some time-discontinuous functions and a δ-like fast component. Cases when the blue sky catastrophe results in a stable relaxation cycle or a stable two-dimensional invariant torus are analyzed. The problem of the appearance of homoclinic structures is also considered.

Showcase of Blue Sky Catastrophes

Let a system of differential equations possess a saddle periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases of the breakdown of such a homoclinic connection that causes the blue sky catastrophe, as well as the onset of complex dynamics. The birth of an invariant torus and a Klein bottle is also described.

Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

Scientific heritage of L.P. Shilnikov

This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.

Cellular mechanisms generating bursting activity in neuronal networks

An open question in neuroscience is how the temporal characteristics of bursting activity are controlled by intrinsic biophysical characteristics. We present two mechanisms organized around the cornerstone bifurcation in a 3D Hodgkin-Huxley style neuronal model. This bifurcation satisfies the criteria for both the Shilnikov blue sky catastrophe and the saddle-node bifurcation on an invariant circle (SNIC) [1-3]. The burst duration (BD) and interburst interval (IBI) increase as the inverse of the square root of the difference between the corresponding parameter and its bifurcation value. The cornerstone bifurcation also determines the stereotypical transient responses of silent and spiking neurons [3]. The mechanisms presented here are based on these transient responses. The first mechanism described half-center oscillator consisting of two intrinsically silent, mutually inhibitory neurons (Figure 1AB). These two neurons were in the silent parameter regime but close to the cornerstone bifurcation. The two coupled neurons showed anti-phase bursting activity without the release and escape mechanisms. We found that if the half-activation voltage of a non-inactivating potassium current (V1/2,mk2) was systematically shifted towards the bifurcation value for the saddle-node bifurcation of periodic orbits, the BDs of both neurons increased in accordance with the inverse-square-root law and exhibited linear dependence on the spike number per burst. Figure 1 The cornerstone bifurcation generates mechanisms controlling bursting in neural networks. (A). Two endogenously silent, mutually inhibitory coupled neurons produce alternating bursting activity. V1/2,mk2 controls BDs. (B). The BDs are depicted as blue ... The second mechanism described the bursting activity of two intrinsically spiking, mutually excitatory neurons. The parameters of the neurons were in vicinity of the cornerstone bifurcation. This network exhibited synchronized bursting (Figure 1CD). Remarkably, excitatory interaction between endogenously spiking neurons essentially led to reduction of excitability of the network. When the half-activation voltage of hyperpolarization-activated current (V1/2,mh) was systematically shifted to the bifurcation value, IBIs of both neurons increased in accordance with the inverse-square-root law and the BDs and the number of spikes per burst stayed constant (6 spikes per burst). This study reveals new mechanisms controlling bursting activity in small neuronal networks based on cellular properties determining transient responses of endogenously spiking and silent neurons. These mechanisms are generic and could govern the bursting regimes in rhythmic neuronal network such as central pattern generators.

A codimension-2 bifurcation controlling endogenous bursting activity and pulse-triggered responses of a neuron model.

The dynamics of individual neurons are crucial for producing functional activity in neuronal networks. An open question is how temporal characteristics can be controlled in bursting activity and in transient neuronal responses to synaptic input. Bifurcation theory provides a framework to discover generic mechanisms addressing this question. We present a family of mechanisms organized around a global codimension-2 bifurcation. The cornerstone bifurcation is located at the intersection of the border between bursting and spiking and the border between bursting and silence. These borders correspond to the blue sky catastrophe bifurcation and the saddle-node bifurcation on an invariant circle (SNIC) curves, respectively. The cornerstone bifurcation satisfies the conditions for both the blue sky catastrophe and SNIC. The burst duration and interburst interval increase as the inverse of the square root of the difference between the corresponding bifurcation parameter and its bifurcation value. For a given set of burst duration and interburst interval, one can find the parameter values supporting these temporal characteristics. The cornerstone bifurcation also determines the responses of silent and spiking neurons. In a silent neuron with parameters close to the SNIC, a pulse of current triggers a single burst. In a spiking neuron with parameters close to the blue sky catastrophe, a pulse of current temporarily silences the neuron. These responses are stereotypical: the durations of the transient intervals–the duration of the burst and the duration of latency to spiking–are governed by the inverse-square-root laws. The mechanisms described here could be used to coordinate neuromuscular control in central pattern generators. As proof of principle, we construct small networks that control metachronal-wave motor pattern exhibited in locomotion. This pattern is determined by the phase relations of bursting neurons in a simple central pattern generator modeled by a chain of oscillators.