Showing papers in "Siam Journal on Applied Dynamical Systems in 2013"

Adaptive and Sequential Gridding Procedures for the Abstraction and Verification of Stochastic Processes

Abstract: This work is concerned with the generation of finite abstractions of general state-space processes to be employed in the formal verification of probabilistic properties by means of automatic techniques such as probabilistic model checkers. The work employs an abstraction procedure based on the partitioning of the state-space, which generates a Markov chain as an approximation of the original process. A novel adaptive and sequential gridding algorithm is presented and is expected to conform to the underlying dynamics of the model and thus to mitigate the curse of dimensionality unavoidably related to the partitioning procedure. The results are also extended to the general modeling framework known as stochastic hybrid systems. While the technique is applicable to a wide arena of probabilistic properties, with focus on the study of a particular specification (probabilistic safety, or invariance, over a finite horizon), the proposed adaptive algorithm is first benchmarked against a uniform gridding approach t...

A reduction method for boolean network models proven to conserve attractors.

Abstract: Boolean models, wherein each component is characterized with a binary (ON or OFF) variable, have been widely employed for dynamic modeling of biological regulatory networks. However, the exponential dependencse of the size of the state space of these models on the number of nodes in the network can be a daunting prospect for attractor analysis of large-scale systems. We have previously proposed a network reduction technique for Boolean models and demonstrated its applicability on two biological systems, namely, the abscisic acid signal transduction network as well as the T-LGL leukemia survival signaling network. In this paper, we provide a rigorous mathematical proof that this method not only conserves the fixed points of a Boolean network, but also conserves the complex attractors of general asynchronous Boolean models wherein at each time step a randomly selected node is updated. This method thus allows one to infer the long-term dynamic properties of a large-scale system from those of the corresponding reduced model.

Wandering Bumps in Stochastic Neural Fields

Abstract: We study the effects of noise on stationary pulse solutions (bumps) in spatially extended neural fields. The dynamics of a neural field is described by an integrodifferential equation whose integral term characterizes synaptic interactions between neurons in different spatial locations of the network. Translationally symmetric neural fields support a continuum of stationary bump solutions, which may be centered at any spatial location. Random fluctuations are introduced by modeling the system as a spatially extended Langevin equation whose noise term we take to be additive. For nonzero noise, bumps are shown to wander about the domain in a purely diffusive way. We can approximate the associated diffusion coefficient using a small noise expansion. Upon breaking the (continuous) translation symmetry of the system using spatially heterogeneous inputs or synapses, bumps in the stochastic neural field can become temporarily pinned to a finite number of locations in the network. As a result, the effective diffusion of the bump is reduced, in comparison to the homogeneous case. As the modulation frequency of this heterogeneity increases, the effective diffusion of bumps in the network approaches that of the network with spatially homogeneous weights.

Time-Scale Separation from Diffusion-Mapped Delay Coordinates

Abstract: It has long been known that the method of time-delay embedding can be used to reconstruct nonlinear dynamics from time series data. A less-appreciated fact is that the induced geometry of time-delay coordinates increasingly biases the reconstruction toward the stable directions as delays are added. This bias can be exploited, using the diffusion maps approach to dimension reduction, to extract dynamics on desired time scales from high-dimensional observed data. We demonstrate the technique on a wide range of examples, including data generated by a model of meandering spiral waves and video recordings of a liquid-crystal experiment.

Power-Law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species

Abstract: We present determinant criteria for the preclusion of nondegenerate multiple steady states in net- works of interacting species. A network is modeled as a system of ordinary differential equations in which the form of the species formation rate function is restricted by the reactions of the network and how the species influence each reaction. We characterize families of so-called power-law kinet- ics for which the associated species formation rate function is injective within each stoichiometric class and thus the network cannot exhibit multistationarity. The criterion for power-law kinetics is derived from the determinant of the Jacobian of the species formation rate function. Using this characterization, we further derive similar determinant criteria applicable to general sets of kinet- ics. The criteria are conceptually simple, computationally tractable, and easily implemented. Our approach embraces and extends previous work on multistationarity, such as work in relation to chemical reaction networks with dynamics defined by mass-action or noncatalytic kinetics, and also work based on graphical analysis of the interaction graph associated with the system. Further, we interpret the criteria in terms of circuits in the so-called DSR-graph.

Dynamics of Stochastically Blinking Systems. Part I: Finite Time Properties ∗

Abstract: We consider dynamical systems whose parameters are switched within a discrete set of values at equal time intervals. Similar to the blinking of the eye, switching is fast and occurs stochastically and independently for different time intervals. There are two time scales present in such systems, namely the time scale of the dynamical system and the time scale of the stochastic process. If the stochastic process is much faster, we expect the blinking system to follow the averaged system where the dynamical law is given by the expectation of the stochastic variables. We prove that, with high probability, the trajectories of the two systems stick together for a certain period of time. We give explicit bounds that relate the probability, the switching frequency, the precision, and the length of the time interval to each other. We discover the apparent presence of a soft upper bound for the time interval, beyond which it is almost impossible to keep the two trajectories together. This comes as a surprise in view of the known perturbation analysis results. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system.

Computation of limit cycles and their isochrons: Fast algorithms and their convergence

Abstract: We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using $N$ points, a Newton step requires $O(N)$ storage and $O(N\log N)$ operations in Fourier discretization or $O(N)$ operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild nondegeneracy conditions, then there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem ...

Dynamics of Stochastically Blinking Systems. Part II: Asymptotic Properties

Abstract: We study stochastically blinking dynamical systems as in the companion paper (Part I). We analyze the asymptotic properties of the blinking system as time goes to infinity. The trajectories of the averaged and blinking system cannot stick together forever, but the trajectories of the blinking system may converge to an attractor of the averaged system. There are four distinct classes of blinking dynamical systems. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or noninvariance under the dynamics of the blinking system. In the case of invariance, we prove that the trajectories of the blinking system converge to the attractor(s) of the averaged system with high probability if switching is fast. In the noninvariant single attractor case, the trajectories reach a neighborhood of the attractor rapidly and remain close most of the time with high probability when switching is fast. In the noninvariant multiple attractor case, the trajectory may escape to another attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities. Each of the four cases is illustrated by a specific example of a blinking dynamical system. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system.

Poisson--Nernst--Planck Systems for Ion Flow with a Local Hard-Sphere Potential for Ion Size Effects

Abstract: In this work, we analyze a one-dimensional steady-state Poisson--Nernst--Planck-type model for ionic flow through a membrane channel with fixed boundary ion concentrations (charges) and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. A local hard-sphere potential that depends pointwise on ion concentrations is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V (current-voltage) relation and identify two critical potentials or voltages for ion size effects. Under electroneutrality ...

Metastability in a Stochastic Neural Network Modeled as a Velocity Jump Markov Process

Abstract: One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synapti- cally coupled neuronal populations. Often noise is incorporated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve ac- cording to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterized by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks.

Multiple Geometric Viewpoints of Mixed Mode Dynamics Associated with Pseudo-plateau Bursting

Abstract: Pseudo-plateau bursting is a type of oscillatory waveform associated with mixed mode dynamics in slow/fast systems and commonly found in neural bursting models. In a recent model for the electrical activity and calcium signaling in a pituitary lactotroph, two types of pseudo-plateau bursts were discovered: one in which the calcium drives the bursts and another in which the calcium simply follows them. Multiple methods from dynamical systems theory have been used to understand the bursting. The classic 2-timescale approach treats the calcium concentration as a slowly varying parameter and considers a parametrized family of fast subsystems. A more novel and successful 2-timescale approach divides the system so that there is only one fast variable and shows that the bursting arises from canard dynamics. Both methods can be effective analytic tools, but there has been little justification for one approach over the other. In this work, we use the lactotroph model to demonstrate that the two analysis techniques...

Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps

Abstract: This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

Nonlinear Stationary States in PT-Symmetric Lattices

Abstract: In the present work we examine both the linear and nonlinear properties of two related parity-time (PT)-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem. Second, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals for a finite PT-dNLS defect in the infinite dNLS lattice are wider than in the case of an isolated PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anticontinuum limit for the dNLS equation. Numerical computations illustrate the existence of nonlinear stationary states as well as the stability and saddle-center bifurcations of discrete solitons.

Bifurcations in Delay Differential Equations and Applications to Tumor and Immune System Interaction Models

Abstract: In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf--Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, Bautin bifurcation, and Hopf--Hopf bifurcation. As an application, we study the dynamical behaviors of a model describing the interaction between tumor cells and effector cells of the immune system. Numerical examples and simulations are presented to illustrate the obtained results.

Dynamics of the Budyko Energy Balance Model

Abstract: A dynamical system derived from a conceptual climate model is investigated. The model is often referred to as the Budyko--Sellers-type energy balance model (EBM), or in short, the Budyko model. The focus of this study is on a mathematical formulation that further develops the model by capturing the ice line dynamics and thereby places the Budyko model within the dynamical systems framework. The classical Budyko EBM is represented here as an integro-difference equation, then coupled with an ice line equation. The resulting infinite dimensional system is shown to possess an attracting one dimensional invariant manifold. The novel formulation captures solutions that others have previously obtained while also making the formulation and the analysis of the model more precise.

Computation of Balanced Equivalence Relations and Their Lattice for a Coupled Cell Network

Abstract: A coupled cell network describes interacting (coupled) individual systems (cells). As in networks from real applications, coupled cell networks can represent inhomogeneous networks where different types of cells interact with each other in different ways, which can be represented graphically by different symbols, or abstractly by equivalence relations. Various synchronous behaviors, from full synchrony to partial synchrony, can be observed for a given network. Patterns of synchrony, which do not depend on specific dynamics of the network, but only on the network structure, are associated with a special type of partition of cells, termed balanced equivalence relations. Algorithms in Aldis [J. W. Aldis, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), pp. 407--427] and Belykh and Hasler [I. Belykh and M. Hasler, Chaos, 21 (2011), 016106] find the unique pattern of synchrony with the fewest clusters. In this paper, we compute the set of all possible patterns of synchrony and show their hierarchy struc...

Local and Global Stability of Equilibria for a Class of Chemical Reaction Networks

Abstract: A class of chemical reaction networks is described with the property that each positive equilibrium is locally asymptotically stable relative to its stoichiometry class, an invariant subspace on which it lies. The reaction systems treated are characterized primarily by the existence of a certain factorization of their stoichiometric matrix and strong connectedness of an associated graph. Only very mild assumptions are made about the rates of reactions, and, in particular, mass action kinetics are not assumed. In many cases, local asymptotic stability can be extended to global asymptotic stability of each positive equilibrium relative to its stoichiometry class. The results are proved via the construction of Lyapunov functions whose existence follows from the fact that the reaction networks define monotone dynamical systems with increasing integrals.

Stability of Relative Equilibria in the Planar $n$-Vortex Problem

Abstract: We study the linear and nonlinear stability of relative equilibria in the planar $n$-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.

Rigorous Numerics in Floquet Theory: Computing Stable and Unstable Bundles of Periodic Orbits ∗

Abstract: In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution $\Phi(t)$ is a canonical decomposition of the form $\Phi(t)=Q(t)e^{Rt}$, where $Q(t)$ is a real periodic matrix and $R$ is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of $Q(t)$ and to simultaneously solve for $R$ and $Q(t)$ with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of $R$ and $Q$ can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields. The new proposed method does not require rigorous numerical integration of the ODE.

Multiparameter Models of Innovation Diffusion on Complex Networks

Abstract: A model, applicable to a range of innovation diffusion applications with a strong peer-to-peer component, is developed and studied, along with methods for its investigation and analysis. A particular application is to individual households deciding whether to install an energy efficiency measure in their home. The model represents these individuals as nodes on a network, each with a variable representing their current state of adoption of the innovation. The motivation to adopt is composed of three terms, representing personal preference, an average of each individual's network neighbors' states, and a system average, which is a measure of the current social trend. The adoption state of a node changes if a weighted linear combination of these factors exceeds some threshold. Numerical simulations have been carried out, computing the average uptake after a sufficient number of time-steps over many randomizations at all model parameter values, on various network topologies, including random (Erdos--Renyi), s...

Spots in the Swift-Hohenberg Equation ∗

Abstract: The existence of stationary localized spots for the planar and the three-dimensional Swift--Hohenberg equations is proved using geometric blow-up techniques. The spots found in this paper have a much larger amplitude than that expected from a formal scaling in the far field. One advantage of the geometric blow-up methods used here is that the anticipated amplitude scaling does not enter as an assumption into the analysis but emerges naturally during the construction. Thus, the approach used here may also be useful in other contexts where the scaling is not known a priori.

Interplay Between Synaptic Delays and Propagation Delays in Neural Field Equations

Abstract: Neural field equations describe the activity of neural populations at a mesoscopic level. Although the early derivation of these equations introduced space dependent delays coming from the finite speed of signal propagation along axons, there have been few studies concerning their role in shaping the (nonlinear) dynamics of neural activity. This is mainly due to the lack of analytical tractable models. On the other hand, constant delays have been introduced to model the synaptic transmission and the spike initiation dynamics. By incorporating the two kinds of delays into the neural field equations, we are able to find the Hopf bifurcation curves analytically, which produces many Hopf--Hopf interactions. We use normal theory to study two different types of connectivity that reveal a surprisingly rich dynamical portrait. In particular, the shape of the connectivity strongly influences the spatiotemporal dynamics.

Singular Dynamics in Gene Network Models

Abstract: Gene regulatory networks are commonly modeled by steep sigmoid interactions or by their limiting step functions. This leads to difficulties in dealing with singular dynamics, i.e., when some of the gene expressions are close to their thresholds. Two methods have been proposed to analyze this situation: the steep sigmoid framework based on singular perturbation techniques and the Filippov theory of differential inclusions. However, these lead to different concepts of solutions. Here we revisit these approaches and show their relationship. For the Filippov framework, we emphasize the use of two different solution concepts, namely, with a convex and a not necessarily convex right-hand side of the differential inclusion. We show that the second case is identical, under certain assumptions, to the solution concept used in the steep sigmoid framework. Even without these assumptions we obtain an existence result in the 2-dimensional case. We present 3-dimensional examples that do not fit the classical singular p...

Pulses in a Gierer--Meinhardt Equation with a Slow Nonlinearity

Abstract: In this paper, we study in detail the existence and stability of localized pulses in a Gierer-Meinhardt equation with an additional "slow" nonlinearity. This system is an explicit example of a general class of singularly perturbed, two component reaction-diffusion equations that goes significantly beyond well-studied model systems such as Gray-Scott and Gierer-Meinhardt. We investigate the existence of these pulses using the methods of geometric singular perturbation theory. The additional nonlinearity has a profound impact on both the stability analysis of the pulse—compared to Gray- Scott/Gierer-Meinhardt-type models a distinct extension of the Evans function approach has to be developed—and the stability properties of the pulse: several (de)stabilization mechanisms turn out to be possible. Moreover, it is shown by numerical simulations that, unlike the Gray-Scott/Gierer- Meinhardt-type models, the pulse solutions of the model exhibit a rich and complex behavior near the Hopf bifurcations.

Amplified Hopf Bifurcations in Feed-Forward Networks

Abstract: In [B. Rink and J. Sanders, Trans. Amer. Math. Soc., to appear] the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like $\sim|\lambda|^{\frac{1}{2}},\sim|\lambda|^{\frac{1}{6}},\sim|\lambda|^{\frac{1}{18}}$, etc. Such amplified Hopf branches were previously found in a subclass of feed-forward networks with three cells, first under a normal form assumption [M. Golubitsky and I. Stewart, Bull. Amer. Math. Soc. (N.S.), 43 (2006), pp. 305--364] and later by explicit computations [T. Elmhirst and M. Golubitsky, SIAM J. Appl. Dyn. Syst., 5 (2006), pp. 205--251], [M. Golubitsky and C. Postlethwaite, Discrete Contin. Dyn. Syst., 32 (2012), pp. 2913--2935]. We explain here how these bifurcations arise gene...

Existence and Stability of Traveling Pulses in a Neural Field Equation with Synaptic Depression

Abstract: We examine the existence and stability of traveling pulse solutions in a continuum neural network with synaptic depression and smooth firing rate function. The existence proof relies on geometric singular perturbation theory and blow-up techniques, as one needs to track the solution near a point on the slow manifold that is not normally hyperbolic. The stability of the pulse is then investigated by computing the zeros of the corresponding Evans function. This study predicts that synaptic depression leads to the formation of stable traveling pulses with algebraic decay along their back. This characteristic feature differs from the exponential decay of traveling pulses of neural field models with linear adaptation.

Influence of Variable Side-Stay Geometry on the Shimmy Dynamics of an Aircraft Dual-Wheel Main Landing Gear ∗

Abstract: Commercial aircraft are designed to fly but also need to operate safely and efficiently as vehicles on the ground. During taxiing, take-off, and landing the landing gear must operate reliably over a wide range of forward velocities and vertical loads. Specifically, it must maintain straight rolling under a wide variety of operating conditions. It is well known, however, that under certain conditions the wheels of the landing gear may display unwanted oscillations, referred to as shimmy oscillations, during ground maneuvers. Such oscillations are highly unwanted from a safety and a ride-comfort perspective. In this paper we conduct a study into the occurrence of shimmy oscillations in a main landing gear (MLG) of a typical midsize passenger aircraft. Such a gear is characterized by a main strut attached to the wing spar with a side-stay that connects the main strut to an attachment point closer to the fuselage center line. Nonlinear equations of motion are developed for the specific case of a two-wheeled M...

Stochastic Regular Grazing Bifurcations

Abstract: A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude $\varepsilon$. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms or, if there exists an attracting period-$n$ solution when $\varepsilon=0$, be well approximated by the sum of $n$ Gaussian densities centered about each point of the deterministic solution, and scaled by $\frac{1}{n}$, for sufficiently small $\varepsilon>0$. We explain the irregular forms an...

Synchronization Properties of Trees in the Kuramoto Model

Abstract: We consider the Kuramoto model of coupled oscillators, specifically the case of tree networks, for which we prove a simple closed-form expression for the critical coupling. For several classes of tree, and for both uniform and Gaussian vertex frequency distributions, we provide tight closed-form bounds and empirical expressions for the expected value of the critical coupling. We also provide several bounds on the expected value of the critical coupling for all trees. Finally, we show that for a given set of vertex frequencies, there is a rearrangement of oscillator frequencies for which the critical coupling is bounded by the spread of frequencies.

Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Abstract: Homoclinic bifurcations are important phenomena that cause global rearrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that the eigenvalues of its Jacobian are real. If such a homoclinic orbit is broken by varying a suitable parameter, then, generically, a single periodic orbit $\Gamma$ bifurcates. We consider the case that the saddle quantity of the equilibrium is negative so that $\Gamma$ is an attractor (rather than of saddle type). At the moment of bifurcation the two-dimensional stable manifold of the saddle, when followed along the homoclinic orbit, may form either an orientable or nonorientable surface, and one speaks of an orientable or a nonorientable homoclinic bifurcation. A change of orientability occurs at two kinds of codim...