One-dimensional Stable Manifold Research Articles

Breakdown of Heteroclinic Connections in the Analytic Hopf-Zero Singularity: Rigorous Computation of the Stokes Constant

Consider analytic generic unfoldings of the three- dimensional conservative Hopf-zero singularity. Under open conditions on the parameters determining the singularity, the unfolding possesses two saddle-foci when the unfolding parameter is small enough. One of them has one-dimensional stable manifold and two-dimensional unstable manifold, whereas the other one has one- dimensional unstable manifold and two-dimensional stable manifold. Baldomá et al. (J Dyn Differ Equ 25(2):335–392, 2013) gave an asymptotic formula for the distance between the one-dimensional invariant manifolds in a suitable transverse section. This distance is exponentially small with respect to the perturbative parameter, and it depends on what is usually called a Stokes constant. The nonvanishing of this constant implies that the distance between the invariant manifolds at the section is not zero. However, up to now there do not exist analytic techniques to check that condition. In this paper we provide a method for obtaining accurate rigorous computer-assisted bounds for the Stokes constant. We apply it to two concrete unfoldings of the Hopf-zero singularity, obtaining a computer-assisted proof that the constant is nonzero.

Shilnikov attractors in three-dimensional orientation-reversing maps.

A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing and orientation-preserving cases. We consider the three-dimensional Mirá map x¯=y,y¯=z,andz¯=Bx+Cy+Az-y2 with the negative Jacobian (B<0) as a basic model demonstrating various types of Shilnikov attractors. We show that depending on values of parameters A,B, and C, such attractors can be of three possible types: hyperchaotic (with two positive and one negative Lyapunov exponent), flow-like (with one positive, one very close to zero, and one negative Lyapunov exponent), and strongly dissipative (with one positive and two negative Lyapunov exponents). We study scenarios of the formation of such attractors in one-parameter families.

A continuation approach for computing parameter-dependent separatrices in SRAM cells

• A novel approach to analyze parameter-dependent separatrices in SRAM cells is proposed. • The method is constructed based on boundary-value problems solved via continuation techniques. • In this way, families of separatrices can be approximated in a computationally efficient manner. • The method is tested in a typical SRAM bit cell via the continuation platform COCO. • A comprehensive modelling of the cell is carried out using the EKV equations for CMOS transistors. The continued scaling of CMOS semiconductor technology, together with the corresponding reduction of operating voltages, pose serious challenges for SRAM arrays regarding susceptibility to parameter variations, reduction of stability margins and reliability of read and write operations. One approach, alternative to various noise margin definitions, is to characterize the robustness of SRAM cells via the computation of stability boundaries, also referred to as separatrices. These curves define threshold voltages producing a flip in the logical state of an SRAM cell, which may lead to data loss due to disturbances, e.g. during read operations. The present work proposes a numerical approach to compute families of parameter-dependent separatrices based on path-following (continuation) methods, exploiting the geometric features of one-dimensional stable manifolds around saddle equilibria. The proposed approach is tested on an SRAM cell whose mathematical description is carried out in detail, based on the EKV (Enz, Krummenacher and Vittoz) equations for MOS transistors. In addition, the present work includes a comprehensive numerical study of the effect of parameter perturbations on separatrices, in a computationally efficient manner.

α-flips and T-points in the Lorenz system

We consider the Lorenz system near the classic parameter regime and study the phenomenon we call an α-flip. An α-flip is a transition where the one-dimensional stable manifolds Ws(p±) of two secondary equilibria p± undergo a sudden transition in terms of the direction from which they approach p±. This is a bifurcation at infinity and does not involve an invariant object in phase space. This fact was discovered by Sparrow in the 1980s but the stages of the transition could not be calculated and the phenomenon was not well understood (Sparrow 1982 The Lorenz equations (New York: Springer)). Here we employ a boundary value problem set-up and use pseudo-arclength continuation in AUTO to follow this sudden transition of Ws(p±) as a continuous family of orbit segments. In this way, we geometrically characterize and determine the moment of the actual α-flip. We also investigate how the α-flip takes place relative to the two-dimensional stable manifold of the origin, which shows no apparent topological change before or after the α-flip. Our approach allows for easy detection and subsequent two-parameter continuation of the first and further α-flips. We illustrate this for the first 25 α-flips and find that they end at terminal points, or T-points, where there is a heteroclinic connection from the secondary equilibria to the origin. It turns out that α-flips must occur naturally near T-points. We find scaling relations for the α-flips and T-points that allow us to predict further such bifurcations and to improve the efficiency of our computations.

TRANSITIONAL DYNAMICS OF AN ENDOGENOUS GROWTH MODEL WITH AN EROSION EFFECT

The convergence features of an endogenous growth model with physical capital, human capital and R&D have been studied. We add an erosion effect (supported by empirical evidence) to this model, and fully characterize its convergence properties. The dynamics is described by a fourth-order system of differential equations. We show that the model converges along a one-dimensional stable manifold and that its equilibrium is saddle-path stable, for most plausible values for the parameters. We also argue that one of the implications of considering this ‘erosion effect’ is the increase in the adherence of the model to data.

Transitional Dynamics of an Endogenous Growth Model with an Erosion Effect

The convergence features of an endogenous growth model with physical capital, human capital and R&D have been studied. We add an erosion effect (supported by empirical evidence) to this model, and fully characterize its convergence properties. The dynamics is described by a fourth-order system of differential equations. We show that the model converges along a one-dimensional stable manifold and that its equilibrium is saddle-path stable, for most plausible values for the parameters. We also argue that one of the implications of considering this erosion effect is the increase in the adherence of the model to data.

Global Dynamics of a Predator-Prey Model with Stage Structure for the Predator

The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle's invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when (4.2) fails to hold and the unique positive steady state $E^*$ becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of $E^*$, albeit with fluctuating population sizes.

The Existence of Shilnikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof

In this paper we present a new topological tool which allows us to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system $y'''+y'+0.5y^2=c^2$ [16]. We prove that there exists a countable set of parameter values $c$ for which a pair of the Shilnikov homoclinic orbits to the equilibrium points $(\pm c\sqrt2,0,0)$ appear. This result was conjectured by Michelson [16]. We also show that there exists a countable set of parameter values for which there exists a heteroclinic orbit connecting the equilibrium $(-c\sqrt2,0,0)$ possessing a one-dimensional unstable manifold with the equilibrium $(c\sqrt2,0,0)$ possessing a one-dimensional stable manifold. The method used in the proof can be applied to other reversible systems. To verify the assumptions of the main topological theorem for the Michelson system, we use rigorous computations based on interval arithmetic.

The Field–Noyes model with periodic orbits

This paper considers the Field–Noyes model. There are two possibilities if we consider only the case that the positive equilibrium of the system p is hyperbolic. Either p is asymptotically stable or it is unstable with a one-dimensional stable manifold. Both cases can occur, depending on the values of the parameters. Condition for the existence of a nontrivial periodic orbit are given in the second case.

Transitional Dynamics in an Endogenous Growth Model with Physical Capital, Human Capital and R&amp;D

We analyze the transitional dynamics of an endogenous growth model with physical capital, human capital, and R&D. We provide conditions for the existence of a feasible steady state equilibrium with positive long-run growth. For appropriate parameter values, the transitional dynamics of the model is represented by a two-dimensional stable manifold. This provides much richer dynamics than that of the standard two-sector endogenous growth model which is characterized by a one-dimensional stable manifold. We also show how the adjustment paths can be correctly computed by noting that the continuity of the shadow prices involves the continuity of transitional paths.

Computing One-Dimensional Stable Manifolds and Stable Sets of Planar Maps without the Inverse

We present an algorithm to compute the one-dimensional stable manifold of a saddle point for a planar map. In contrast to current standard techniques, here it is not necessary to know the inverse or approximate it, for example, by using Newton's method. Rather than using the inverse, the manifold is grown starting from the linear eigenspace near the saddle point by adding a point that maps back onto an earlier segment of the stable manifold. The performance of the algorithm is compared to other methods using an example in which the inverse map is known explicitly. The strength of our method is illustrated with examples of noninvertible maps, where the stable set may consist of many different pieces, and with a piecewise-smooth model of an interrupted cutting process. The algorithm has been implemented for use in the DsTool environment and is available for download with this paper.

Dynamo effect in a driven helical flow.

The Roberts flow, a helical flow in the form of convectionlike rolls, is known to be capable of both kinematic and nonlinear dynamo action. We study the Roberts dynamo with particular attention being paid to the spatial structure of the generated magnetic field and its back-reaction on the flow. The dynamo bifurcation is decisively determined by the symmetry group of the problem, which is given by a subgroup of discrete transformations and a continuous translational invariance of the flow. In the bifurcation the continuous symmetry is broken while the discrete subgroup symmetry completely survives. Its actions help in understanding the spatial structures of the magnetic field and of the modified flow. In accordance with experimental observations, the magnetic field component perpendicular to the originally invariant direction is much stronger than the component in this direction. Furthermore, the magnetic field is largely concentrated in layers separating the convectionlike rolls of the flow and containing, in particular, its stagnation points, which are isolated for the modified flow while they are line filling for the original Roberts flow. The magnetic field is strongest near beta-type stagnation points, with a two-dimensional unstable and a one-dimensional stable manifold, and is weak near alpha-type stagnation points, with a two-dimensional stable and a one-dimensional unstable manifold. This contrasts with the usual picture that dynamo action is promoted at the alpha points and impeded at the beta points. Both the creation of isolated stagnation points and the concentration of strong fields at the beta points may be understood as a result of the way in which the Roberts dynamo saturates. It is also found that, while the original Roberts flow is regular, the modified flow is chaotic in the layers between the convectionlike rolls where the magnetic field is concentrated. This chaoticity, which results from the back-reaction of the magnetic field on the flow, appears to merely enhance magnetic diffusion rather than to strengthen the dynamo effect.

Research of the stability region in a power system

A power system is employed to illustrate how we can apply singular perturbation theory to decompose a full system into two subsystems, slow and fast subsystems. Then, we study the qualitative properties of their solutions and finally obtain the stability region and an analytical expression of the approximate stability boundary of the operation point of the full system by numerical simulation and by computing the local quadratic approximation of the one-dimensional stable manifold at the saddle point. Furthermore, we consider the effects of changing the parameters on the size of the stability region.

More on the Difference Equation y n + 1 = ( p + y n −1 )/( qy n + y n −1 )

From previous studies of the equation in the title with positive parameters p and q and positive initial conditions we know that if q h 4 p + 1 then the equilibrium is a global attractor. We also know that if q > 4 p + 1 then every solution eventually enters and remains in the interval [ p / q , 1]. In this strip there exists a "unique" prime period two solution that is locally asymptotically stable. In this paper, we provide more insight as to the behavior of solutions of the equation in the title in the strip [ p / q , 1], where a one-dimensional stable manifold lives.

On the Asymmetric May--Leonard Model of Three Competing Species

In this paper we analyze the global asymptotic behavior of the asymmetric May--Leonard model of three competing species:\, $\frac{dx_i}{dt}=x_i(1-x_i-\beta_ix_{i-1}-\alpha_ix_{i+1})$, $x_i(0) > 0$, $i=1,2,3$ with $x_{0}=x_3$, $x_4=x_1$ under the assumption \, $0 < \alpha_i < 1 < \beta_i$, $i=1,2,3$. Let $A_i=1-\alpha_i$ and $B_i=\beta_i-1$, $i=1,2,3$. The linear stability analysis shows that the interior equilibrium P=(p1 , p2 , p3 ) is asymptotically stable if A1A2A3 > B1B2B3 and P is a saddle point with one-dimensional stable manifold $\Gamma$ if A1A2A3 < B1B2B3 . Hopf bifurcation occurs when A1A2A3 = B1B2B3 . For the case A1A2A3 ne ,B1B2B3 we eliminate the possibility of the existence of periodic solutions by applying the Stokes theorem. Then, from the Poincar{é}--Bendixson theorem for three-dimensional competitive systems, we show that (i) if A1A2A3 > B1B2B3 then P is global asymptotically stable in Int(R3+ ), (ii) if A1A2A3 < B1B2B3 then for each initial condition $x_0\not\in\Gamma$, the solution $\varphi(t,x_0)$ cyclically oscillates around the boundary of the coordinate planes as the trajectory of the symmetric May--Leonard model does, and (iii) if A1A2A3 = B1B2B3 then there exists a family of neutrally stable periodic orbits.

An Algorithm for the Computation of Preimages in Noninvertible Mappings

For discrete dynamical systems generated by iterating a diffeomorphism, every point in the phase space has a unique preimage and it is straightforward to compute geometric structures such as inverse orbits and one-dimensional stable manifolds of periodic points. For noninvertible mappings, however, some points have multiple preimages; others may have no preimages. This makes the computation of inverse orbits difficult, because accurate computations require global knowledge about the way the mapping folds and pleats phase space. In this article we use ideas from singularity theory to examine the geometry of noninvertible mappings. We use the geometry to derive a computational algorithm for efficiently computing preimages in noninvertible mappings.

Large homogeneous communication networks with Markovian access control II. Global behavior and residence time

A solution to the problem of dynamic analysis of large homogeneous communication networks with Markovian access control disciplines is given. For the case of multiple steady states and buffered users it is shown that: (i) the state space portrait of the system contains a one-dimensional stable manifold; (ii) the dynamics of the system can be reduced to a birth and death process defined on this manifold; and (iii) the residence time of each metastable steady state of the above birth and death process can be calculated on the basis of a new asymptotic procedure. Simulations show that this procedure results in acceptable precision.

Properties of solutions of certain two-dimensional nonlinear systems of ordinary differential equations on and outside a stable manifold

Some two-dimensional nonlinear systems with an irregular singularity at infinity are investigated. Properties of their solutions on and outside a one-dimensional stable manifold are studied. Representations for solutions on the manifold are derived in the form of one-parameter exponential series. It is shown how solutions not tending to zero at infinity deviate from the stable manifold.