Saddle Equilibrium Research Articles

Basins of Attraction of Equilibrium Points of Monotone Difference Equations

We investigate the global character of the difference equation of the form $$ x_{n+1} = f(x_n, x_{n-1},\ldots, x_{n-k+1}), \quadn=0,1, \ldots $$ with several equilibrium points, where $f$ is increasing in all its variables. We show that a considerable number of well known difference equations can be embeded into this equation through the iteration process. We also show that a negative feedback condition can be used to determine a part of the basin of attraction of different equilibrium points, and that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium points. 2000 Mathematics Subject Classification. 39A10, 39A11

Robust hierarchical games of linear discrete-time systems based on off-policy model-free reinforcement learning

An off-policy model-free reinforcement learning (RL) algorithm is proposed for a robust hierarchical game while considering incomplete information and input constraints. The robust hierarchical game exhibits characteristics of a Stackelberg–Nash (SN) game, where equilibrium points are designated as Stackelberg–Nash–Saddle equilibrium (SNE) points. An off-policy method is employed for the RL algorithm, addressing input constraints by using excitation input instead of real-time update polices as control inputs. Moreover, a model-free method is implemented for the off-policy RL algorithm, accounting for the challenge posed by incomplete information. The goal of this paper is to develop an off-policy model-free RL algorithm to obtain approximate SNE polices of the robust hierarchical game with incomplete information and input constraints. Furthermore, the convergence and effectiveness of the off-policy model-free RL algorithm are guaranteed by proving the equivalence of Bellman equation between nominal SNE policies and approximate SNE policies. Finally, a simulation is provided to verify the advantage of the developed algorithm.

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

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

On the reduction of the topological classification of gradient-like flows problem to the classification of polar flows I. A. Saraev

In this paper we consider a class G(Mn) of gradient-like flows on connected closed manifolds of dimension n≥4 such that for any flow ft∈G(Mn) stable and unstable invariant manifolds of saddle equilibria do not intersect invariant manifolds of other saddle equilibria. It is known that the ambient manifold of any flow from the class G(Mn) can be splitted into connected summ of the sphere Sn , gft≥0 copies of direct products Sn−1×S1 , and a simply connected manifold which is not homeomorphic to the sphere. The number gft is determined only by the number of nodal equilibria and the number of saddle equilibria such that one of their invariant manifolds has the dimension (n−1) (we call such equilibria trivial saddles). A simply connected manifold which is not homeomorphic to the sphere presents in the splitting if and only if the set of saddle equilibria contains points with unstable manifolds of dimension i∈{2,…,n−2} (we call such equilibria non-trivial saddles). Moreover, the complete topological classification was obtained for flows from the class G(Mn) without non-trivial saddles. In this paper we prove that for any flow ft∈G(Mn) the carrier manifold can be splitted into a connected sum along pairwise disjoint smoothly embedded spheres (separating spheres) that do not contain equilibrium states of the flow ft and transversally intersect its trajectories. The restriction of the flow ft to the complements to these spheres uniquely (up to topological equivalence and numbering) defines a finite set of flows ft1,…,ftl defined on the components of a connected sum. Moreover, for any j∈1,…,l , the set of saddle equilibria of the flow ftj consists either only of trivial saddles or only of of non-trivial ones and then the flow ftj is polar. We introduce the notion of consistent topological equivalence for flows ft1,…ftj and show that flows ft,f′t∈G(Mn) are topologically equivalent if and only if for each of these flows the set of separating spheres exists that defines consistently topologically equivalent flows on the components of the connected sum.

Uncertain stochastic hybrid differential game system with V-n jumps: Saddle point equilibrium strategies and application to advertising duopoly game

This paper focuses on the saddle point equilibrium strategies for uncertain stochastic hybrid differential game systems under chance theory. We consider systems with dynamics described by both an uncertain differential equation with V-n jumps (UDEV) and a stochastic differential equation (SDE), with linear and quadratic performance index functions. Based on chance theory and the equation of optimality of uncertain stochastic hybrid optimal control model, the saddle equilibrium equation for general uncertain stochastic hybrid differential game system is established. Furthermore, three classes of differential game models are discussed by using equilibrium equation. Finally, an advertising duopoly game problem as an application is provided, the corresponding optimal profit difference value and advertising strategies are analyzed.

Saddle transport and chaos in the double pendulum

Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting.

Conjoined Lorenz twins-a new pseudohyperbolic attractor in three-dimensional maps and flows.

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane. We also found a discrete-time analog of this "conjoined-twins" attractor in a cubic three-dimensional Hénon map with a central symmetry. We show numerically that both attractors are pseudohyperbolic, which guarantees that each orbit of the attractor has a positive maximal Lyapunov exponent, and this property is preserved under small perturbations. We also describe bifurcation scenarios for the emergence of the attractors in one-parameter families of three-dimensional flows and maps possessing the symmetries.

Synchronization and anti-synchronization of a novel fractional order chaotic system with a quadratic term

ABSTRACT This paper introduces a new three-dimensional chaos system that is different from any existing chaotic systems. The basic dynamic properties of the suggested system have been analyzed through theoretical and numerical studies and the rich chaotic behavior has been confirmed using phase portraits, Lyapunov analysis and bifurcation diagram. A simple model with a nonlinear quadratic term, unstable saddle equilibrium points and broadband chaotic behavior are some of the interesting features of the suggested system. Then, due to higher accuracy of fractional order models than integer order models, a novel fractional chaotic system has been extracted based on the suggested chaotic system and new nonlinear methods planned to achieve the synchronization and anti-synchronization goals for the system. A backstepping approach is designed to synchronize and ensure stability in Lyapunov’s concept. Also, the anti-synchronization between the two novel fractional systems is achieved by applying the active control technique, and subsequently Lyapunov stability is shown under the proposed scheme. The simulation results in MATLAB environment show the synchronization and anti-synchronization efficiency for the proposed innovative fractional order turbulent system.

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

On a topological classification of multidimensional polar flows

The work solves the classification problem for structurally stable flows, which goes back to the classical works of Andronov, Pontryagin, Leontovich and Mayer. One of important examples of such flows is so-called Morse-Smale flow, whose non-wandering set consists of a finite number of fixed points and periodic trajectories. To date, there are exhaustive classification results for Morse-Smale flows given on manifolds whose dimension does not exceed three, and a very small number of results for higher dimensions. This is explained by increasing complexity of the topological problems that arise while describing the structure of the partition of a multidimensional phase space into trajectories. In this paper authors investigate the class G(Mn) of Morse-Smale flows on a closed connected orientable manifold Mn whose non-wandering set consists of exactly four points: a source, a sink, and two saddles. For the case when the dimension n of the supporting manifold is greater or equal than four, it is additionally assumed that one of the invariant manifolds for each saddle equilibrium state is one-dimensional. For flows from this class, authors describe the topology of the supporting manifold, estimate minimum number of heteroclinic curves, and obtain necessary and sufficient conditions of topological equivalence. Authors also describe an algorithm that constructs standard representative in each class of topological equivalence. One of the surprising results of this paper is that while for n=3 there is a countable set of manifolds that admit flows from class G(M3), there is only one supporting manifold (up to homeomorphism) for dimension n>3.

Homoclinic Boundary-Saddle Bifurcations in Planar Nonsmooth Vector Fields

In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interactions with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection to cross or slide along the discontinuity. Even the simplest case of connection to a regular saddle, which hits a discontinuity as a parameter is varied, is surprisingly complex. In this paper, we construct bifurcation diagrams for nonresonant saddles in the plane, unfolding the homoclinic connection to a boundary saddle in a nonsmooth dynamical system. As an application, we exhibit such diagrams for a model of a forced pendulum.

On classification of Morse-Smale flows on projective-like manifolds

In this paper, the problem of topological classification of gradient-like flows without heteroclinic intersections, given on a four-dimensional projective-like manifold, is solved. We show that a complete topological invariant for such flows is a bi-color graph that describes the mutual arrangement of closures of three-dimensional invariant manifolds of saddle equilibrium states. The problem of construction of a canonical representative in each topological equivalence class is solved.

On bifurcations of Lorenz attractors in the Lyubimov–Zaks model

We provide numerical evidence for the existence of the Lorenz and the Rovella (contracting Lorenz) attractors in the generalization of the Lorenz model proposed by Lyubimov and Zaks. The Lorenz attractor is robustly chaotic (pseudohyperbolic) in contrast to the Rovella attractor, which is only measure-persistent (it exists for a set of parameter values, which is nowhere dense but has a positive Lebesgue measure). It is well known that in this model, for certain values of parameters, there exists a homoclinic butterfly (a pair of homoclinic loops) to the symmetric saddle equilibrium, which is neutral, i.e., its eigenvalues λ2<λ1<0<γ are such that the saddle index ν=−λ1/γ is equal to ∼1. The birth of the Lorenz attractor at this codimension-two bifurcation is established by means of numerical verification of the Shilnikov criterion. For the birth of the Rovella attractor, we propose a new criterion, which is also verified numerically.

Spatiotemporal stability of periodic travelling waves in a heteroclinic-cycle model

We study a rock–paper–scissors model for competing populations that exhibits travelling waves in one spatial dimension and spiral waves in two spatial dimensions. A characteristic feature of the model is the presence of a robust heteroclinic cycle that involves three saddle equilibria. The model also has travelling fronts that are heteroclinic connections between two equilibria in a moving frame of reference, but these fronts are unstable. However, we find that large-wavelength travelling waves can be stable in spite of being made up of three of these unstable travelling fronts. In this paper, we focus on determining the essential spectrum (and hence, stability) of large-wavelength travelling waves in a cyclic competition model with one spatial dimension. We compute the curve of transition from stability to instability with the continuation scheme developed by Rademacher et al (2007 Physica D 229 166–83). We build on this scheme and develop a method for computing what we call belts of instability, which are indicators of the growth rate of unstable travelling waves. Our results from the stability analysis are verified by direct simulation for travelling waves as well as associated spiral waves. We also show how the computed growth rates accurately quantify the instabilities of the travelling waves.

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

Entropy charts and bifurcations for Lorenz maps with infinite derivatives.

This paper deals with one-dimensional factor maps for the geometric model of Lorenz-type attractors in the form of two-parameter family of Lorenz maps on the interval I=[-1,1] given by Tc,ν(x)=(-1+c⋅|x|ν)⋅sign(x). This is the normal form for splitting the homoclinic loop with additional degeneracy in flows with symmetry that have a saddle equilibrium with a one-dimensional unstable manifold. Due to L. P. Shilnikov' results, such a bifurcation (under certain conditions) corresponds to the birth of the Lorenz attractor. We indicate those regions in the parameter plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

Smale-Williams solenoids in autonomous system with saddle equilibrium.

We construct an autonomous low-dimensional system of differential equations by replacement of real-valued variables with complex-valued variables in a self-oscillating system with homoclinic loops of a saddle. We provide analytical and numerical indications and argue that the emerging chaotic attractor is a uniformly hyperbolic chaotic attractor of Smale-Williams type. The four-dimensional phase space of the flow consists of two parts: a vicinity of a saddle equilibrium with two pairs of equal eigenvalues where the angular variable undergoes a Bernoulli map, and a region which ensures that the trajectories return to the origin without angular variable changing. The trajectories of the flow approach and leave the vicinity of the saddle equilibrium with the arguments of complex variables undergoing a Bernoulli map on each return. This makes possible the formation of the attractor of a Smale-Williams type in Poincaré cross section. In essence, our model resembles complex amplitude equations governing the dynamics of wave envelops or spatial Fourier modes. We discuss the roughness and generality of our scheme.

Intentional power system islanding under cascading outages using energy function method

In this study, the ‘where to island’ issue under the transient stability constraints is addressed. Without considering the transient stability, the islanding strategy may fail to stop the propagation of harmful dynamics throughout the network. This study promotes the current available controlled islanding model to handle the transient stability criteria, which is the most important issue during network splitting. Based on the wide area measurements, a two-stage transient stability constrained network splitting model is developed using a proper transient energy function. In the first stage, the conventional intentional splitting problem is formulated as a mixed-integer linear programming (MILP) optimisation model with considering operational, coherency and linear AC load flow constraints. The boundary of each island is determined using an optimisation model to achieve the minimum total power imbalance. To assess the transient stability, the network splitting plan obtained from the first stage is then evaluated in the second stage using a transient energy function. In the second stage, to satisfy the transient stability constraint of the critical island, a linear constraint is constructed and added to the MILP formulation of the first stage. In the second stage, the saddle or control unstable equilibrium points are determined using an optimisation model.

Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

Abstract We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T c,ɛ (x) = (−1 + c|x| 1−ɛ ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.

Homoclinic Bifurcations and Chaos in the Fishing Principle for the Lorenz-like Systems

In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios.