Fixed Saddle Points Research Articles

The Birth of a Hidden Attractor Through Boundary Crisis

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.

Numerical investigation of lagrangian coherent structures in steady rotation vortex shedding control

In this paper, vortex shedding and suppression are numerically investigated as autonomous and non-autonomous dynamical systems respectively. Lagrangian coherent structures (LCSs) are used as a numerical tool to analyze these systems. These structures are ridges of Finite time Lyapunov exponent (FTLE) which act as material surfaces that are transport barriers within the flow. Initially, the utility of LCSs is explored for revealing the coherent structures of these systems. Finally, an active flow control method, steady rotation is applied to the non-autonomous dynamical system with different speed ratios to mitigate vortex shedding magnitude. This will eventually turn the system into an autonomous system. Fixed saddle points, separation profiles essentially as unstable time variant manifolds attached to cylinder wall and evolution of other unstable manifolds with variant speed ratios are analyzed with reference to LCSs. It is revealed that speed ratio of 2.1 fully suppresses the von Karman vortex street at Reynolds number of 100 and system turns into an autonomous dynamical system with fixed saddle points and time-invariant manifolds.

On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

Phase saddles in light beams

In a section through a monochromatic light beam the contour map of phase contains saddlepoints. It has already been shown that a two-dimensional model of two superposed coaxialGaussian beams, in antiphase and having different waist sizes, contains two saddles thatperform an elaborate dance as the ratio of the amplitudes of the beams is altered. Thepresent paper explains why this choreography is qualitatively identical to that found in asymmetrical version of an earlier and simpler two-dimensional model, where a plane wave ismodulated by a quadratic polynomial. If wavefronts are defined as lines of equal phase,successive wavefronts pinch together in these models, and change their connectedness asthey pass through two fixed saddle points on the axis. Although the idea of aphase saddle is not generally applicable in three dimensions, it can be extendedto three dimensions in axially symmetric models, for example, two superposedcoaxial Gaussian beams. The saddles are features of the set of azimuthal planes,and can either form rings around the axis or be on the axis itself. The actionhere as a parameter changes takes a more dramatic form, because it involvesboth a vortex ring and two saddle points on the axis, which collide and explodeinto a concentric saddle ring. The physical significance of saddles is that theychange the topology of the wavefronts. In two dimensions a moving wavefront linepassing through a fixed saddle point on the axis undergoes reconnection. As itmeets an off-axis saddle the same process occurs but in a different orientation.In three dimensions as a wavefront passes through a saddle point on the axis,its local form changes from a hyperboloid of two sheets to a hyperboloid of onesheet, or vice versa, via a cone of angle . As it passes through a saddle ring a similar transition occurs simultaneously at all pointsof the ring. The changes in the topology of a wavefront as it encounters a monkey saddleare also interesting. The behaviour of the saddle points and rings in two superposedthree-dimensional Gaussian beams is also similar to that found in a simpler quadraticpolynomial model.

Diffeomorphisms with infinitely many irrational invariant curves

AbstractFor a surface diffeomorphism f∈Diff l(M), with l≥8, we prove that if f exhibits a non-transversal heteroclinic cycle composed of two fixed saddle points Q1 and Q2, one dissipative and the other expansive, then there exists an open set 𝒱⊂Diff l(M) such that $ f \in \overline {\mathcal {V}}$ and there exists a dense set 𝒟⊂𝒱 such that for all g∈𝒟, g exhibits infinitely many invariant periodic curves with irrational rotation numbers. Moreover, these curves are C1 conjugated to an irrational rotation on 𝕊1.

Nonlinear Evolution of a Balloon Satellite Orbit

The motion of a spherically symmetric balloon satellite near the equatorial plane is considered. Taking the Earth's oblateness and solar light pressure into account, the integral of motion can be obtained under certain simplifications. The eccentricity is related to the solar angle which represents an angle between pericenter and the Sun. This analytical approximation describes a large and complicated evolution of the eccentricity in corresponding areas of the phase space and the space of parameters. Phase portraits contain fixed saddle points and separatrices that divide different types of oscillations of the eccentricity. In the unsimplified problem, separatrices break down, and specific stochastic motions arise. The aims of the present study are (1) evaluation of the accuracy of analytical approximation with the help of numerical integration using a sufficiently complete model of motion and (2) numerical investigation of stochastic motions and dimensions of stochastic zones in the region of broken separatrices for an adequate model of motion. For a balloon satellite with a semimajor axis of 2.15 Earth's radii and a windage of 30 cm2/g the dimensions of a stochastic zone in eccentricity and solar angle are 10–5and 0.1°, respectively. The analytical approximation describes the orbit evolution in the right way, except for the cases of large eccentricities, e> 0.4, which corresponds to a pericenter height of less than 1400 km, where the atmospheric drag is already significant.

Attractor properties of physical branches of the solution to the renormalization group equation

We investigate the global phase-portrait structure of a local version of the exact renormalization group (RG) equation for a fluctuating scalar field of the order parameter. All the physical branches of the RG equation solution for the fixed points belong to the attractor subspace to which the local density of the Ginzburg-Landau-Wilson functional is attracted for largely arbitrary initial configurations. The solution of the RG equation corresponding to the nontrivial fixed point determining the critical behavior under the second-order phase transition is a fixed saddle point of this attractor subspace separating the attraction domains of two stable solutions corresponding to the high- and low-temperature thermodynamic regimes.

A singular perturbation method for saddle connections and subitarmonie of certain nonlinear differential equations with fixed saddle points

Saddle connections and subharmonics are investigated for a class of forced second order differentiai equations which haVea fixed saddie point. In titese equations, which have linear damping and a nonlinear restoring term, tite ampiitude of the forcing terrn depends on displacemeni in tite system. Saddie connections are significant in nonlinear systems since their appearance signals a homociinic bifurca- tion. Tite approach uses a singular perturbation asetitod which has a fairiy broad appiication to saddle connections and also to Various subharmonics. The singular perturbation is unusual in that it uses a time-scaie which has to be constructed over an inflnite interVal. The system with a cubie restoring term and a quadratic amplitude is Iooked at in sorne detaii.

Closing stable and unstable manifolds on the two sphere

Let f be a diffeomorphism of the two sphere. In this note we prove that if the unstable manifold of a fixed point p forf accumulates on the stable manifold of p, then/can be approximated arbitrarily closely Cr, r> 1, such that they intersect. 1. The results. Let d be a distance on the two sphere S2 coming from a Riemannian metric. The stable inanifold of a point p is defined to be Ws(p,f) = {x S2: d(fnx,fnp) 0 as n o co}. The unstable mnanifold of p is the stable manifold for f-1, Wu(p,f)= WS(p,f-'). Let W`(p,f)-{p}= Ws(p,f)' and Wu(p,f)-{p}= Wu(p,f)'. A fixed point p is called a saddle point if the eigenvalues 2, ,u of the derivative Df(p) satisfy O , and p a fixed saddle point off such that W8(p, f )' nclosure WU(p, f) 0 0 . Then f can be approximated arbitrarily closely Cr byf' such thatf'(p)=p and WS(p,f')' n Wu (p, f )# O. COROLLARY. There is a residual subset (complement of a first category set) R of the set of all Cr diffeomorphisms, Diff r(S2), such that iff e R, p isa saddlefixedpoint off, and Ws(p,f )' nclosure Wu(p,f) $ 0 then Ws(p,f)' r) WU(p,f)$ QS The reason for restricting to S2 is to use the Jordan separation theorem. Therefore analogous results are true on the two disk and R2. In the case of the disk, the above theorem should prove useful to prove a conjecture of Smale in [6]. For this use it would be nice to prove the result assuming only that {p}$closure Ws(p,f) r\closure Wu(p,f). Next, I would hope the results would be true for periodic points but was unable to prove it. There are all sorts of related closing lemma conjectures. See [5]. Received by the editors February 24, 1973. AMS (MOS) subject classifications (1970). Primary 58F10. 1 This research was partially supported by the Organization of American States and the National Science Foundation, GP 19815. (a) American Mathematical Society 1973