Number Of Unstable Periodic Orbits Research Articles

Control of Chaos in the Rössler System

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

Scattering one step from chaos

An infinite set of aligned hard discs of decreasing radius and distance is an example of a scattering system with a compact scatterer that has an infinite number of periodic orbits with the special property that no trajectory leads from one of these orbits to the next. This makes the system integrable; that is, there is no topological chaos despite the infinite number of unstable periodic orbits. We shall see that some of the usual features of chaotic scattering are present while others are not.

Crisis transitions in excitable cell models

It is believed that sudden changes both in the size of chaotic attractor and in the number of unstable periodic orbits on chaotic attractor are sufficient for interior crisis. In this paper, some interior crisis phenomena were discovered in a class of physically realizable dissipative dynamical systems. These systems represent the oscillatory activity of membrane potentials observed in excitable cells such as neuronal cells, pancreatic β-cells, and cardiac cells. We examined the occurrence of interior crises in these systems by two means: (i) constructing bifurcation diagrams and (ii) calculating the number of unstable periodic orbits on chaotic attractor. Bifurcation diagrams were obtained by numerically integrating the simultaneous differential equations which simulate the activity of excitable membranes. These bifurcation diagrams have shown an apparent crisis activity. We also demonstrate in terms of the associated Poincaré maps that the number of unstable periodic orbits embedded in a chaotic attractor suddenly increases or decreases at the crisis.

Secure communications and unstable periodic orbits of strange attractors

Every chaotic attractor is thought to be composed of the collection of unstable manifolds of all its unstable periodic orbits (UPOs). These UPOs can be used for communication by amplitude or phase modulation just as one would modulate stable periodic orbits (resonant frequencies of a linear system or limit cycles-in a nonlinear system). Using UPOs for communications has a variety of benefits. Since a chaotic attractor contains an infinite number of UPOs, a single time series can carry many different messages by placing each one on a different UPO. Because UPOs are topological properties of a chaotic attractor, the signal has a natural noise immunity, and receipt of a signal should be robust to moderate differences between the transmitter and receiver parameters. >

Unstable periodic orbits and the dimensions of multifractal chaotic attractors.

The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily fine-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and Chaotic repellers are considered.

The Lorenz System: II. The Homoclinic Convolution of the Stable Manifolds

The unstable manifold of the origin becomes homoclinic orbit when r = rho. For r > rho, the two-dimensional stable manifold of the origin develops an inhomogeneous structure in local regions of the phase space. Due to a convoluted structure of only one half of this stable manifold, the set remains R2 in some local regions, whereas in other regions it locally becomes R2 × S(r), where S(r) is an uncountable, but non-Cantor set. S(r) can be put into correspondence with the infinite set of unstable periodic points of the logistic map. The open intervals between the points of S(r) are connected to form the basins of attraction, B±, of the stable fixed points C±. The future itinerary of the trajectories which start in these intervals can be put into correspondence with any semi-infinite Bernoulli sequence. The topological structure of the basins B± is determined, including the spatial ordering of the Bernoulli sequences. This is illustrated in more detail for a particular value of r. It is shown that the spatial ordering of these itineraries can be produced by a series of maps, for all 0 < r < rt (where C± become unstable). The α-limit set (i.e., the limit set of (x(t), y(t), z(t)) as t → – ∞) of points on the stable manifold of the origin, Ws, contains an infinite number of unstable periodic orbits. As r approaches a value rhe ≃ 24.06 (in the case σ = 10, b = 8/3; Kaplan and Yorke [2]) the unstable manifold of the origin, Wu, becomes heteroclinic to the simplest of these unstable periodic orbits (the ω-limit sets of Wu± are the periodic orbits nearest C±). The topology of Ws is studied only for r < rhe in the present study.

Correlations and spectra of an intermittent chaos near its onset point

A one-parameter family of piecewise-linear discontinuous maps, which bifurcates from a periodic state of periodm, (m=2, 3,...) to an intermittent chaos, is studied as a new model for the onset of turbulence via intermittency. The onset of chaos of this model is due to the excitation of an infinite number of unstable periodic orbits and hence differs from Pomeau-Manneville's mechanism, which is a collapse of a pair of stable and unstable periodic orbits. The invariant density, the time-correlation function, and the power spectrum are analytically calculated for an infinite sequence of values of the bifurcation parameterβ which accumulate to the onset point βc from the chaos sideɛ ≡ β-β c > 0. The power spectrum nearɛ=0 is found to consist of a large number of Lorentzian lines with two dominant peaks. The highest peak lies around frequencyω=2π/m with the power-law envelope l/¦ω-(2π/m)¦4. The second-highest peak lies around ωo = 0 with the envelope l/¦ω¦2. The width of each line decreases asɛ, and the separationΔω between lines decreases asɛ/lg3−1. It is also shown that the Liapunov exponent takes the formλ∼-ɛ/m and the mean lifetime of the periodic state in the intermittent chaos is given bymɛ−1(lnɛ−1+1).