Deterministic 3D strange attractors are beautiful…but very rare !

No more ten.

Lorenz, Rossler, Chua, Gilpin, Ueta & Chen…and I !

Indeed, I built a class of Strange attractors with a system of three ODE ( only one is nonlinear).

Research intitled:

Feedback Loop in Extended van der Pol’s Equation Applied to an Economic Model of Cycles, and published in the “International Journal of Bifurcation and Chaos”, Vol. 9, N°4 (1999), pp. 745-756.
In the same volume of the Chen ODE System article.

Presentation ( pictures and graphics ) in :

http://chaos-3d.e-monsite.com

Kindest Regards,

Safieddine Bouali

The two papers by N. A. Magnitskii and S. V. Sidorov [2001, 2007] are very interesting. Especially consider their Item number 7 from the Conclusions of the 2001 paper:

“7. How to distinguish effectively between real chaos determined by the attractor of the Lorenz
system and erroneous chaos formed by errors in computations?”

And this additional comment following the above:

“In addition to the above questions, the results of the present paper obviously lead to one more
global problem dealing not only with the Lorenz system but with an arbitrary nonlinear dynamical
system as well: in general, is it possible to accept any result obtained by numerical experiments
but not justied by analytic proofs?”

I think these guys are onto something important.

More Literature Sources.

Christoph Dellago and Wm. G. Hoover, “Finite-Precision Stationary States At And Away From Equilibrium�, Physical Review E Volume 62, Number 5 November 2000.
Abstract
We study the precision dependence of equilibrium and nonequilibrium phase-space distribution functions for time-reversible dynamical systems simulated with finite, computational precision. The conservative and dissipative cases show different behavior, with substantially reduced period lengths in the dissipative case. The main effect of finite precision is to change the phase-space fraction occupied by the distributions. The convergence of thermodynamic averages is only slightly affected. We introduce and discuss a simple stochastic model which is nicely consistent with all of our numerical results. This model links the length of periodic orbits to the strange attractor’s correlation dimension.

Wm. G. Hoover, Carol G. Hoover, and H. A. Posch, “Dynamical Instabilities, Manifolds, And Local Lyapunov Spectra Far From Equilibrium�, Computational Methods in Science and Technology Vol. 7, No. 1, pp. 55-65, 2001.
Abstract
We consider an harmonic oscillator in a thermal gradient far from equilibrium. The motion is made ergodic and fully time-reversible through the use of two thermostat variables. The equations of motion contain both linear and quadratic terms. The dynamics is chaotic. The resulting phase-space distribution is not only complex and multifractal, but also ergodic, due to the time-reversibility property. We analyze dynamical time series in two ways. We describe local, but comoving, singularities in terms of the “local Lyapunov spectrum” {λ}. Local singularities at a particular phase-space point can alternatively be described by the local eigenvalues and eigenvectors of the “dynamical matrix” D=∂ν/∂r =Δν. D is the matrix of derivatives of the equations of motion dr/dt=ν(r). We pursue this eigenvalue-eigenvector description for the oscillator. We find that it breaks down at a dense set of singular points, where the four eigenvectors span only a three-dimensional subspace. We believe that the concepts of stable and unstable global manifolds are problematic for this simple nonequilibrium system.

Wm. G. Hoover, Carol G. Hoover, and Florian Grond, “Phase-Space Growth Rates, Local Lyapunov Spectra, And Symmetry Breaking For Time-Reversible Dissipative Oscillators�, Communications in Nonlinear Science and Numerical Simulation, Vol. 13, pp. 1180–1193, 2008.
Abstract
We investigate and discuss the time-reversible nature of phase-space instabilities for several flows, dx/dt=f(x). The flows describe thermostated oscillator systems in from two through eight phase-space dimensions. We determine the local extremal phase-space growth rates, which bound the instantaneous comoving Lyapunov exponents. The extremal rates are point functions which vary continuously in phase space. The extremal rates can best be determined with a ‘‘singular-value decomposition’’ algorithm. In contrast to these precisely time-reversible local ‘‘point function’’ values, a time-reversibility analysis of the comoving Lyapunov spectra is more complex. The latter analysis is nonlocal and requires the additional storing and playback of relatively long (billion-step) trajectories.
All the oscillator models studied here show the same time reversibility symmetry linking their time-reversed and time-averaged ‘‘global’’ Lyapunov spectra. Averaged over a long time-reversed trajectory, each of the long-time-averaged Lyapunov exponents simply changes signs. The negative/positive sign of the summed-up and long-time-averaged spectra in the forward/backward time directions is the microscopic analog of the Second Law of Thermodynamics. This sign changing of the individual global exponents contrasts with typical more-complex instantaneous ‘‘local’’ behavior, where there is no simple relation between the forward and backward exponents other than the local (instantaneous) dissipative constraint on their sum. As the extremal rates are point functions, they too always satisfy the sum rule.

Wm. G. Hoover, C.G. Hoover, H.A. Posch, and J.A. Codelli, “The Second Law Of Thermodynamics And Multifractal Distribution Functions: Bin Counting, Pair Correlations, And The Kaplan–Yorke Conjecture�, Communications in Nonlinear Science and Numerical Simulation xxx (2005) xxx–xxx
Abstract
We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 .lt. D .lt. 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.

N. A. Magnitskii and S. V. Sidorov, “A New View of the Lorenz Attractor�, Differential Equations, Vol. 37, No. 11, pp. 1568-1579, 2001.
Translated from Differentsial’nye Uravneniya, Vol. 37, No. 11, pp. 1494-1506, 2001.

No abstract available.

N. A. Magnitskii and S. V. Sidorov, “Transition To Chaos In Nonlinear Dynamical Systems Described By Ordinary Differential Equations�, Computational Mathematics and Modeling, Vol. 18, No. 2, pp. 128-146, 2007.
Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 73 – 98, 2003.
Abstract
We investigate scenarios that create chaotic attractors in systems of ordinary differential equations (Vallis, Rikitaki, Rossler, etc.). We show that the creation of chaotic attractors is governed by the same mechanisms. The Feigenbaum bifurcation cascade is shown to be universal, while subharmonic and homoclinic cascades may be complete, incomplete, or not exist at all depending on system parameters. The existence of a saddle – focus equilibrium plays an important and possibly decisive role in the creation of chaotic attractors in dissipative nonlinear systems described by ordinary differential equations.

What’s your definition of convergence?

1. The numerical method converges to the solution of the differential equations

2. The numerical method converges to the solution of the discretized equations

3. The numerical method converges even when large time steps and coarse grids are used. As a consequence, one can get an accurate solution in a reasonable amount of time without expensive computers.

Convergence usually depends on the accuracy you want, the discretization you use, the numerical method and on the computer power you have.