Zaslavsky web map

• George Zaslavsky, Department of Physics and Courant Institute of Mathematical Sciences, New York University, NY

The stochastic web is a thin net of fibers of finite width in the phase space of a Hamiltonian system with chaotic trajectories within the web and with regular dynamics outside one, at least in the web's vicinity. A paradigm example of the stochastic web is the Arnold web (Arnold, 1964; Arnold et al., 2006) along which a particle can perform unbounded Arnold diffusion. Such a web exists for the number of degrees of freedom \(N>2\) and non-degeneracy condition for the Hessian\[Hess H_0 \equiv | \partial^2 H_0 / \partial I \partial I | \neq0\ ,\] where \(H_0=H_0(I)\) is unperturbed Hamiltonian, and action \( I \in {\mathbb R}^N\ .\)

Stochastic web map

The stochastic web map, known as Zaslavsky web map, occurs in Hamiltonian systems with \(N>1\) satisfying the degeneracy condition \(Hess H_0=0\) (Zaslavsky et al., 1986; Zaslavsky et al., 1991).

The web map is generated by a periodically kicked linear oscillator with the Hamiltonian: \[\tag{1} H=(1/2)(\dot{x}^2+\omega_0^2x^2)-(\omega_0K/T) \cos x \sum_{m=-\infty}^{\infty} \delta(t/T-m) \]

where \(T\) is the period and \(K\) is the intensity of the kicks. The corresponding map \(\hat{T_{\alpha}}\) connects the dimensionless coordinates \((u=\dot{x}/\omega_0, v=-x)\) between two successive kicks: \[ u_{n+1}= (u_n+K\sin v_n)\cos \alpha + v_n \sin \alpha \] \(\tag{2} \hat{T_{\alpha}}\ :\)

\[ v_{n+1}= -(u_n+K\sin v_n)\sin \alpha + v_n \cos \alpha. \] The most interesting case occurs when the oscillator and the kicks are in resonance\[\alpha = \alpha_q = 2 \pi /q\ ,\] \(q \in {\mathbb N}\ ,\) \(q \geq 3\ .\) Then the stochastic web tiles the phase plane \((u,v)\) (see Figs. 1-5).

Figure 1: Thin stochastic web for \(q=4\ ,\) \(K=1.5\) is filled by the trajectory of the map \(\hat{T_{\alpha}}\ .\) There are invariant curves and isolated from the web stochastic layers inside the cells created by the web.

Figure 2: Magnification of the stochastic web in Fig.1 shows that area of the web is non-uniformly filled and has very complex pattern with islands and subislands.

Figure 3: Stochastic web for \(q=6\ ,\) \(K=2\) by the only trajectory of the map \(\hat{T_{\alpha}}\ .\) Nonuniform density of points along the web represents a result of the random walk process along the web for a finite time.

Figure 4: Animation of the random walk process created by the only trajectory of the map \(\hat{T_{\alpha}}\) for \(q=5\ ,\) \(K=0.8\ .\) The 5-fold symmetry web emerges in the phase space as a result of the random walk.

Figure 5: The same as in Fig.4 but for \(q=8\ ,\) \(K=.8108745\ .\)

Figure 6: "Thick isolines" of the web skeleton Hamiltonian \(H_q(u,v)\ ,\) for \(q=4\ .\) Different colors correspond to the sets of points \((u,v)\) that belong to different layers of \(H_q \in (E_m-\Delta E/2, E_m+\Delta E/2)\ .\)

Figure 7: The web skeleton , as in Fig.6, but for \(q=3\ .\)

Figure 8: The web skeleton , as in Fig.6, but for \(q=5\ .\) The pattern is similar to the 5-fold symmetry 2-dimensional quasicrystal.

Figure 9: The web skeleton , as in Fig.6, but for \(q=8\ .\)

Figure 10: Animation of a particle trajectory along the 4-fold symmetry stochastic web for \(K=6.349\) demonstrates anomalous (non-Gaussian) diffusion with Levy flights.

Some properties of the stochastic webs are: (a) The stochastic web can appear for \(\alpha\) rational or fairly close to a rational for finite \(K\ ;\) (b)The structure of the web can be characterized by the web skeleton, i.e. by some net of channels filled by any trajectory that starts within the web. Evidently, the skeleton is an invariant, i.e. its shape doesn't depend on time. (c)The web structure, i.e. its skeleton, has a symmetry of the crystalline type for \(q \in \{q_c\} \equiv (3,4,6) \) and of the quasicrystal type if \(q \not\in \{q_c\}\ ;\) (d) The stochastic web exists for an arbitrary small \(K\) if \(q \in \{q_c\}\) and the size of the meshes is independent of \(K\ .\) It is conjectured that for \(q \not\in \{q_c\}\) the web exists for any small \(K\) but the smaller is \(K\ ,\) the larger are the web's meshes.

The Web Skeleton

The invariant structure of the stochastic web can be obtained by averaging of (1) over the period \(T\ .\) Then, \(H=H(u,v,t) \rightarrow H_q(u,v)\) and \[\tag{3} H_q(u,v)=-(K/q)\ \sum_{j=1}^{q} \cos(\rho \cdot e_j) \]

where \(\rho =(u,v)\) and \(e_j=(\cos (2\pi j/q)-\sin (2\pi j/q))\) are the unit vectors that form a regular \(q\)-star. Isolines generated by (3) (satisfying \(H_q=\)const) are shown in Figs 6-9. The web skeleton can be considered as a "thick isoline" obtained from (3) when \(H_q \in (E_m-\Delta E/2, E_m+\Delta E/2)\ ,\) \(\Delta E\) is small, and \(E_m\) corresponds to the value of \(H_q\) for which the distribution function of the number of saddle points as a function of energy has a maximum. For \(q \in \{q_c\}\) such skeleton exists even if \( \Delta E \rightarrow 0\ .\) The web skeleton can be used as a stencil for different kinds of art patterns such as 5-fold Penrose tiling. Different oriental ornaments with 5-fold symmetric stencils were found in decorations of 11-14 centuries in Iran, Granada, Cordoba (see Zaslavsky et al., 1991 and references therein).

The map \(\hat{T_{\alpha}}\) for \(\alpha=\alpha_q\) can be considered as a dynamical generator of the \(q\)-fold symmetry for arbitrary integer \(q\ .\) It appeared first in the description of a charged particle dynamics in a constant magnetic wave packet. In general, the map (2) is an alternative to the Chirikov-Taylor standard map since \(H_0=(1/2)(\dot{x}^2+\omega_0^2x^2)\) is degenerate and thus KAM theory cannot be directly applied to (1).

Important developments on the stochastic web map were obtained in (Pekarsky and Rom-Kedar, 1997; Dana and Amit, 1995; Lowenstein, 1993 and 1995). Particles dynamics along the stochastic web generated by the map (2) is diffusive and unbounded contrary to the Arnold web along which the unbounded diffusion can be only for \(N>2\ .\) The diffusion is, in general, anomalous and can be described by fractional kinetics (Zaslavsky, 2005). Particularly, it can be superdiffusion. An example for superdiffusion with \(q=4\) (\(\hat{T_{\pi /2}}: u_{n+1}=v_n, v_{n+1}=-u_n-K \sin (v_n)\)) is shown in Fig. 10.

Updates for quantum chaos (from 2026)

The quantum evolution of Zaslavsky web map have been studied by different groups (see e.g. Shepelyansky and Sire, 1992; Dana, 1994; Gardiner et al., 1997; Kells et al. 2004; Billam and Gardiner, 2009). For the ratio between oscillator period to period of kicks being R=4 the quantum dynamics is reduced to the Kicked Harper model and there is no quantum localization. For R=5 and irrational R values there are localization and delocalization regimes.

Figure 11: Time reversal in kicked oscillator done after 30 kicks (return at time 60); top panel shows energy time dependence for quantum (full curves) and classical (dashed curves) cases; different levels of noise are shown. Bottom panel shows the quantum fidelity dependence on time (after Ermann et al. 2026)

Figure 12: Time reversal in kicked oscillator; left column shows classical distribution in phase space at times 0, 30, 60; right column shows quantum Husimi distributions at same times t Plack constant being unity; chaos parameter K=3 (after Ermann et al. 2026)

The Boltzmann–Loschmidt dispute of 1876 questioned the possibility of a statistical irreversible description by time-reversible classical equations of motion of atoms. In Ermann et al. (2026) it is shown analytically and numerically that the quantum chaos diffusion of cold atoms, or ions, in a harmonic trap and pulsed optical lattice can be inverted back in time with up to 100% efficiency. This is in sharp contrast to classical evolution, where exponentially small errors break time reversibility (see Figs. 11, 12).

February 2026: for works of quantum dynamics of this unitary and dissipative system see Dissipative quantum chaos and Refs. in Ref.[4] there.

References

Arnold V. I. (1964) Instability of dynamical systems with several degrees of freedom. Sov. Math Dokl., 5:581-585

Arnold V. I., Kozlov V. V., and Neishtadt A. I. (2006) Mathematical Aspects of Classical and Celestial Mechanics (Dynamical Systems III. Encyclopedia of Mathematical Sciences), 3rd ed. Springer, New York

Zaslavsky G. M., Zakharov M. Yu., Sagdeev R. Z., Usikov D. A., and Chernikov A. A. (1986) Stochastic web and diffusion of particles in magnetic field. Sov. Phys. JETP 64:294-303

Zaslavsky G. M., Sagdeev R. Z., Usikov D. A., and Chernikov A. A. (1991) Weak Chaos and Quasiregular Patterns. Cambridge University Press, Cambridge

Zaslavsky G. M. (2005) Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford

Dana I. and Amit M. (1995) General-approach to diffusion of periodically kicked charges in a magnetic-field. Phys. Rev. E, 51:R2731-R2734

Pekarsky S. and Rom-Kedar V. (1997) Uniform stochastic web in two-dimensional Hamiltonian systems. Phys. Lett. A, 225:274-286

Lowenstein J. H. (1993) Quasiperiodic structure of the stochastic web map. Phys. Rev. E, 47:R3811-R3814

Lowenstein J. H. (1995) Fixed-point densities for a quasiperiodic kicked-oscillator map. Chaos, 5:566-577

Shepelyansky D. and Sire C. (1992) Quantum evolution in a dynamical quasi-crystal. Europhys. Lett. 20(2): 95

Dana I. (1994) Quantum suppression of diffusion on stochastic web. Phys. Rev. Lett. 73: 1609

Gardiner S.A., Cirac J.L. and Zoller P. (1997) Quantum chaos in an ion trap: the delta-kicked harmonic oscillator. Phys. Rev. Lett. 79: 4790

Kells G.A., Twamley J. and Hefferman D.M. (2004) Dynamical properies of the delta-kicked harmonic oscillator. Phys. Rev. E 79: 015203(R)

Billam T.P. and Gardiner S.A. (2009) Quantum resonances in an atom-optical δ-kicked harmonic oscillator. Phys. Rev. A 80: 023414

Ermann L., Chepelianskii A.D. and Shepelyansky D.L. (2026) Boltzmann–Loschmidt Dispute Reloaded: Quantum 150 Years Later. Physics 28: 594; https://doi.org/10.3390/e28060594

Internal references

• Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• George Zaslavsky (2007) Zaslavsky map. Scholarpedia, 2(5):2662.

See also

Arnold Diffusion, Chaos, Hamiltonian Systems, KAM Theory, Zaslavsky Map, Dissipative quantum chaos

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Accepted on: 2007-10-11 20:54:20 GMT