Homoclinic Tangles Research Articles

Homoclinic tangles in the DIII-D tokamak from the map equations in natural canonical coordinates*

ABSTRACTTrajectories of magnetic field lines are a 1½ degree of freedom Hamiltonian system. The unperturbed separatrix and the perturbed separatrix in a divertor tokamak are fundamentally different. Magnetic asymmetries cause the separatrix to form extremely complicated structures known as homoclinic tangles. After each toroidal circuit, the perturbed separatrix manifolds meet in a fixed poloidal plane and intersect to form homoclinic tangle in order to preserve the topological invariants. This tangle becomes extremely complicated as the magnetic field lines take more and more toroidal turns. This effect is most pronounced near the X-point. The homoclinic tangles of the DIII-D tokamak separatrix from the magnetic perturbation representing the peeling-ballooning modes are studied. The homoclinic tangles can have important implications for the edge physics in divertor tokamaks.

Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study

Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.

Modeling non-stationary, non-axisymmetric heat patterns in DIII-D tokamak

Non-axisymmetric stationary magnetic perturbations lead to the formation of homoclinic tangles near the divertor magnetic saddle in tokamak discharges. These tangles intersect the divertor plates in static helical structures that delimit the regions reached by open magnetic field lines reaching the plasma column and leading the charged particles to the strike surfaces by parallel transport. In this article we introduce a non-axisymmetric rotating magnetic perturbation to model the time evolution of the three-dimensional magnetic field of a single-null DIII-D tokamak discharge developing a rotating tearing mode. The non-axiymmetric field is modeled using the magnetic signals to adjust the phases and currents of a set of internal filamentary currents that approximate the magnetic field in the plasma edge region. The stable and unstable manifolds of the asymmetric magnetic saddle are obtained through an adaptive calculation providing the cuts at a given poloidal plane and the strike surfaces. For the modeled shot, the experimental heat pattern and its time development are well described by the rotating unstable manifold, indicating the emergence of homoclinic lobes in a rotating frame due to the plasma instabilities.

Computing Melnikov Curves for Periodically Perturbed Piecewise Smooth Oscillators

Curves dividing the parameter plane into regions according to the presence or absence of homoclinic or heteroclinic tangle corresponding to the periodically perturbed saddle of the piecewise smooth oscillator are studied using Melnikov analysis. The analysis is not simplified by choosing the discontinuity plane at a convenient location. Separatrix of the unperturbed system is parametrized exactly in a piecewise manner. Switching times, i.e. parameter values at which the separatrix crosses the discontinuity plane, are obtained. Switching times split the Melnikov integral into various subintegrals which are evaluated either exactly using term-wise integration of the infinite series of the integrand or approximately using a finite-term series approximation of the integrand, the latter being computationally an extensive task. Integral evaluations though approximate, are purely analytical expressions in terms of special functions such as digamma and hypergeometric. Melnikov plots show that the boundary between three regions in the parameter plane differ qualitatively in case of parametric and external excitations, however; adding self-excitation to the external one does not much alter the boundary qualitatively and quantitatively.

Jumping mechanisms of Trojan asteroids in the planar restricted three- and four-body problems

We explore minimal dynamical mechanisms for the transport of Trojan asteroids between the vicinities of the stable Lagrange points \(L_{4}\) and \(L_{5}\) within the framework of the planar restricted three- and four-body problems. This transport, called “jumping” of Trojan asteroids, has been observed numerically in the sophisticated Solar System models. However its dynamical mechanisms have not been fully explored yet. The present study shows that invariant manifolds emanating from an unstable periodic orbit around the unstable Lagrange point \(L_{3}\) mediate the jumping of Trojan asteroids in the Sun–Jupiter planar restricted three-body problem. These invariant manifolds form homoclinic tangles and lobes when projected onto the configuration space through a discrete mapping. Thus the resulted lobe dynamics explains the mechanism for the jumping of Jupiter’s Trojan asteroids. In the Sun–Earth planar restricted three-body problem, on the other hand, invariant manifolds of an unstable periodic orbit around \(L_{3}\) do not exhibit clear homoclinic tangles nor lobes, indicating that the jumping is very difficult to occur. It is then shown that the effect of perturbation of Venus is important for the onset of the jumping of Earth’s Trojan asteroids within the framework of the Sun–Earth–Venus planar restricted four-body problem. The results presented here could shed new insights into the transport mechanism as well as trajectory design associated with \(L_{3}\), \(L_{4}\), and \(L_5\).

Homoclinic tangle of the ideal separatrix in the DIII-D tokamak from (30, 10) + (40, 10) perturbation

Trajectories of magnetic field lines are a 1½ degree of freedom Hamiltonian system. The perturbed separatrix in a divertor tokamak is radically different from the unperturbed one. This is because magnetic asymmetries cause the separatrix to form extremely complicated structures called homoclinic tangles. The shape of flux surfaces in the edge region of divertor tokamaks such as the DIII (J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)) is fundamentally different from near-circular. Recently, a new method is developed to calculate the homoclinic tangle and lobes of the separatrix of divertor tokamaks in physical space (A. Punjabi and A. Boozer, Phys. Lett. A 378, 2410 (2014)). This method is based on three elements: preservation of the two invariants—symplectic and topological neighborhood—and a new set of canonical coordinates called the natural canonical coordinates. The very complicated shape of edge surfaces can be represented very accurately and very realistically in these new coordinates (A. Punjabi and H. Ali, Phys. Plasmas 15, 122502 (2008); A. Punjabi, Nucl. Fusion 49, 115020 (2009)). A symplectic map in the new coordinates can advance the separatrix manifold forward and backward in time. Every time the two manifolds meet in a fixed poloidal plane, they intersect and form homoclinic tangle to preserve the two invariants. The new coordinates can be mapped to physical space and the dynamical evolution of the homoclinic tangle can be seen and pictured in physical space. Here, the new method is applied to the DIII-D tokamak to study the basic features of the homoclinic tangle of the unperturbed separatrix from two Fourier components, which represent the peeling-ballooning modes of equal amplitude and no radial dependence, and the results are analyzed. Homoclinic tangle has a very complicated structure and becomes extremely complicated for as the lines take more toroidal turns, especially near the X-point. Homoclinic tangle is the most complicated near the X-point and forms the largest lobes there. On average, the field lines cover a distance of about 9 m per turn. Poloidal rotation of the lines has large gradients in the poloidal direction. The average normal displacement of the lines on the separatrix varies from 5 mm to 7 cm. Average outward displacement of the lines is considerably larger than the inward displacement; however, on the average more lines are displaced inside than outside of the separatrix. The field line diffusion normal to the separatrix has extremely wide variation and very large poloidal gradients. Half of all the lines are lost in less than 6 turns. Complicated electric potentials will be required to maintain the neutrality of the plasma, and the E × B drifts from these fields can modify plasma confinement and influence the edge physics (A. Punjabi and A. Boozer, Phys. Lett. A 378, 2410 (2014)).

Delineating the magnetic field line escape pattern and stickiness in a poloidally diverted tokamak

We analyze a Hamiltonian model with five wire loops that delineates magnetic surfaces of tokamaks with poloidal divertor. Non-axisymmetric magnetic perturbations are added by external coils, similar to the correction coils that have been installed or designed in present tokamaks. To show the influence of magnetic perturbations on the field line escape, we integrate numerically the field line differential equations and obtain the footprints and deposition patterns on the divertor plate. Moreover, we show that the homoclinic tangle describes the deposition patterns in the divertor plate, agreeing with results observed in sophisticated simulation codes. Additionally, we show that while chaotic lines escape to the divertor plates, some of them are trapped, for many toroidal turns, in complex structures around magnetic islands, embedded in the chaotic region, giving rise to stickiness evidences characteristic of chaotic Hamiltonian systems. Finally, we introduce a random collisional term to the field line mapping to investigate stickiness alterations due to particle collisions. Within this model, we conclude that, even reduced by collisions, the observed trapping still influences the field line transport. The results obtained for our numerical estimations indicate that the reported trapping may affect the transport in present tokamaks.

Homoclinic tangle in tokamak divertors

Magnetic field lines are the trajectories of a 112 degree of freedom Hamiltonians. Plasmas in tokamaks are confined in regions where the magnetic field lines form closed toroidal surfaces. These surfaces are bounded by a separatrix, and outside the separatrix the magnetic field lines and the plasma flow to special regions of the walls called divertors. Both the confinement of the plasma and the feasibility of divertors are sensitive to the behavior of the magnetic field lines near the separatrix in the presence of non-axisymmetric magnetic perturbations. Separatrix manifold forms homoclinic tangle to preserve the symplectic invariant and topological neighborhood as the manifold evolves in canonical time. A scheme is developed based on these two invariants to calculate homoclinic tangle for actual tokamak equilibria that are subjected to non-axisymmetric perturbations. This scheme is used to study homoclinic tangles of a specific tokamak configuration. How non-ideal effects fill in the lobes formed by homoclinic tangles is demonstrated using a radial expansion operator to simulate plasma diffusion. It is found that for sufficiently rapid plasma diffusion the effects of the tangle on plasma footprint on collector plate are washed out.

Analytical invariant manifolds near unstable points and the structure of chaos

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain,the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as i) the order of truncation of the series increases, and ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.

Early chaos theory

The article “Chaos at fifty” by Adilson Motter and David Campbell (Physics Today, May 2013, page 27) pays a well-deserved homage to Edward Lorenz for his contribution to chaos theory and meteorology. However, I take exception to the statement at the beginning of the article that “in 1963 an MIT meteorologist revealed deterministic predictability to be an illusion and gave birth to a field that still thrives.” On the contrary, the exponential growth of errors in some deterministic systems and the practical consequences for predictability have been appreciated by scientists for more than a century. I am particularly fond of what Henri Poincaré says in “Le hasard,” chapter 4 of his book Science et Méthode (Ernest Flammarion, 1908; my translation):A small cause, that escapes us, determines a considerable effect that we cannot ignore, and we then say that this effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial instant, we could predict exactly the situation of this same universe at a later instant.But Poincaré remarks that we know the initial situation only approximately, and that it may happen that small differences in initial conditions generate large differences later, so that prediction becomes impossible. Poincaré offers meteorology as an example:Why do meteorologists have such difficulty in predicting the weather with any certainty? … [They see] that a cyclone will appear, but they are unable to say where; a tenth of a degree added or subtracted at some arbitrary place, the cyclone appears here and not there, and causes destructions in countries which it would have spared. If one had been aware of this tenth of a degree, one could have known it in advance, but the observations were not spaced closely enough, and were not precise enough, and this is why everything seems due to chance. Here again we find the same contrast between a tiny cause, that the observer cannot measure, and considerable effects, which may be appalling disasters.So, Poincaré knew about the butterfly effect 50 years before Lorenz. Lorenz’s contribution is not so much the knowledge of the Lorenz attractor (a somewhat abstract model for atmospheric convection) as it is the demonstration that with computers, meteorologists can progressively improve their modeling of the dynamics of the atmosphere.Instead of a serendipitous discovery giving birth to a new field of science out of the blue, I see the blooming of chaos theory as a consequence of the progress in mathematical, experimental, and computational techniques, which over several decades have given rise to a formidable self-organized multidisciplinary effort.Using the mathematical theory of dynamical systems developed after Poincaré and Jacques Hadamard, and based on their work, Floris Takens and I, for instance, showed that Landau’s quasi-periodic theory of turbulence was unstable and led to hyperbolic dynamics and “strange attractors.”11. D. Ruelle, F. Takens, Comm. Math. Phys. 20, 167 (1971). https://doi.org/10.1007/BF01646553 That was an early contribution to what was not yet called chaos theory.A fundamental problem Poincaré explicitly left open was that of the stability of the solar system. The problem was not solved by the discovery of homoclinic tangles, because they may involve only sets of measure zero. The Kolmogorov-Arnold-Moser theory gave the hope that one could prove the stability of the solar system. But delicate computational work by Jack Wisdom and Jacques Laskar in the 1980s finally proved instability and thus solved the important classical problems of stability22. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris (1899). (or long-term predictability). Laskar’s contributions in particular are chaos theory at its best:33. J. Laskar, Nature 338, 237 (1989). https://doi.org/10.1038/338237a0 They provide new views on the history of climates and other important geological questions.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. D. Ruelle, F. Takens, Comm. Math. Phys. 20, 167 (1971). https://doi.org/10.1007/BF01646553, Google ScholarCrossref2. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris (1899). Google ScholarCrossref3. J. Laskar, Nature 338, 237 (1989). https://doi.org/10.1038/338237a0, Google ScholarCrossref© 2014 American Institute of Physics.

Continuation and collapse of homoclinic tangles

By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The new bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. These bifurcation equations consist of a finite or infinite set of hilltop normal forms known from singularity theory. For the Henon family we determine numerically the connected components of branches with multi-humped homoclinic orbits that pass through several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.

Non-linear MHD simulation of ELM energy deposition

The mechanisms of edge-localized mode (ELM) energy deposition are studied by means of non-linear magnetohydrodynamic (MHD) simulation of ELMs. The footprint of the ELM heat flux at the divertor is found to increase approximately linearly with the total ELM energy loss for JET-scale plasmas, which is similar to the experimentally observed broadening of the ELM energy deposition with ELM energy loss. For these relatively large ELMs, in which conductive losses dominate, the divertor footprint broadening is due to an increase in the magnetic perturbation of the ballooning mode with increasing ELM energy loss, which results in a widening of the homoclinic tangles intersecting the target. The first results from ELM simulations in the ITER Q = 10 scenario indicate that on the ITER scale the broadening is similar for conductive and convective ELMs at least up to an ELM energy loss of 4 MJ. For the larger conductive-type ELMs the magnetic perturbation and its homoclinic tangles determine the pattern of the ELM heat flux at the divertor target similar to the JET-scale results. For the smaller convective ELMs, the ELM footprint is determined by the radial distance travelled by plasma filaments expelled by the ELM and the loss of the plasma energy in the filaments along the magnetic field lines.

Multifractality, stickiness, and recurrence-time statistics

We identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics (RTS), which is based on the concept of Poincaré recurrences. The projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near-integrable system, the annular billiard. These microstructures interfere in the effective transport of a particle in the phase space, which can be observed through RTS. This technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea.

Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations

We implement a semi-analytic scheme for numerically computing high order polynomial approximations of the stable and unstable manifolds associated with the fixed points of the normal form for the family of quadratic volume-preserving diffeomorphisms with quadratic inverse. We use this numerical scheme to study some hyperbolic dynamics associated with an invariant structure called a vortex bubble. The vortex bubble, when present in the system, is the dominant feature in the phase space of the quadratic family, as it encloses all invariant dynamics. Our study focuses on visualizing qualitative features of the vortex bubble such as bifurcations in its geometry, the geometry of some three-dimensional homoclinic tangles associated with the bubble, and the “quasi-capture” of homoclinic orbits by neighboring fixed points. Throughout, we couple our results with previous qualitative numerical studies of the elliptic dynamics within the vortex bubble of the quadratic family.

Non-linear MHD simulation of ELM energy deposition

The mechanisms for the broadening of the energy deposition footprint during ELMs are studied using non-linear MHD simulations of ELMs with the JOREK code. For conductive type ELMs it is found that the footprint broadening is due to the formation of perturbations in the magnetic field structure (homoclinic tangles) due to the magnetic component of the ballooning instability. The MHD simulations yield a scaling of the broadening with ELM size (and with the amplitude of the magnetic perturbation) similar to the experimentally observed trend. In simulations of ELMs in the convective ELM regime, in the ITER Q=10 scenario, the ELM footprint broadening is due to filaments being expelled from the plasma. As the filaments travel radially, the energy and density is lost in the parallel direction to the divertor and first wall.

PROBING THE LOCAL DYNAMICS OF PERIODIC ORBITS BY THE GENERALIZED ALIGNMENT INDEX (GALI) METHOD

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

Nonuniformly expanding 1D maps with logarithmic singularities

For a certain parametrized family of maps on a circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behaviour. This implies the existence of “chaotic” dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.

Chaotic escape from an open vase-shaped cavity. II. Topological theory

We present part II of a study of chaotic escape from an open two-dimensional vase-shaped cavity. A surface of section reveals that the chaotic dynamics is controlled by a homoclinic tangle, the union of stable and unstable manifolds attached to a hyperbolic fixed point. Furthermore, the surface of section rectifies escape-time graphs into sequences of escape segments; each sequence is called an epistrophe. Some of the escape segments (and therefore some of the epistrophes) are forced by the topology of the dynamics of the homoclinic tangle. These topologically forced structures can be predicted using the method called homotopic lobe dynamics (HLD). HLD takes a finite length of the unstable manifold and a judiciously altered topology and returns a set of symbolic dynamical equations that encode the folding and stretching of the unstable manifold. We present three applications of this method to three different lengths of the unstable manifold. Using each set of dynamical equations, we compute minimal sets of escape segments associated with the unstable manifold, and minimal sets associated with a burst of trajectories emanating from a point on the vase's boundary. The topological theory predicts most of the early escape segments that are found in numerical computations.

Dynamics of a Hysteretic Relay Oscillator with Periodic Forcing

The dynamics of a hysteretic relay oscillator with simple harmonic forcing is studied in this paper. Even though there are no bounded solutions in the absence of forcing, periodic excitation gives rise to more complex responses including periodic, quasi-periodic, and chaotic behavior. A Poincaré map is introduced to facilitate mathematical analysis. Families of period-one solutions are determined as fixed points of the Poincaré map. These represent coexisting subharmonic responses. Conditions on the amplitude and frequency of the forcing for the existence of periodic solutions have been obtained. Linear stability analysis reveals that these solutions can be classified as centers or saddles. The presence of higher periodic or quasi-periodic motions together with homoclinic and heteroclinic tangles implies the existence of chaotic solutions.

Complex dynamics associated with the appearance/disappearance of invariant closed curves

Abstract In this paper we study some global bifurcations arising in a heterogeneous financial model with fundamentalists and imitators. Such bifurcations which cause the appearance and disappearance of closed invariant curves (attracting or repelling) involve the stable and unstable sets of a saddle cycle with consequent changes in their dynamic behavior. Numerical investigations show that the transition between two qualitatively different regimes are characterized by the occurrence of homoclinic tangles with chaotic dynamics.