From attractor to chaotic saddle: a tale of transverse instability

Ashwin, Peter; Buescu, Jorge; Stewart, Ian

doi:10.1088/0951-7715/9/3/006

Cited by 338 publications

(315 citation statements)

References 49 publications

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“…The figure suggests that the solution trajectory spirals within this subspace before being ejected in a direction normal to it. These ejections out of the invariant subspace appear to oceur intermittently, much as might be expected of dynamics associated with a blow-out bifurcation and the so-called in-out intermitteney [27,28]; similar behavior occurs in other partial differential equations with reflection-invariant subspaces such as those studied by Proctor and Lega [29] or Covas et al [30]. Also included in Fig.…”

Section: Dynamics In the General Casementioning

confidence: 70%

“…Note, however, that despite the chaotic motion about the A = B subspace both A(x,x) and B(x,x) spend sometimes quite long periods of time very cióse to reflection-invariant subspaces before escaping again. As already mentioned, behavior of this type may be associated with the onset of in-out intermitteney [27,28]. This behavior is not revealed unambiguously by time series shown in Fig.…”

Section: Dynamics In the General Casementioning

confidence: 73%

“…9c). Such sporadic excursions from the reflection-invariant subspaces are again likely to be the consequence of a blow-out bifurcation [27,28]. Fig.…”

Section: Dynamics In the General Casementioning

confidence: 97%

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Dynamics of counterpropagating waves in parametrically forced systems

Martel

Knobloch

Vega

2000

Physica D: Nonlinear Phenomena

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Parametrically driven waves in weakly dissipative systems with one extended dimensión are considered. Múltiple scale techniques are used to derive amplitude equations describing the interaction between counterpropagating waves. Dissipation, detuning and forcing are all assumed to be weak and any coupling to mean fields (such as large scale flows in fluid systems) is ignored. If the aspect ratio is moderately large the system is described by a pair of nonlocal equations for the (complex) amplitudes of the waves. The dynamics of these equations are studied both in annular and bounded geometries with lateral walls. The equations admit spatially uniform solutions in the form of standing waves and spatially nonuniform solutions with both simple and complex time-dependence. Transitions among these states are investigated as a function of the driving in three particular cases.

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Section: Dynamics In the General Casementioning

confidence: 70%

Section: Dynamics In the General Casementioning

confidence: 73%

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Dynamics of counterpropagating waves in parametrically forced systems

Martel

Knobloch

Vega

2000

Physica D: Nonlinear Phenomena

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“…None the less, the states appear to be linearly stable attractors (we verified stability numerically by computing Lyapunov exponents). We attribute this feature to the geometry of the phase space near these attractors, which must be unusually complicated [36]. y 4 , z 4 ).…”

Section: Time Dependent States For Small N ; Clustersmentioning

confidence: 98%

“…In those systems, an invariant, chaotic set loses stability in a transverse direction (a "blow-out" bifurcation [36]), and the system desynchronizes. An important feature of these systems is that the transverse stability is determined by some complicated average over this set.…”

Section: Preliminary Initial-value Problemsmentioning

confidence: 99%

The Lorenz–Fermi–Pasta–Ulam experiment

Balmforth

Pasquero

Provenzale

2000

Physica D: Nonlinear Phenomena

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We consider a chain of Lorenz '63 systems connected through a local, nearest-neighbour coupling. We refer to the resulting system as the Lorenz-Fermi-Pasta-Ulam lattice because of its similarity to the celebrated experiment conducted by Fermi, Pasta and Ulam. At large coupling strengths, the systems synchronize to a global, chaotic orbit of the Lorenz attractor. For smaller coupling, the synchronized state loses stability. Instead, steady, spatially structured equilibrium states are observed. These steady states are related to the heteroclinic orbits of the system describing stationary solutions to the partial differential equation that emerges on taking the continuum limit of the lattice. Notably, these orbits connect saddle-foci, suggesting the existence of a multitude of such equilibria in relatively wide systems. On lowering the coupling strength yet further, the steady states lose stability in what appear to be always subcritical Hopf bifurcations. This can lead to a variety of time-dependent states with fixed time-averaged spatial structure. Such solutions can be limit cycles, tori or possibly chaotic attractors. "Cluster states" can also occur (though with less regularity), consisting of lattices in which the elements are partitioned into families of synchronized subsystems. Ultimately, for very weak coupling, the lattice loses its time-averaged spatial structure. At this stage, the properties of the lattice are probably chaotic and approximately scale with the lattice size, suggesting that the system is essentially an ensemble of elements that evolve largely independent of one another. The weak interaction, however, is sufficient to induce widespread coherent phases; these are ephemeral states in which the dynamics of one or more subsystems takes a more regular form. We present measures of the complexity of these incoherent lattices, and discuss the concept of a "dynamical horizon" (that is, the distance along the lattice that one subsystem can effectively influence another) and error propagation (how the introduction of a disturbance in one subsystem becomes spread throughout the lattice).

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Enhancing synchronizability of dynamical networks using the connection graph stability method

Jalili

Rad

Hasler

2007

Circuit Theory & Apps

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Dynamical networks with diffusive couplings are investigated from the point of view of synchronizability. Arbitrary connection graphs are admitted but the coupling is symmetric. Networks with equal interaction coefficients for all edges of the interaction graph are compared with networks where the interaction coefficients vary from edge to edge according to the bounds for global synchronization obtained by the connection graph stability method. Synchronizability is tested numerically by establishing the time to decrease the synchronization error from 1 to 10 −5 in the case of networks of identical Lorenz or Rössler systems. Synchronizability from the point of view of phase synchronization is also tested for networks of non-identical Lorenz or Rössler systems. In this case the phase-order parameters are compared, as a function of the mean interaction strength. Throughout, as network structures, scale-free and Watts-Strogatz small-world networks are used. ENHANCING SYNCHRONIZABILITY OF DYNAMICAL NETWORKS 613Whether or not the network synchronizes depends on three causes:• The dynamics of the individual system, expressed by F(.) in Equation (1).• The type and strength of the interaction between the individual systems. In Equation (1) this is represented by the fact that the interaction is linear and the interaction matrix D is of diffusive type, and by the order of magnitude of the coefficients d i j . • The network structure, represented by the connection graph.

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From attractor to chaotic saddle: a tale of transverse instability

Cited by 338 publications

References 49 publications

Dynamics of counterpropagating waves in parametrically forced systems

Dynamics of counterpropagating waves in parametrically forced systems

The Lorenz–Fermi–Pasta–Ulam experiment

Enhancing synchronizability of dynamical networks using the connection graph stability method

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