Resonance bifurcations of robust heteroclinic networks

Robust heteroclinic cycles are known to change stability in resonance bifurcations, which occur when an algebraic condition on the eigenvalues of the system is satisfied and which typically result in the creation or destruction of a long-period periodic orbit. Resonance bifurcations for heteroclinic networks are more complicated because different subcycles in the network can undergo resonance at different parameter values, but have, until now, not been systematically studied. In this article we present the first investigation of resonance bifurcations in heteroclinic networks. Specifically, we study two heteroclinic networks in $\R^4$ and consider the dynamics that occurs as various subcycles in each network change stability. The two cases are distinguished by whether or not one of the equilibria in the network has real or complex contracting eigenvalues. We construct two-dimensional Poincare return maps and use these to investigate the dynamics of trajectories near the network. At least one equilibrium solution in each network has a two-dimensional unstable manifold, and we use the technique developed in [18] to keep track of all trajectories within these manifolds. In the case with real eigenvalues, we show that the asymptotically stable network loses stability first when one of two distinguished cycles in the network goes through resonance and two or six periodic orbits appear. In the complex case, we show that an infinite number of stable and unstable periodic orbits are created at resonance, and these may coexist with a chaotic attractor. There is a further resonance, for which the eigenvalue combination is a property of the entire network, after which the periodic orbits which originated from the individual resonances may interact. We illustrate some of our results with a numerical example.Comment: 46 pages, 20 figures. Supplementary material (two animated gifs) can be found on http://www.maths.leeds.ac.uk/~alastair/papers/KPR_res_net_abs.htm

Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

Bifurcations of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems

Towards Convectons in the Supercritical Regime: Homoclinic Snaking in Natural Doubly Diffusive Convection

Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. A variety of patterns can arise from these buoyancy-driven flows, including spatially localised states known as convectons, which consist of convective fluid motion localised within a background of quiescent fluid. We consider these states in a vertical slot with the horizontal temperature and solutal gradients providing competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, convectons have been studied with specific parameter values where the onset of convection is subcritical, and the states have been found to lie on a pair of secondary branches that undergo homoclinic snaking in a parameter regime below the onset of linear instability. In this paper, we show that convectons persist into parameter regimes in which the primary bifurcation is supercritical and there is no bistability, despite coexistence between the stable conduction state and large-amplitude convection. We detail this transition by considering spatial dynamics and observe how the structure of the secondary branches becomes increasingly complex owing to the increased role of inertia at low Prandtl numbers

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms below which there can be no persistent switching. Our results are illustrated by a numerical example

Near-onset dynamics in natural doubly diffusive convection

Doubly diffusive convection is considered in a vertical slot where horizontal temperature and solutal variations provide competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, the linear stability of the conductive state is known, but the convection patterns arising from the primary instability have only been studied for specific parameter values. We have extended this by determining the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers using a weakly nonlinear analysis. The resulting convection branches are extended using numerical continuation and we find large-amplitude steady convection states can coexist with the stable conduction state for sub- and supercritical primary bifurcations. The stability of the convection states is investigated and attracting travelling waves and periodic orbits are identified using time-stepping when these steady states are unstable

Chaos in the Takens-Bogdanov bifurcation with O(2) symmetry

The Takens–Bogdanov bifurcation is a codimension two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and traveling and standing waves in the nonlinear regime and predicts the presence of modulated traveling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated traveling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil’nikov type. Thus chaos is present arbitrarily close to the codimension two point