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Register a new userHomoclinic Bifurcations of the Merging Strange Attractors in the Lorenz-like System
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In this article we construct the parameter region where the existence of a homoclinic orbit to a zero equilibrium state of saddle type in the Lorenz-like system will be analytically proved in the case of a nonnegative saddle value. Then, for a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the detected parameter region is carried out to discover several new interesting bifurcation scenarios.
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We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.
For every $rinmathbb{N}_{geq 2}cup{infty}$, we show that the space of ergodic measures is path connected for $C^r$-generic Lorenz attractors while it is not connected for $C^r$-dense Lorenz attractors. Various properties of the ergodic measure space for Lorenz attractors have been showed. In particular, a $C^r$-connecting lemma ($rgeq2$) for Lorenz attractors also has been proved. In $C^1$-topology, we obtain similar properties for singular hyperbolic attractors in higher dimensions.
A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can describe chaotic fluctuations. We identify a criterion, for the appearance of such non-linear terms. This depends on whether an invariant, symmetric tensor of the algebra can vanish or not. This proposal is studied in detail for the fundamental representation of $mathfrak{u}(2)$. We find a knotted structure for the attractor, a bimodal distribution for the largest Lyapunov exponent and that the dynamics takes place within the Cartan subalgebra, that does not contain only the identity matrix, thereby can describe the quantum fluctuations.
The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of emph{strange attractors}. However Lorenz research was mainly based on (non-rigourous) numerical simulations and, until recently, the proof of the existence of the Lorenz attractor remained elusive. To address that problem some authors introduced geometric Lorenz models and proved that geometric Lorenz models have a strange attractor. In 2002 it was shown that the original Lorenz model behaves like a geometric Lorenz model and thus has a strange attractor. In this paper we show that geometric Lorenz attractors are computable, as well as their physical measures.
We analyse a periodically-forced SIR model to investigate the influence of seasonality on the disease dynamics and we show that the condition on the basic reproduction number $mathcal{R}_0<1$ is not enough to guarantee the elimination of the disease. Using the theory of rank-one attractors, for an open subset in the space of parameters of the model for which $mathcal{R}_0<1$, the flow exhibits persistent strange attractors, producing infinitely many periodic and aperiodic patterns. Although numerical experiments have already suggested that periodically-forced SIR model may exhibit observable chaos, a rigorous proof was not given before. Our results agree well with the empirical belief that intense seasonality induces chaos. This should serve as a warning to all doing numerics (on epidemiological models) who deduce that the disease disappears merely because $mathcal{R}_0<1$.
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