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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.
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
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
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
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
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.