Alternatively, mixed mode patterns can appear near a Turing-Hopf bifurcation [ 66].
In our numerical simulations, we show that the system exhibits patterns like rhombic pattern, mixed-mode pattern, super-squares pattern, roll, and hexagonal pattern.
Finally, numerical simulations are performed to illustrate the weakly nonlinear theoretical predictions and through these simulations some patterns (single mode pattern, mixed-mode pattern, super-squares pattern, roll pattern, hexagonal pattern) are found.
Therefore, the predicted asymptotic solution given by the positive equilibrium (E_{3}^) is the following mixed-mode pattern: {mathbf {w}}=varepsilon{boldsymbol{varrho}} bigl( A_{1infty}cos (x)cos(0.5y)+A_{2infty}cos 0.5x cos y) bigr) +O bigl epsilon^{2}bigr), (4.33) where (A_{1infty}) and (A_{2infty}) are the coordinates of the positive equilibrium (E_{3}^).
Thus, we shall distinguish two kinds of stationary patterns: (i): a single mode stationary pattern when any of the equilibria (E_{1}^{pm}) or (E_{2}^{pm}) is stable; (ii): a mixed-mode stationary pattern when any of the equilibria (E_{3}^{pm}) is stable.
Miura et al. [47] used a mixed-mode Turing pattern to explain the Doublefoot mutant mouse limb, and Turing models of the development of Arabidopsis trichomes on the leaf blade were recently tested against a suite of mutants and transgenic lines [48], [49].
The bi-material characteristics of film systems make the nature of interfacial cracks as mixed mode, with cracks exhibiting various complex patterns such as telephone cord blisters.
We propose that Nasonia uses a mixed mode of segmentation wherein pair-rule genes pattern the embryo in a manner resembling Drosophila at the anterior and ancestral Tribolium at the posterior.
The three-dimensional pattern of crack opening shows substantial mixed mode I III contact on the faces of asperities behind the crack tip, even at the maximum load of the fatigue cycle.
In addition, the study illustrates that a heterogeneous distribution of fast K+ conductances could cause heterogeneous lactotroph firing patterns". "We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting.
Principles, characteristics and design of ligands for mixed mode chromatography have been discussed previously [ 74].
