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## Interior structural bifurcation of 2D symmetric incompressible flows

 1 Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey 2 Department of Mathematics, Marmara University, 34722 Istanbul, Turkey

* Corresponding author: Taylan Şengül

Received  May 2019 Published  February 2020

The structural bifurcation of a 2D divergence free vector field $\mathbf{u}(\cdot, t)$ when $\mathbf{u}(\cdot, t_0)$ has an interior isolated singular point $\mathbf{x}_0$ of zero index has been studied by Ma and Wang [23]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when $\mathbf{u}(\cdot, t_0)$ is anti-symmetric with respect to $\mathbf{x}_0$, or symmetric with respect to the axis located on $\mathbf{x}_0$ and normal to the unique eigendirection of the Jacobian $D\mathbf{u}(\cdot, t_0)$, the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when $\mathbf{u}(\cdot, t_0)$ has an interior isolated singular point $\mathbf{x}_0$ with index -1, 1. In particular, we show that if such a vector field with its acceleration at $t_0$ both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.

Citation: Deniz Bozkurt, Ali Deliceoğlu, Taylan Şengül. Interior structural bifurcation of 2D symmetric incompressible flows. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020032
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##### References:
The topological structure of ${\mathbf u}_0$ near the origin, under the assumptions: (a) (S2), (b) (S7), $k$ is even, $\alpha \lambda < 0$, (c) (S7), $k$ is even, $\alpha \lambda > 0$, (d) (S7), $k$ is odd
Structural bifurcation diagram of the unfolding of codimension-one singularities for the $\operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1$ case in Theorem 3.2: (a) $t = t_0+\epsilon$, (b) $t = t_0$, (c) $t = t_0 - \epsilon$
Structural bifurcation diagram of the unfolding of codimension-one singularities for the $\operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = 1$ case in Theorem 3.2: (a) $t = t_0+\epsilon$, (b) $t = t_0$, (c) $t = t_0 - \epsilon$
Schematic illustration of the unfolding of codimension-one singularities for flows with reflectional symmetry (18) in the $\operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1$ case. (a) $t = t_{0}-\epsilon$, (b) $t = t_{0}$, (c) $t = t_{0}+\epsilon$
Schematic illustration of the unfolding of codimension-one singularities for flows with anti-symmetry (19) in the $\operatorname{ind}( {\mathbf u}^0, {\mathbf x}_0) = -1$ case. (a) $t = t_{0}-\epsilon$, (b) $t = t_{0}$, (c) $t = t_{0}+\epsilon$
The illustration of the dimensionless boundary value problem
The streamlines patterns in rectangular cavities of various aspect ratios $A$ and various lid speed ratios $S$. (a) $(A, S) = (0.5, -1)$, (b) $(A, S) = (1, -1)$, (c) $(A, S) = (0.3, -1)$, (d) $(A, S) = (0.16, -1).$
The bifurcated solutions for $k=2$, $n=3$, $2 \lambda \lambda_1 + \alpha \lambda_2 > 0$
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta + 2 \lambda^2> 0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center $\alpha \beta + 2 \lambda^2< 0$ ${\mathbf x}_{0}(\epsilon)$ is a center ${\mathbf x}_{\pm}(\epsilon)$ are centers, ${\mathbf x}_{0}(\epsilon)$ is a saddle
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta + 2 \lambda^2> 0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center $\alpha \beta + 2 \lambda^2< 0$ ${\mathbf x}_{0}(\epsilon)$ is a center ${\mathbf x}_{\pm}(\epsilon)$ are centers, ${\mathbf x}_{0}(\epsilon)$ is a saddle
The bifurcated solutions for $k=2$, $n>3$, $2 \lambda \lambda_1 + \alpha \lambda_2 > 0$
 $\epsilon<0$ $\epsilon>0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center
 $\epsilon<0$ $\epsilon>0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center
The bifurcated solutions for $k>2$, $2k<n+1$, $\alpha \lambda_2 >0$
 $\epsilon<0$ $\epsilon>0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center
 $\epsilon<0$ $\epsilon>0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center
The bifurcated solutions for $k>2$, $2k>n+1$, $\alpha \lambda_2>0$
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta> 0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center $\alpha \beta< 0$ ${\mathbf x}_{0}(\epsilon)$ is a center ${\mathbf x}_{\pm}(\epsilon)$ are centers, ${\mathbf x}_{0}(\epsilon)$ is a saddle
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta> 0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center $\alpha \beta< 0$ ${\mathbf x}_{0}(\epsilon)$ is a center ${\mathbf x}_{\pm}(\epsilon)$ are centers, ${\mathbf x}_{0}(\epsilon)$ is a saddle
The bifurcated solutions for $k>2$, $2k=n+1$, $\alpha \lambda_2 > 0$
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta + k \lambda^2> 0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center $\alpha \beta + k \lambda^2< 0$ ${\mathbf x}_{0}(\epsilon)$ is a center ${\mathbf x}_{\pm}(\epsilon)$ are centers, ${\mathbf x}_{0}(\epsilon)$ is a saddle
 $\epsilon<0$ $\epsilon>0$ $\alpha \beta + k \lambda^2> 0$ ${\mathbf x}_{0}(\epsilon)$ is a saddle ${\mathbf x}_{\pm}(\epsilon)$ are saddles, ${\mathbf x}_{0}(\epsilon)$ is a center $\alpha \beta + k \lambda^2< 0$ ${\mathbf x}_{0}(\epsilon)$ is a center ${\mathbf x}_{\pm}(\epsilon)$ are centers, ${\mathbf x}_{0}(\epsilon)$ is a saddle
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