Dynamical systems $\leftrightarrow$ Babylon of terms

This section discusses some of the often used terms in combination with dynamical system analysis. Most of the terms might be well-known to the reader, but often several differing terms are used in literature to denote the same concept or object. To avoid possible confusion about these sometimes interchangeably used terms a clarifying survey is appropriate.

We start with operator $\nabla$, which is often used to define other important terms for the analysis of dynamical systems. It builds up a vector of the partial derivatives of its operand and is defined as shown in Eq. 3.3 [13]. If $\nabla$'s operand f(x) is a scalar function, then $\nabla{}f(\mathbf{x})$ is called the gradient of f [13]. If $\nabla$'s operand f(x) is a vector function, then $\nabla\mathbf{f}$ is the Jacobian matrix $\mathbf{J}=\partial\mathbf{f}/\partial\mathbf{x}$ of f(x) [19].

$$\nabla = \left( \begin{array}{c} \frac{\partial}{\partial\ma... ...mathbf{f}(\mathbf{x}) = \partial\mathbf{f}/\partial\mathbf{x}$$ (3.2)

An often used (scalar) term is the divergence of a flow $\mathrm{div}\,\mathbf{f}(\mathbf{x})$. It can be written as $\nabla\cdot\mathbf{f}(\mathbf{x})$ or as the trace $\mathit{Tr}$ of f's Jacobian $\nabla\mathbf{f}$ [13]:

$$\mathrm{div}\,\mathbf{f}(\mathbf{x}) = \nabla\cdot\mathbf{f}(... ...m_i \left( \partial\mathbf{f}/\partial\mathbf{x} \right)_{i,i}$$ (3.3)

The divergence basically describes the local amount of outgoing or incoming flow at a specific location of the dynamical system. It is 0, if the amount of incoming flow is equal to the amount of outgoing flow.

Another important term for the local analysis of dynamical systems is the rotation vector of a flow: $\mathrm{rot}\,\mathbf{f}(\mathbf{x})$ [61,75]. This attribute of a flow is often named vorticity instead of rotation and abbreviated by $\omega$ [29]. As a third term sometimes curl is used instead of rotation [29]. The vorticity/rotation/curl of a flow is defined as follows:

$$\omega = \mathrm{rot}\,\mathbf{f}(\mathbf{x}) = \mathrm{curl}\,\mathbf{f}(\mathbf{x}) = \nabla\!\times\!\mathbf{f}(\mathbf{x})$$ (3.4)

Vector $\mathrm{rot}\,\mathbf{f}(\mathbf{x})$ describes the rotation axis and its length the rotation velocity, which is given at state x. Note, that some references define the vorticity slightly different as $\omega=(1/2)\cdot\mathrm{rot}\,\mathbf{f}(\mathbf{x})$.

A scalar term related to the vorticity as defined above is the stream vorticity $\Omega$ [29,75]. It is the cosine of the angle enclosed by the vorticity vector and the flow vector f(x). This term characterizes the type of rotation in the system. If $\Omega$ is 1, the flow rotates around the flow vector f(x), whereas a value of 0 implies, that either there is no vorticity or the flow rotates in a plane which also contains the direction of the flow.

$$\Omega = \frac{\mathbf{f}\cdot\omega} {\left\vert\mathbf{f}\... ...\vert\!\cdot\!\left\vert\nabla\!\times\!\mathbf{f}\right\vert}$$ (3.5)

Just slightly different from the above definition is the specification of helicity [19]. Furthermore the helicity density H_d as given in the literature is just the same as helicity [67]. A value of 0 means exactly the same as no stream vorticity, but helicity increases proportional to the length of $\omega$ and f. It is defined by:

$$H_d = \Omega\cdot\left\vert\mathbf{f}\right\vert\cdot\left\ve... ...bf{f}\cdot\omega = \mathbf{f}\cdot(\nabla\!\times\!\mathbf{f})$$ (3.6)

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