Computing of B-series by automatic differentiation
Ferenc A. Bartha Hans Z. Munthe-Kaas
Discrete & Continuous Dynamical Systems - A 2014, 34(3): 903-914 doi: 10.3934/dcds.2014.34.903
We present an algorithm based on Automatic Differentiation for computing general B-series of vector fields $f\colon \mathbb{R}^n\rightarrow \mathbb{R}^n$. The algorithm has a computational complexity depending linearly on $n$, and provides a practical way of computing B-series up to a moderately high order $d$. Compared to Automatic Differentiation for computing Taylor series solutions of differential equations, the proposed algorithm is more general, since it can compute any B-series. However the computational cost of the proposed algorithm grows much faster in $d$ than a Taylor series method, thus very high order B-series are not tractable by this approach.
keywords: elementary differentials automatic differentiation multivariate derivatives B-series rooted trees.
Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model
Ferenc A. Bartha Ábel Garab
Journal of Computational Dynamics 2014, 1(2): 213-232 doi: 10.3934/jcd.2014.1.213
We consider the global asymptotic stability of the trivial fixed point of the difference equation $x_{n+1}=m x_n-\alpha \varphi(x_{n-1})$, where $(\alpha,m) \in \mathbb{R}^2$ and $\varphi$ is a real function satisfying the discrete Yorke condition: $\min\{0,x\} \leq \varphi(x) \leq \max\{0,x\}$ for all $x\in \mathbb{R}$. If $\varphi$ is bounded then $(\alpha,m) \in [|m|-1,1] \times [-1,1]$, $(\alpha,m) \neq (0,-1), (0,1)$ is necessary for the global stability of $0$. We prove that if $\varphi(x) \equiv \tanh(x)$, then this condition is sufficient as well.
keywords: neural networks. Global stability Neimark–Sacker bifurcation rigorous numerics strong resonance graph representations interval analysis

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