Lyapunov function as potential function: A dynamical equivalence

Wang

et al. 2013

Phys. Rev. E

Self Cite

Based on conventional Ito or Stratonovich interpretation, zero-mean multiplicative noise can induce shifts of attractors or even changes of topology to a deterministic dynamics. Such phenomena usually introduce additional complications in analysis of these systems. We employ in this paper a new stochastic interpretation leading to a straightforward consequence: The steady state distribution is Boltzmann-Gibbs type with a potential function severing as a Lyapunov function for the deterministic dynamics. It implies that an attractor corresponds to the local extremum of the distribution function and the probability is equally distributed right on an attractor. We consider a prototype of nonequilibrium processes, noisy limit cycle dynamics. Exact results are obtained for a class of limit cycles, including a van der Pol type oscillator. These results provide a new angle for understanding processes without detailed balance and can be verified by experiments.

“…[41] [the binary operator of two n-dimensional vectors is defined as x × y = (x i y j − x j y i ) n×n , and the result is an n × n matrix]:…”

Section: -3mentioning

confidence: 99%

“…The singularity problem for this construction has been discussed in Ref. [41]. Previous works focus more on the diffusion matrix, ignoring the important role played by the friction matrix S(q) and the Lorentz force matrix A(q).…”

Section: -3mentioning

confidence: 99%

Exploring a noisy van der Pol type oscillator with a stochastic approach

Wang

et al. 2013

Phys. Rev. E

Self Cite

“…If the drift term f(q) = Fq and any two eigenvalues λ i and λ j of the matrix F satisfying λ i + λ j 0, the existence and uniqueness of [S + A] (S and A are independent of state variable provided as a boundary condition) in the whole state space are guaranteed by a theorem on Lyapunov equation. 2,3 The condition λ i + λ j 0 is actually not essential as already demonstrated 4 and an explicit expression was obtained. Hence, the speculation on the general non-uniqueness by ZL is incorrect.…”

mentioning

confidence: 78%

Comment on “Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond” [J. Chem. Phys. 144, 094109 (2016)]

The Journal of Chemical Physics

Tang

2016

Self Cite

Connections between a “SDE decomposition” to other frameworks constructing landscape in non-equilibrium processes were discussed by Zhou and Li [J. Chem. Phys. 144, 094109 (2016)]. It was speculated that the SDE decomposition would not be generally unique. In this comment, we demonstrate both mathematically and physically that the speculation is incorrect and the uniqueness is guaranteed under appropriate conditions. A few related issues are also clarified, such as the limitation of obtaining potential function from steady state distribution. Current demonstration may lead to a better understanding on the structure and robustness of the decomposition framework.

“…The stochastic dynamical decomposition leads to a stochastic integration (A-type) which is different from traditional Ito's and Stratonovich's types. The approach adds a unique advantage connecting determinacy and stochasticity through dual roles of the potential functions [27,38]. Experiments confirm that some processes in nature do correspond to the A-type integration [27].…”

Section: Introductionmentioning

confidence: 89%

“…Let us look at this regulation when the dynamics of the metabolites x and the parameters V are uncoupled. Note that φ is in fact a Lyapunov function of the original network when V remains constant [37,38]. On the trajectory of x(t), as a property of Lyapunov function, φ monotonically decreases in time.…”

Section: Construction Of Stable Metabolic Dynamicsmentioning

confidence: 99%

Towards stable kinetics of large metabolic networks: Nonequilibrium potential function approach

Chen

et al. 2016

Phys. Rev. E

Self Cite

While the biochemistry of metabolism in many organisms is well studied, details of the metabolic dynamics are not fully explored yet. Acquiring adequate in vivo kinetic parameters experimentally has always been an obstacle. Unless the parameters of a vast number of enzyme-catalyzed reactions happened to fall into very special ranges, a kinetic model for a large metabolic network would fail to reach a steady state. In this work we show that a stable metabolic network can be systematically established via a biologically motivated regulatory process. The regulation is constructed in terms of a potential landscape description of stochastic and nongradient systems. The constructed process draws enzymatic parameters towards stable metabolism by reducing the change in the Lyapunov function tied to the stochastic fluctuations. Biologically it can be viewed as interplay between the flux balance and the spread of workloads on the network. Our approach allows further constraints such as thermodynamics and optimal efficiency. We choose the central metabolism of Methylobacterium extorquens AM1 as a case study to demonstrate the effectiveness of the approach. Growth efficiency on carbon conversion rate versus cell viability and futile cycles is investigated in depth.