Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations

We developed a potential flux landscape theory to investigate the dynamics and the global stability of a chemical Lorenz chaotic strange attractor under intrinsic fluctuations. Landscape was uncovered to have a butterfly shape. For chaotic systems, both landscape and probabilistic flux are crucial to the dynamics of chaotic oscillations. Landscape attracts the system down to the chaotic attractor, while flux drives the coherent motions along the chaotic attractors. Barrier heights from the landscape topography provide a quantitative measure for the robustness of chaotic attractor. We also found that the entropy production rate and phase coherence increase as the molecular numbers increase. Power spectrum analysis of autocorrelation function provides another way to quantify the global stability of chaotic attractor. We further found that limit cycle requires more flux and energy to sustain than the chaotic strange attractor. Finally, by detailed analysis we found that the curl probabilistic flux may provide the origin of the chaotic attractor.     

The potential and flux landscape theory of evolution

We established the potential and flux landscape theory for evolution. We found explicitly the conventional Wright's gradient adaptive landscape based on the mean fitness is inadequate to describe the general evolutionary dynamics. We show the intrinsic potential as being Lyapunov function(monotonically decreasing in time) does exist and can define the adaptive landscape for general evolution dynamics for studying global stability. The driving force determining the dynamics can be decomposed into gradient of potential landscape and curl probability flux. Non-zero flux causes detailed balance breaking and measures how far the evolution from equilibrium state. The gradient of intrinsic potential and curl flux are perpendicular to each other in zero fluctuation limit resembling electric and magnetic forces on electrons. We quantified intrinsic energy, entropy and free energy of evolution and constructed non-equilibrium thermodynamics. The intrinsic non-equilibrium free energy is a Lyapunov function. Both intrinsic potential and free energy can be used to quantify the global stability and robustness of evolution. We investigated an example of three allele evolutionary dynamics with frequency dependent selection (detailed balance broken). We uncovered the underlying single, triple, and limit cycle attractor landscapes. We found quantitative criterions for stability through landscape topography. We also quantified evolution pathways and found paths do not follow potential gradient and are irreversible due to non-zero flux. We generalized the original Fisher's fundamental theorem to the general (i.e., frequency dependent selection) regime of evolution by linking the adaptive rate with not only genetic variance related to the potential but also the flux. We show there is an optimum potential where curl flux resulting from biotic interactions of individuals within a species or between species can sustain an endless evolution even if the physical environment is unchanged. We offer a theoretical basis for explaining the corresponding Red Queen hypothesis proposed by Van Valen. Our work provides a theoretical foundation for evolutionary dynamics.     

Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: curvature, gauge field, and generalized fluctuation-dissipation theorem

The driving force of the dynamical system can be decomposed into the gradient of a potential landscape and curl flux (current). The fluctuation-dissipation theorem (FDT) is often applied to near equilibrium systems with detailed balance. The response due to a small perturbation can be expressed by a spontaneous fluctuation. For non-equilibrium systems, we derived a generalized FDT that the response function is composed of two parts: (1) a spontaneous correlation representing the relaxation which is present in the near equilibrium systems with detailed balance and (2) a correlation related to the persistence of the curl flux in steady state, which is also in part linked to a internal curvature of a gauge field. The generalized FDT is also related to the fluctuation theorem. In the equal time limit, the generalized FDT naturally leads to non-equilibrium thermodynamics where the entropy production rate can be decomposed into spontaneous relaxation driven by gradient force and house keeping contribution driven by the non-zero flux that sustains the non-equilibrium environment and breaks the detailed balance. On any particular path, the medium heat dissipation due to the non-zero curl flux is analogous to the Wilson lines of an Abelian gauge theory.     

Mathematical analysis of the smallest chemical reaction system with Hopf bifurcation

Recently we presented an up to now unstudied three-dimensional dynamical system which is, according to our given definition, the smallest chemical reaction system with Hopf bifurcation. We here study the Hopf bifurcation in detail and prove that near the bifurcation point a stable limit cycle arises. In the analysis we use the methods of local bifurcation theory, especially the center manifold and the normal form theorem. In a similar way we analyse the also occurring transcritical bifurcation. Besides studying local stability, we give the proofs for global stability of the trivial steady state in the whole positive phase space and for the nontrivial steady state in a closed domain containing the steady state point.     

介观化学体系中的动力学尺度效应

以生命和表面催化体系为对象,研究了介观化学体系中内涨落对体系非线性动力学行为的调控作用。内涨落可以诱导随机振荡,其强度在体系处于最佳尺度时会出现一个甚至多个极大值,并且在耦合体系中会得到进一步增强,表现为尺度共振效应、尺度选择效应和双重尺度效应,揭示了介观化学体系中尺度效应的新机制。     

介观化学体系中的动力学尺度效应

以生命和表面催化体系为对象,研究了介观化学体系中内涨落对体系非线性动力学行为的调控作用。内涨落可以诱导随机振荡,其强度在体系处于最佳尺度时会出现一个甚至多个极大值,并且在耦合体系中会得到进一步增强,表现为尺度共振效应、尺度选择效应和双重尺度效应,揭示了介观化学体系中尺度效应的新机制。     

Conditions for Mixed Mode Oscillations and Deterministic Chaos in Nonlinear Chemical Systems

An approach has been proposed for finding the conditions for the existence of mixed-mode oscillations and deterministic chaos in a kinetic scheme after reduction to a simple system of equations. Analysis of the position and stability of the steady states of this system suggested simple conditions for the existence of mixed-mode oscillations and deterministic chaos. The boundaries for monostability, bistability, and oscillations were also found. The results obtained were completely confirmed by numerical modelling.     

Wave patterns driven by chemomechanical instabilities in responsive gels

The first experimental evidence of a chemomechanical mechanism leading to morphogenetic instabilities is demonstrated experimentally. The system consists of a pH-responsive gel that swells at high pH and shrinks at low pH, and a bistable reaction system exhibiting an acid steady state (pH approximately 2) and an alkaline steady state (pH approximately 10). Within the gel, the steady state selection depends on the gel size. We show that in a constant and uniform nonequilibrium chemical environment, the responsive gel undergoes large amplitude dynamical deformations under the form of travelling contraction waves and complex spatio-temporal volume oscillations. These deformations are coupled to concentration patterns of protons. We present different sequences of dynamical behaviors observed under various controlled chemical conditions. A simple heuristic model is proposed to account for the observations. These experiments open a new route for pattern formation driven by chemical energy, in soft matter systems.     

Non-meanfield deterministic limits in chemical reaction kinetics

A general mechanism is proposed by which small intrinsic fluctuations in a system far from equilibrium can result in nearly deterministic dynamical behaviors which are markedly distinct from those realized in the meanfield limit. The mechanism is demonstrated for the kinetic Monte Carlo version of the Schnakenberg reaction where we identified a scaling limit in which the global deterministic bifurcation picture is fundamentally altered by fluctuations. Numerical simulations of the model are found to be in quantitative agreement with theoretical predictions.     

Isochronicity and limit cycle oscillation in chemical systems

Chemical oscillation is an interesting nonlinear dynamical phenomenon which arises due to complex stability condition of the steady state of a reaction far away from equilibrium which is usually characterised by a periodic attractor or a limit cycle around an interior stationary point. In this context Lienard equation is specifically used in the study of nonlinear dynamical properties of an open system which can be utilized to obtain the condition of limit cycle. In conjunction with the property of limit cycle oscillation, here we have shown the condition for isochronicity for different chemical oscillators with the help of renormalisation group method with multiple time scale analysis from a Lienard system. When two variable open system of equations are transformed into a Lienard system of equation the condition for limit cycle and isochronicity can be stated in a unified way. For any such nonlinear oscillator we have shown the route of a dynamical transformation of a limit cycle oscillation to a periodic orbit of centre type depending on the parameters of the system.     

Experimental study of the molecular reorientation induced by the ordinary wave in a nematic liquid crystal film

When an s-polarized laser beam impinges on a homeotropically aligned film of nematic liquid crystal at small incident angle, undamped oscillations of the molecular director may be produced. At high beam intensities the oscillations break up into deterministic chaos. Although this effect has been known for a long time and the route to chaos has been qualitatively analysed, no detailed study of the actual molecular director motion has been carried out. We have designed an experimental apparatus to monitor the dynamics of the molecular director, as described by its two polar angles. We observed different dynamical regimes, depending on the laser intensity: steady states, ocillations, rotation and, at the highest laser intensities, deterministic chaos. Moreover, the transitions between the oscillation and rotation regimes are characterized by intermittency.     

Energy Propagation Through a Protometabolism Leading to the Local Emergence of Singular Stationary Concentration Profiles

Living systems rely on chains of energy transfer from an energy source to maintain their metabolism. This task requires functionally identified components and organizations. However, propagation of a sustained energy flux through a cascade of reaction cycles has never been reproduced at a steady state in a simple chemical system. By using energy patterning and a diffusing hub reactant, we achieved the transfer of energy through an abiotic protometabolism. Patterned illumination was applied to a liquid solution of a reversible photoacid. It resulted in the local onset of a proton pump, which subsequently drove an extended reaction–diffusion cycle that involved pH‐sensitive reactants. Thus, light has been used for locally setting out of chemical equilibrium a reaction involving “blind” reactants. The spontaneous onset of an energy‐transfer chain notably drives the local generation of singular dissipative chemical structures; continuous matter fluxes are dynamically maintained at boundaries between spatially and chemically segregated zones, in the absence of any membrane or predetermined material structure.     

Temperature-induced route to chaos in the H2O2-HSO3(-)-S2O3(2-) flow reaction system

Low-frequency, high-amplitude pH-oscillations observed experimentally in the H2O2-HSO3(-)-S2O3(-) flow reaction system at 21.0 degrees C undergo period-doubling cascades to chemical chaos upon decreasing the temperature to 19.0 degrees C in small steps. Period-4 oscillations are observed at 20.0 degrees C and can be calculated on the basis of a simple model. A reverse transition from chaos to high-frequency limit cycle oscillations is also observable in the reaction system upon decreasing further the temperature step by step to 15.0 degrees C. Period-2 oscillations are measured at 18.0 degrees C. Such a temperature-change-induced transition between periodic and chaotic oscillatory states can be understood by taking into account the different effects of temperature on the rates of composite reactions in the oscillatory system. Small differences in the activation energies of the composite reactions are responsible for the observed transitions. Temperature-change-induced period doubling is suggested as a simple tool for determining whether an experimentally observed random behavior in chemical systems is of deterministic origin or due to experimental noise.     

非平衡定态化学反应体系对细致平衡偏离的不可逆性之随机热力学量度

在随机描述的层次上,成功地构筑了非平衡定态化学反应体系对细致平衡偏离的不可逆性之随机热力学判据.基于Fokker-Planck方程建立了连续Markov过程系统的随机熵平衡方程,发现在随机态空间中随机熵的时间变率亦可分为源项和流项两部分贡献.对于化学反应体系,作为态空间源项的随机熵产生可作为偏离细致平衡不可逆性的合适度量,此泛函量的零值标志着细致平衡.进一步借助按系统广度量的Ω展开法,通过对随机力及共轭的随机流的分析揭示态空间中的随机熵产生仅源于状态对细致平衡的偏离,并主要来自非Poisson涨落的贡献,因而可作为随机热力学量去判别和量度化学反应体系对细致平衡的偏离.     

Introduction to Chemical Oscillations Using a Modified Lotka Model

In teaching chemical kinetics most textbooks use the Lotka-Volterra Model to introduce the concept of chemical oscillations. Unfortunately, the Lotka-Volterra Model yields neutrally stable limit cycles for any initial conditions, which is a nonphysical property not observed in chemical kinetics. A more physical, two-variable model with simple linear stability analysis is, therefore, desirable. Here, we consider a Modified Lotka-Volterra Model that shows multiple physical steady states and both damped and stable oscillations. We can also study a stable node bifurcation to a saddle point and a stable node bifurcation to a stable limit cycle. This dynamically richer model can be analyzed through a simple linear stability analysis and numerical integration of the system of ordinary differential equations. Both methods, in particular the analytical analysis, are accessible to undergraduate students.     

Identification of the laminar-turbulent transition process in a plasma plume

This paper illustrates the complete dynamics of the laminar-turbulent transition process in a plasma plume using a simple measurement of the magnation-point heat flux correlated with acoustic, optical, and voltage drop fluctuations. In the laminar flow regime a steady jet is produced and the heat fluxes are accurately predicted hr laminar correlations. The initial stage of transition is characterized by the formation of axisymmetric structures and velocity fluctuations which increase the heat flux over laminar correlations. This is followed by a rapid decrease in heat flux as the vortex structures become more intense and rapidly entrain external air into the plume. The plume oscillations (acoustical and voltage drop) become most intense and are identical in frequency at the point of minimum heat flux. The transition is complete which the transition to small-scale turbulence in the exiting boundary layer which results in dec reased entrainment and increased heat flux.     

Entropy production in a mesoscopic chemical reaction system with oscillatory and excitable dynamics

Stochastic thermodynamics of chemical reaction systems has recently gained much attention. In the present paper, we consider such an issue for a system with both oscillatory and excitable dynamics, using catalytic oxidation of carbon monoxide on the surface of platinum crystal as an example. Starting from the chemical Langevin equations, we are able to calculate the stochastic entropy production P along a random trajectory in the concentration state space. Particular attention is paid to the dependence of the time-averaged entropy production P on the system size N in a parameter region close to the deterministic Hopf bifurcation (HB). In the large system size (weak noise) limit, we find that P ～ N(β) with β = 0 or 1, when the system is below or above the HB, respectively. In the small system size (strong noise) limit, P always increases linearly with N regardless of the bifurcation parameter. More interestingly, P could even reach a maximum for some intermediate system size in a parameter region where the corresponding deterministic system shows steady state or small amplitude oscillation. The maximum value of P decreases as the system parameter approaches the so-called CANARD point where the maximum disappears. This phenomenon could be qualitatively understood by partitioning the total entropy production into the contributions of spikes and of small amplitude oscillations.     

Artificial Supramolecular Pumps Powered by Light

The development of artificial nanoscale motors that can use energy from a source to perform tasks requires systems capable of performing directionally controlled molecular movements and operating away from chemical equilibrium. Here, the design, synthesis and properties of pseudorotaxanes are described, in which a photon input triggers the unidirectional motion of a macrocyclic ring with respect to a non-symmetric molecular axle. The photoinduced energy ratcheting at the basis of the pumping mechanism is validated by measuring the relevant thermodynamic and kinetic parameters. Owing to the photochemical behavior of the azobenzene moiety embedded in the axle, the pump can repeat its operation cycle autonomously under continuous illumination. NMR spectroscopy was used to observe the dissipative non-equilibrium state generated in situ by light irradiation. We also show that fine changes in the axle structure lead to an improvement in the performance of the motor. Such results highlight the modularity and versatility of this minimalist pump design, which provides facile access to dynamic systems that operate under photoinduced non-equilibrium regimes.     

Transmission coefficient calculation for proton transfer in triosephosphate isomerase based on the reaction path potential method

A global potential energy surface has been constructed through interpolation of our recently developed reaction path potential for chemical reactions in enzymes which is derived from combined ab initio quantum mechanical and molecular mechanical calculations. It has been implemented for the activated molecular dynamics simulations of the initial proton transfer reaction catalyzed by triosephosphate isomerase. To examine the dynamical effects on the rate constants of the enzymatic reaction, the classical transmission coefficient kappa(t) is evaluated to be 0.47 with the reactive flux approach, demonstrating considerable deviations from transition state theory. In addition, the fluctuations of protein environments have small effects on the barrier recrossing, and the transmission coefficient kappa(t) strongly depends on the fluctuations of atoms near the active site of the enzyme.     

外控弱周期电流约束下电极B-Z反应体系中的时间自组织

The coupled dynamical model consisting of the electrode B-Z system, and its related bulk phase B-Z system externally controlled by a periodical current is proposed on the basis of the Oregonator model. Dynamical behaviors of the electrode B-Z system, when subjected to external current constraint have been investigated systematically under the condition that the bulk phase is at a steady state. Furthermore, by means of the analytic method of slow manifold the regimes favorable to the appearance of limit-cycle oscillation have been determined on both the current~concentration of BrO3- and the current~model parameter plane respectively. The results are similar to those of experiments controlled externally by weak-periodical potential constraint, as reported in former works. It turns out that a limited cycle oscillatory regime degenerates under external periodical current constraint. Meanwhile, a kind of forced oscillations emerges in the regime where limit-cycle oscillations can t appear under constant current constraint.