The chaotic behaviour of piecewise smooth differential equations on two-dimensional torus and sphere

This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.     

Switched flow systems: pseudo billiard dynamics

We study a class of dynamical systems which generalizes and unifies some models arising in the analysis of switched flow systems in manufacturing. The general properties of these dynamical systems, called pseudo billiards, as well as some of their perturbations are discussed.     

非光滑碰撞振动系统动力学分析的Lyapunov指数判据

对n维非光滑(刚性约束和分段光滑)碰撞振动系统引进局部映射,利用Poincaré映射分析方法,建立了该类系统的Lyapunov指数谱与Floquet特征乘子之间的解析关系,提出了非光滑碰撞振动系统动力学分析的Lyapunov指数判据。以一类刚性约束的非线性碰撞振动系统为例,给出该系统的Lyapunov指数谱随参数大范围变化的规律,并将此规律与相应的Poincaré映射分岔图进行仔细对照,得到了一致的结论,验证了上述动力学分析的Lyapunov指数判据的正确性和有效性。     

Bow-tie architecture of gene regulatory networks in species of varying complexity

The gene regulatory network (GRN) architecture plays a key role in explaining the biological differences between species. We aim to understand species differences in terms of some universally present dynamical properties of their gene regulatory systems. A network architectural feature associated with controlling system-level dynamical properties is the bow-tie, identified by a strongly connected subnetwork, the core layer, between two sets of nodes, the in and the out layers. Though a bow-tie architecture has been observed in many networks, its existence has not been extensively investigated in GRNs of species of widely varying biological complexity. We analyse publicly available GRNs of several well-studied species from prokaryotes to unicellular eukaryotes to multicellular organisms. In their GRNs, we find the existence of a bow-tie architecture with a distinct largest strongly connected core layer. We show that the bow-tie architecture is a characteristic feature of GRNs. We observe an increasing trend in the relative core size with species complexity. Using studied relationships of the core size with dynamical properties like robustness and fragility, flexibility, criticality, controllability and evolvability, we hypothesize how these regulatory system properties have emerged differently with biological complexity, based on the observed differences of the GRN bow-tie architectures.     

Relative rotation number for stochastic systems: dynamical and topological applications

We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.     

A counterexample to the multifractal rigidity conjecture of piecewise linear Markov maps of the interval

A multifractal rigidity is the phenomena that the coincidence with the two multifractal spectra implies more strong relation between the given two dynamical systems. For the dimension spectrum of a Bernoulli measure defined on a repeller of a piecewise linear Markov map, Barreira, Pesin and Schmeling proved that a rigidity holds if the repeller is modelled by the full shift of two symbols. It was an open problem whether this rigidity holds in the case of three or more symbols. We give a negative answer to this problem by constructing a counter-example and we show that our counter-example is the unique one in the case of three symbols.     

两自由度干摩擦振动系统的粘滑运动分析

干摩擦结构广泛存在于机械系统中,由此引起的振动行为属于典型的非光滑动力学研究范畴。通过将含干摩擦非线性振动系统看作一分段线性系统,提出了求解各粘滑状态阶段解析解的方法,并以两自由度干摩擦振动系统的力学模型为例进行了数值模拟,进而分析了由干摩擦导致的粘滑振动行为。     

New variational principles for locating periodic orbits of differential equations

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes equations, simulated using the lattice Boltzmann equation.     

On the global stability of sets for impulsive differential systems by Lyapunov's direct method

In the present paper the question of global stability of sets of sufficiently general type with respect to systems of impulsive differential equations is considered

It is proved that the existence of piecewise continuous functions of the type of Lyapunov's functions with certain properties is a sufficient condition for various types of global asymptotic stability     

A dynamical system perspective on mathematical programming

We explore how to build a vector field from the various functions involved in a given mathematical programme, and show that equilibria of the corresponding dynamical system are precisely the solutions of the underlying optimality conditions for the original optimization problem. The general situation in which explicit inequality constraints are present is especially interesting as the vector field has to be discontinuous, and so one is led to consider discontinuous dynamical systems and their equilibria.     

Dynamical intricacy and average sample complexity

We propose a new way to measure the balance between freedom and coherence in a dynamical system and a new measure of its internal variability. Based on the concept of entropy and ideas from neuroscience and information theory, we define intricacy and average sample complexity for topological and measure-preserving dynamical systems. We establish basic properties of these quantities, show that their suprema over covers or partitions equal the ordinary entropies, compute them for many shifts of finite type, and indicate natural directions for further research.     

Modeling of complex dynamical behaviours using a state representation technique based on a vector ARMA approach

We present an original method for constructing linear stochastic models of state representation type able to describe the second-order random behaviour of complex mechanical systems submitted to random dynamical excitations. The situation is the following: the dynamical behaviour to investigate is fundamentally nonlinear and it expresses the dynamics of a system whose mechanical behaviour is complex, poorly known, and consequently hard, even impossible to model. The proposed method is based on the spectral linearization principle and on the use of a vector ARMA formulation. The application considered here concerns the two-dimensional nonlinear stationary dynamical response ξ=(ξ₁, ξ₂) of structural elements of a nuclear power plant. We observe a very good fitting between the target and the result obtained by the proposed method.     

Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity, which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network, and study how the frequency distributions of oscillatory modes of such a system depend on the average connectivity.     

The Plant Location and Technology Acquisition Problem

This paper presents an analytical approach for simultaneous optimization of the plant location, capacity acquisition and technology selection decisions in a multi-product environment. The proposed approach can be useful when there is considerable interaction between these structural decisions e.g., in global manufacturing companies. We present a formal definition of the plant location and technology acquisition problem and provide a mathematical model. We describe the analytical properties of the model, which lead to the development of a solution algorithm. Progressive piecewise linear underestimation constitutes the backbone of our solution algorithm. The arising subproblems are amenable to a dual-based approach. We report on a set of experiments that improved our understanding of the interaction among facility design decisions and showed that the computational performance of the proposed solution procedure is quite satisfactory.     

Self-organization and complexity: a new age for theory,computation and experiment

I first describe the notion of self-organization as a property of far-from-equilibrium nonlinear dissipative dynamical systems. Rather than describing such complex systems at a purely phenomenological level, however, I focus attention on the emergent nature of this complexity, by analysing a few examples of physical and physicochemical systems with simple underlying microscopic dynamics yet complex, self-organizing macroscopic properties. These include several mesoscopic models of fluid dynamics as well as a modern approach to nucleation and growth phenomena. Finally, I discuss how the advent of computational grids is set to provide a major boost to the study of such complex, self-organizing systems.     

An entropic characterization of protein interaction networks and cellular robustness.

The structure of molecular networks is believed to determine important aspects of their cellular function, such as the organismal resilience against random perturbations. Ultimately, however, cellular behaviour is determined by the dynamical processes, which are constrained by network topology. The present work is based on a fundamental relation from dynamical systems theory, which states that the macroscopic resilience of a steady state is correlated with the uncertainty in the underlying microscopic processes, a property that can be measured by entropy. Here, we use recent network data from large-scale protein interaction screens to characterize the diversity of possible pathways in terms of network entropy. This measure has its origin in statistical mechanics and amounts to a global characterization of both structural and dynamical resilience in terms of microscopic elements. We demonstrate how this approach can be used to rank network elements according to their contribution to network entropy and also investigate how this suggested ranking reflects on the functional data provided by gene knockouts and RNAi experiments in yeast and Caenorhabditis elegans. Our analysis shows that knockouts of proteins with large contribution to network entropy are preferentially lethal. This observation is robust with respect to several possible errors and biases in the experimental data. It underscores the significance of entropy as a fundamental invariant of the dynamical system, and as a measure of structural and dynamical properties of networks. Our analytical approach goes beyond the phenomenological studies of cellular robustness based on local network observables, such as connectivity. One of its principal achievements is to provide a rationale to study proxies of cellular resilience and rank proteins according to their importance within the global network context.     

An alternative numerical procedure for simulating the dynamical response of non-linear elastic rods

In this paper there is presented an alternative numerical procedure for obtaining approximations to non-linear conservation laws like those that describe the dynamical behaviour of elastic rods (composed of materials whose stress–strain relation is non-linear). The above-mentioned procedure consists of approximating the solution of the Riemann problem (associated with the considered conservation law) by a piecewise constant function (satisfying the jump conditions) and using Glimm's scheme for advancing in time, step by step. The proposed numerical approach eliminates the necessity of solving (in a complete way) the associated Riemann problem, easing and cheapening its computational implementation. This procedure is employed for simulating the dynamical response of an elastic-non-linear rod, fixed at its edges, that is left in a non-equilibrium state. There is presented a comparison between results obtained through a classical procedure and through the procedure proposed in this work.