Nonsmooth Systems Research Articles

Parallel Subdomain Synthesis of Cell Mapping for Capturing Global Invariant Sets in Higher-Dimensional Dynamical Systems

Unaffordable computational cost and memory storage induced by the curse of dimensionality has become the bottleneck of numerical methods in different fields. In the global analysis of nonlinear dynamical systems, the capability of numerical methods, like cell mapping methods, are mostly feasible only to a system dimension less than four. Although cell mappings are naturally parallelizable that may be used to greatly enhance the computational efficiency, it is still not enough to release the computational burden on a higher-dimensional system of greater than seven, not to mention the memory in dealing with millions of billion cells. In this paper, the subdomain synthesis method, which partitions the chosen region in state space into subdomains suitable for operating in a computational unit and then synthetizes the so-called virtual invariant sets to get the underlying global invariant sets, is promoted to be parallelizable on the subdomains so as to build a two-layer massively parallel architecture in both cell and subdomain levels. The proposed approach can be implemented by GPU Cluster that can maximize the powerful computation capability of hardwares. Examples with global invariant sets in very fine distances of a Jerk system and in bifurcations of a twelve-dimensional nonsmooth rotor system are presented for the first time to demonstrate then the feasibility of the proposed approach.

Process safety consequence modeling using artificial neural networks for approximating heat exchanger overpressure severity

One challenge in accounting for process safety incidents is that accurate modeling is complex, time-intensive, and requires many inputs. Process safety consequence modeling using first principles can be complicated. At the same time, setting up experiments is not always practical. This work proposes an artificial neural network (ANN) framework to predict process safety metrics to prevent overpressure during tube rupture scenarios with reasonable accuracy. Specifically, we apply a feed forward neural network to predict heat exchanger safety rating that is proportional to the heat exchanger pressure normalized with respect to the maximum allowable pressure. By training ANN to a set of tube rupture simulation data, we are able to bypasses the need for solving tedious dynamic and non-smooth system of equations. The ANN-based models yield safety rating predictions that comply with API 521 overpressure standards. We further demonstrate how these predictions can be used to perform real-time monitoring for a network of heat exchangers in a plant setting.

Multielement polynomial chaos Kriging-based metamodelling for Bayesian inference of non-smooth systems

This paper presents a surrogate modelling technique based on domain partitioning for Bayesian parameter inference of highly nonlinear engineering models. In order to alleviate the computational burden typically involved in Bayesian inference applications, a multielement Polynomial Chaos Expansion based Kriging metamodel is proposed. The developed surrogate model combines in a piecewise function an array of local Polynomial Chaos based Kriging metamodels constructed on a finite set of non-overlapping subdomains of the stochastic input space. Therewith, the presence of non-smoothness in the response of the forward model (e.g. nonlinearities and sparseness) can be reproduced by the proposed metamodel with minimum computational costs owing to its local adaptation capabilities. The model parameter inference is conducted through a Markov chain Monte Carlo approach comprising adaptive exploration and delayed rejection. The efficiency and accuracy of the proposed approach are validated through two case studies, including an analytical benchmark and a numerical case study. The latter relates the partial differential equation governing the hydrogen diffusion phenomenon of metallic materials in Thermal Desorption Spectroscopy tests.

Command filter‐based adaptive neural network controller design for uncertain nonsmooth nonlinear systems with output constraint

SummaryThis paper investigates the command filter‐based adaptive neural network tracking control problem for uncertain nonsmooth nonlinear systems. First, an integral barrier Lyapunov function is introduced to deal with the symmetric output constraint and make the output comply with prescribed restrictions. Second, by the Filippov's differential inclusion theory and approximation theorem, the considered nonsmooth nonlinear system is converted to an equivalent smooth nonlinear system. Third, the Levant's differentiator is used to deal with the “explosion of complexity” problem. An error compensation mechanism is established to attenuate the effect of the filtering error on control performance. Then, an adaptive neural network controller is set up by resorting to the backstepping technique. It is strictly mathematically proved that the tracking error can converge to an arbitrarily small neighborhood of the origin and all the signals in the closed‐loop system are semi‐globally uniformly ultimately bounded. Finally, a numerical example and an application example of the robotic manipulator system are provided to demonstrate the availability of the proposed control strategy.

The two-process model for sleep–wake regulation: A nonsmooth dynamics perspective

Since its inception four decades ago the two-process model introduced by Borbély has provided the conceptual framework to explain sleep–wake regulation across many species, including humans. At its core, high level notions of circadian and homeostatic processes are modelled with a low dimensional description in the form of a one dimensional nonautonomous and nonsmooth flow, with the rate of change of homeostatic sleep pressure switching at specific times. These events in time can be described by an implicit map from one switching time to another and have given rise to an elegant mathematical description of periodic orbits and their instabilities using the theory of iterated maps. In this paper we show that an equivalent description can be obtained from a direct analysis of the underlying nonsmooth flow. We further show how to construct the Lyapunov exponent of the nonsmooth flow and use this to uncover a more detailed picture of the Arnol’d tongue structure of the model.Given the growing interest in studying networks of sleepers, where interactions may occur continuously throughout the day–night cycle and not just at event times, we advocate for the future use of techniques from nonsmooth dynamical systems in studying networks of the two-process model.

Novel slow–fast behaviour in an oscillator driven by a frequency-switching force

When an oscillator switches abruptly between different frequencies, there is some ambiguity in deciding how the system should be modelled at the switch. Here we describe two seemingly natural models of a switch in a simple periodically-forced harmonic oscillator, which disagree starkly in their predictions of its long time behaviour. Attempting to resolve the disagreement by ‘regularizing’ the switch not only preserves the disagreement, but shows it increases with time. One of the models corresponds to a conventional ‘Filippov’ description of a nonsmooth system, while the second exhibits a structure that irreversibly ages, developing a number of novel multi-scale behaviours that we believe have not been reported before. These include slow–fast staircases, novel mixed-mode oscillations, and a synchronized canard explosion. These features are proven to exist using asymptotic analysis, but as they involve a slow–fast time-scale separation that increases with time, they lie beyond the reach of numerical methods.

Transient response prediction of randomly excited vibro-impact systems via RBF neural networks

Systems exposed to random excitations may fail long before stationarity is achieved, which necessitates consideration of the transient responses of the system. On the other hand, the transient response prediction of non-smooth systems may face more obstacles than that of smooth systems. One of the foremost challenges lies in the handling of vector fields with singularities in non-smooth systems. In this work, the radial basis function neural networks (RBFNN) is utilized for the first time to analyze the transient response of randomly excited vibro-impact systems (VIS), a class of typical non-smooth systems. The solution of the system response is expressed in terms of a serious of Gaussian activation functions (GAFs) with time-dependent weights. These time-dependent weights are determined by minimizing a loss function, which involves the residual of the differential equations and constraint conditions. To avoid the singularity of the initial condition being a Dirac delta function, a strategy of short-time Gaussian approximation is presented to obtain the initial weights. Two typical VISs exposed to stochastic excitations are presented to verify the suggested scheme. Appropriate comparisons to the data obtained by digital simulation show that the method yields reliable results even for strongly nonlinear systems. With the merits of high computational accuracy and satisfactory efficiency, this approach is expected to be an effective method for solution of random vibration problems of high-dimensional complex non-smooth systems.

A minimum-type nonlinear complementarity simulator with constrained pressure residual (CPR) methods for wormhole propagation in carbonate acidization

Due to the high computational complexity of wormhole propagation in carbonate acidization, nonphysical oscillations of the numerical solutions often appear, which seriously ruins the physical accuracy or even breaks down the whole process of reservoir flows at large-scale simulation. In this paper, we introduce and study a family of recently developed nonlinear complementarity formulations for handling the nonphysical oscillations. Typically, this bound-preserving process is modeled by the corresponding nonlinear partial differential equations (PDEs) along with the inequality restrictions as a non-smooth system of equations under the application of a minimum-type complementary function. Because of the non-smoothness of the nonlinear complementarity system, the nonlinear algebraic system after the fully implicit discretization is solved by using a semismooth Newton method combined with a generalized Jacobian matrix. To accelerate the convergence and enhance the robustness of the corresponding linear iterations, we employ an improved class of constrained pressure residual (CPR) preconditioners with different combinations of physics-based and domain decomposition methods. Experiments on two- and three-dimensional wormhole propagation problems are presented to demonstrate the applicability of the aforementioned algorithms. Large-scale reservoir simulation with more than one billion degrees of freedom is provided to show the algorithmic scalability by using tens of thousands of processors.

Bifurcation analysis of a non-smooth prey–predator model by a differential linear complementarity system

The aim of this paper is to investigate a prey–predator system with threshold harvesting for both species, whereas most researchers are devoted to studying a threshold policy for one population to avoid intractable mathematical simulation occurred by their coupling. Due to the equivalence between differential inclusion and differential complementarity system, we reformulate this ecosystem as a differential linear complementarity system. Then specialized algorithms for the complementarity problem allow us to handle such non-smooth structure, thereby performing a numerical examination of the dynamics and bifurcations of our proposed system with success. As a result, we can observe that this system exhibits many peculiar bifurcation patterns that are inherent to a non-smooth dynamic system, including boundary node bifurcation, pseudo-saddle–node bifurcation, touching bifurcation, and sliding homoclinic bifurcation. Specifically, we observe a multiple crossing bifurcation that results from the superposition of a sliding homoclinic bifurcation and a boundary saddle bifurcation. They are originated from the coupling of their own independent non-smooth structures of the two species. This system also admits some conventional bifurcations like saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, which are identified previously in a prey–predator system without threshold policy. Both the theoretical and numerical results indicate that the non-smooth structure of the threshold harvesting policy increases equilibria, as well as the dynamical complications of the system.

Bursting oscillations with bifurcations of chaotic attractors in a modified Chua’s circuit

This paper aims at the bursting oscillations as well as the dynamical mechanism in non-smooth systems. A fourth-order Chua’s circuit is modified to establish a non-smooth system by introducing a periodic exciting current and a piecewise nonlinear resistor. Under the condition that the dimensionless excitation frequency is far less than 1, the system is a slow–fast system, and five representative bursting patterns are observed via changing the excitation amplitude. By regarding the whole term of periodic exciting current as a bifurcation parameter, several kinds of smooth or non-smooth bifurcations are obtained in the fast subsystem via numerical or theoretic method. Particularly, a so-called saddle-focus-induced catastrophic bifurcation(SFIC), which is generated by collision between a non-smooth chaotic attractor and a saddle-focus, is observed. In addition, bifurcation diagrams and the associated maximal Lyapunov exponent spectrum are also obtained numerically, which can illustrate the scenario from periodic solutions to chaotic solutions via period-doubling bifurcations. Based on the bifurcation analysis, the generation mechanism of the bursting patterns is revealed. Research shows that the SFIC bifurcation in the fast subsystem may result in a sudden jumping behavior in the slow–fast system from a chaotic attractor to another. A spiking attractor may be constructed by chaotic attractors and limit cycle attractors together, which can lead to chaotic–periodic bursting oscillations. Slow passage effect induced by Hopf bifurcation can lead to delay phenomenon of other bifurcation in the slow–fast system.

Random vibration analysis of vibro-impact systems: RBF neural network method

The recent success of the radial basis function neural networks (RBFNN) method for random vibration analysis of smooth systems fuels in speculations that this approach may be extended to non-smooth problems. However, very little is known on the applicability of this approach to non-smooth systems. This work generalizes the RBFNN method to the randomly excited non-smooth vibro-impact system (VI-S). We first transform the non-smooth VI-S system to a continuous nonlinear system. Then, the solution of the reduced Fokker–Planck–Kolmogorov (FPK) equation for the transformed VI-S is expressed in terms of the RBFNN with Gaussian activation functions. The weights of the RBFNN are determined by solving an optimization problem to minimize the reduced FPK equation residual subjected to the constraint of the normalization condition. Three examples are presented to demonstrate the validity of the suggested scheme. Several remarks on the solution process also presented. All the results confirm the applicability and validity of the RBFNN method in dealing with the randomly excited non-smooth VI-S.

On the subcritical period doubling of a non-smooth network system by incremental harmonic balance method

This paper presents an investigation on an unusual bifurcation, i.e., subcritical period doubling (PD), of the non-smooth network system of a multi-modular floating structure. It is cumbersome to deduce the incremental harmonic balance iteration scheme, as non-smooth nonlinearities are resulted from the connectors between adjacent modules. To solve this problem, a method of undetermined coefficient is implemented plus a difference scheme. Periodic solutions are obtained accurately and a series of PDs are tracked as the controlling parameter varies. Interestingly, an unusual case is found to the Period-2 (with 2 times the fundamental period) solution, that is, the Period-4 solution appearing at the bifurcation point grows as the controlling parameter decreases backward. According to this feature, this case is called as subcritical PD compared to a traditional one. At the bifurcation point, the Period-2 solution loses stability featured by the leading Floquet multiplier leaving the unit cycle at +1, which is just opposite to −1 in the traditional cases. Differently, the arising Period-4 solution is unstable and it finally gains stability through cyclic bifurcations.

Double grazing bifurcations of the non-smooth railway wheelset systems

There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the grazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system

In this paper, a double pendulum model with multi-point collision is established to study the sub-harmonic bifurcation of high-dimensional coupled non-smooth systems. Considering the coupling and non-smoothness of the system, a two-step decoupling method is proposed to detect the sub-harmonic bifurcation of a two-degree-of-freedom non-smooth coupled system. The core view is to introduce energy-time scale transformation to overcome the obstacle of the system coupled term. In the first step, a reversible transformation is introduced to decouple the system. This transformation enables the coupled form of the impact term, which presents novel obstacles to the high-dimensional non-smooth system. By introducing energy-time scale transformation in the second step, the system is expressed as a smooth decoupling form of the energy coordinate, and the trouble of impact term coupled is solved. Furthermore, the sub-harmonic Melnikov function which depends on frequency, amplitude of excitation and impact recovery coefficient is derived by using the two-step decoupling method. Hence, the sub-harmonic Melnikov function is extended to the high-dimensional non-smooth system, which reveals the influence of the impact recovery coefficient on the existence of sub-harmonic periodic orbits. The innovation of this method is that it solves the coupled problem of non-smooth terms, quantifies the impact of impact recovery coefficient on the dynamic behavior of the system, and provides a theoretical basis for the actual parameter design and control in engineering. The obtained theoretical results are verified through the numerical simulations.

Birth of strange nonchaotic attractors in a piecewise linear oscillator.

Nonsmooth systems are widely encountered in engineering fields. They have abundant dynamical phenomena, including some results on the complex dynamics in such systems under quasiperiodically forced excitations. In this work, we consider a quasiperiodically forced piecewise linear oscillator and show that strange nonchaotic attractors (SNAs) do exist in such nonsmooth systems. The generation and evolution mechanisms of SNAs are discussed. The torus-doubling, fractal, bubbling, and intermittency routes to SNAs are identified. The strange properties of SNAs are characterized with the aid of the phase sensitivity function, singular continuous spectrum, rational frequency approximation, and the path of the partial Fourier sum of state variables in a complex plane. The nonchaotic properties of SNAs are verified by the methods of maximum Lyapunov exponent and power spectrum.

Boundary dynamics of a non-smooth memristive Hindmarsh-Rose neuron system.

In this paper, the complex dynamics of non-smooth Hindmarsh-Rose (HR) neuron system with local active memristors is discussed through the theory of discontinuous dynamical system. Six mapping structures through two boundaries are employed to illustrate the various switching behaviors of the system. The analytical conditions of the HR system for the switching motions, such as passable motion and grazing motion, are investigated, and the extreme multistability of the improved HR neuron is revealed through switching bifurcations, parameter maps, and attraction basins. The firing activities of various modes in the HR system and the coexistence of various firing behaviors are observed by changing the initial values. Furthermore, the hardware circuit implement of the HR system is established to validate the numerical results.

Complex Dynamics and Sliding Bifurcations of the Filippov Lorenz–Chen System

In this paper, we propose a Filippov switching model which is composed of the Lorenz and Chen systems. By employing the qualitative analysis techniques of nonsmooth dynamical systems, we show that the new Filippov system not only inherits the properties of the Lorenz and Chen systems but also presents new dynamics including new chaotic attractors such as four-wing butterfly attractor, Lorenz attractor with sliding segments, etc. In particular, we find that different new attractors can coexist such as the coexistence of two-point attractors and chaotic attractor, the coexistence of two-point attractors and quasi-periodic solution, the coexistence of transient transition chaos and quasi-periodic solution. Furthermore, nonsmooth bifurcations and numerical analyses reveal that the proposed Filippov system has a series of new sliding bifurcations including a symmetric pair of sliding mode bifurcations, a symmetric pair of sliding Hopf bifurcations, and a symmetric pair of Hopf-like boundary equilibrium bifurcations.

Non-smooth dynamics of Anti Stick–Slip Tool

This paper investigates non-smooth dynamics of an Anti Stick–Slip Tool (ASST) and its effects on a simulated downhole drilling. An ASST is a mechanical tool mounted on a Bottom Hole Assembly and designed to suppress stick–slip phenomena arising in drill-strings. In this paper the ASST is considered as a non-smooth dynamic system, where the presented modelling accounts for switching between Activated and Non-Activated states and proposes quantitative conditions for observing each state. In this tool, a helical spline couples axial and rotational movements acting together with an internal preloaded spring. The helical spline governs a relative motion of coupled parts and the preloaded spring determines a threshold value for activation of the ASST. The internal friction between coupled components contributes to conversion of excessive loads to a relative motion of two coupled parts. The proposed model describes the effect of internal friction forces on the operation of the tool whilst the tool is subject to nonlinear cutting forces from bit–rock interactions. Prescribed kinematics imposed on the tool allows to simplify the model of entire system and to provide an insight into the complex behaviour of the tool. Nonlinear responses of the developed model for a range of system parameters were numerically simulated to explore the dynamic behaviour of the ASST.

Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

Effects of Predator-Driven Prey Dispersal on Sustainable Harvesting Yield

Dispersal of organisms between patches is a common phenomenon in ecology and plays an important role in predator–prey population dynamics. We propose a nonsmooth Filippov predator–prey model in a two-patch environment characterized by a generalist predator-driven intermittent refuge protection of an apprehensive prey along with a balanced dispersal of the prey between refuge and nonrefuge areas. By employing qualitative techniques of nonsmooth dynamical systems, we see that the switching surface is a repeller whenever the interior equilibria are virtual, causing long-term population fluctuations. We find that the level of prey vigilance and the rate of prey dispersal play pivotal roles in the total harvesting yield. We observe that a sustainable high harvesting yield is possible when the prey is less vigilant and obtain the harvesting efforts for maximum sustainable total yield (MSTY). We further modify the model by considering a continuous threshold predator-driven prey dispersal and show that the model exhibits a Hopf bifurcation when the level of prey vigilance exceeds some critical threshold value. By comparing the dynamics of the two models we see that for a sustainable high harvesting yield of the system with continuous threshold dispersal, the prey needs to be highly vigilant compared to that of the system with intermittent dispersal of the prey. Further, we find numerically that the estimated MSTY from both models remains the same.