Discontinuous Dynamics Research Articles

Hidden-like Attractors in a Class of Discontinuous Dynamical Systems

In continuous dynamical systems, a hidden attractor occurs when its basin of attraction does not connect with small neighborhoods of equilibria. This research aims to investigate the presence of hidden-like attractors in a class of discontinuous systems that lack equilibria. The nature of non-smoothness in Filippov systems is critical for producing a wide variety of interesting dynamical behaviors and abrupt transient responses to dynamic processes. To show the effects of non-smoothness on dynamic behaviors, we provide a simple discontinuous system made of linear subsystems with no equilibria. The explicit closed-form solutions for each subsystem have been derived, and the generalized Poincaré maps have been established. Our results show that the periodic orbit can be completely established within a sliding region. We then carry out a mathematical investigation of hidden-like attractors that exhibit sliding-mode characteristics, particularly those associated with grazing-sliding behaviors. The proposed system evolves by adding a nonlinear function to one of the vector fields while still preserving the condition that equilibrium points do not exist in the whole system. The results of the linear system are useful for investigating the hidden-like attractors of flow behavior across a sliding surface in a nonlinear system using numerical simulation. The discontinuous behaviors are depicted as motion in a phase space governed by various hidden attractors, such as period doubling, period-m segments, and chaotic behavior, with varying interactions with the sliding mode.

Mixed Thermal and Renewable Energy Generation Optimization in Non-Interconnected Regions via Boolean Mapping

Global efforts aiming to shift towards renewable energy and smart grid configurations require accurate unit commitment schedules to guarantee power balance and ensure feasible operation under different complex constraints. Intelligent systems utilizing hybrid and high-level techniques have arisen as promising solutions to provide optimum exploration–exploitation trade-offs at the expense of computational complexity. To ameliorate this requirement, which is extremely expensive in non-interconnected renewable systems, radically different approaches based on enhanced priority schemes and Boolean encoding/decoding have to take place. This compilation encompasses various mappings that convert multi-valued clausal forms into Boolean expressions with equivalent satisfiability. Avoiding any need to introduce prior parameter settings, the solution utilizes state-of-the-art advancements in the field of artificial intelligence models, namely Boolean mapping. It allows for the efficient identification of the optimal configuration of a non-convex system with binary and discontinuous dynamics in the fewest possible trials, providing impressive performance. In this way, Boolean mapping becomes capable of providing global optimum solutions to unit commitment utilizing fully tractable procedures without deteriorating the computational time. The results, considering a non-interconnected power system, show that the proposed model based on artificial intelligence presents advantageous performance in terms of generating cost and complexity. This is particularly important in isolated networks, where even a-not-so great deviation between production and consumption may reflect as a major disturbance in terms of frequency and voltage.

The complicated behaviours of a novel model of smooth and discontinuous dynamics with quasi-zero-stiffness property

The complicated behaviours of a novel model of smooth and discontinuous dynamics with quasi-zero-stiffness property

Spike and hike modeling for interest rate derivatives: with an application to SOFR caplets

After the replacement of Libor with alternative reference rates across the world, market participants must now adjust to a variety of changes to both contract terms and market rate dynamics. Focusing mostly on the new SOFR rate in the US, this paper considers the interaction of two important characteristics of SOFR derivatives: the backward-looking settlement style of SOFR floating rate payments; and the historically ‘jagged’ nature of SOFR time series. We introduce mechanisms that allow us to modify existing term structure models to incorporate rich dynamics for both ‘surprising’ and anticipated spikes and hikes in overnight rates. Numerical tests on SOFR-style caplets show that such model enhancements can produce significant effects on implied caplet volatility levels and skews. Besides establishing a practitioner-friendly framework for derivatives pricing, our paper is broadly applicable to scenario generation and risk management in any market with discontinuous rates dynamics.

Online parameter identification based predictive pressure control for train electro-pneumatic braking systems with thermal effect

The electro-pneumatic braking system with ON/OFF solenoid valves has been widely used in trains due to its advantages and superiority. The undesirable impact of the thermal effect on the electro-pneumatic braking system leads to frequent valve switching, degradation of the pressure tracking performance and sometimes instability. This article presents an adaptive model predictive control approach to solve the pressure control problem under temperature uncertainty based on a switched unscented Kalman filter. First, a nonlinear switched dynamical model with the uncertain temperature parameter is derived for the electro-pneumatic braking system by comprehensively integrating its nonlinear, discontinuous dynamics and thermal effect. Using a switched unscented Kalman filter on the presented model of the system, the temperature parameter is accurately estimated to improve the model’s accuracy. Based on the corrected system model and the designed adaptive model predictive control method, the pressure tracking performance and the valves’ switchings of the electro-pneumatic braking system are improved, and the stability is guaranteed. The simulations and the experiments conducted for a braking system prototype confirm the performance validity of the proposed method.

Nonuniqueness Phenomena in Discontinuous Dynamical Systems and Their Regularizations

Nonuniqueness Phenomena in Discontinuous Dynamical Systems and Their Regularizations

Positioning adolescence in the developmental timeline.

Adolescence, the second decade of life, bridges childhood and adulthood, but also represents a host of unique experiences that impact health and well-being. Lifespan theories often emphasize the continuity of individual characteristics and their contexts from childhood to adolescence, underscoring the distal influence of childhood experiences. Yet, adolescence is marked by transitions that may provoke discontinuities, particularly within individuals, their contexts, and their interactions within those contexts. These discontinuities occur at varied times, orders, and intensities for individualyouth, suggesting that adolescence may be a developmental turning point where earlier life experiences may be mediated, reversed, or transformed by proximal events. This perspective piece emphasizes the importance of considering transitions, discontinuities, and developmental turning points in adolescence as well as their potential to explain heterogeneity in adolescent and adult outcomes. We explore one biological and one contextual transition in adolescence and highlight innovative theories and methods for investigating continuity and discontinuity dynamics across development, which could lead to new insights related to the adolescent period and its importance in shaping future life trajectories.

Assessing the dissipative capacity of particle impact dampers based on nonlinear bandwidth characteristics

The dissipative capacity as quantified by the nonlinear bandwidth measure of impulsively loaded linear primary resonators or primary structures (PSs) coupled to particle impact dampers (PIDs) is assessed. The considered PIDs are designed by initially placing different numbers of spherical, linearly viscoelastic granules at various 2D initial topologies and clearances. The strongly nonlinear and highly discontinuous dynamics of the PIDs are simulated via the discrete element method taking Hertzian interactions, slipping friction caused by granular rotations into account. An extended definition of nonlinear bandwidth is used to evaluate the energy dissipation capacity of the integrated PS-PID systems. To this end, the time-bandwidth (T-B) product is defined by nonlinear bandwidth in tandem with characteristic time. The T-B product is studied as a measure of the capacity of these systems to store or dissipate vibration energy. It is found that the initial topologies of the granules in the PID drastically affect the T-B product, which, depending on shock intensity, may break the classical limit of unity of linear time-invariant dissipative resonators. The optimal PS-PID systems composed of multiple granules produce large nonlinear bandwidths, indicating strong dissipative capacity of broadband input energy by the PIDs. Moreover, the granular collect-and-collide regime yields high nonlinear bandwidth and efficient energy dissipation capacity, whereas the opposite is observed for the granular gaseous state regime. The relationship between energy dissipation by the PID and nonlinear bandwidth of the PS are discussed, and it is found that as the shock intensity increases these two measures tend to vary similarly. The implications of these findings on the study of the dissipative capacity of the PS-PID system are discussed, yielding a predictive methodology for designing PIDs to act as highly effective nonlinear energy sinks capable of rapid and efficient suppression of vibration induced by shocks.

Comments on “Global leader-following consensus in finite time for fractional-order multi-agent systems with discontinuous inherent dynamics subject to nonlinear growth”

Despite the proposed results on the non-existing of finite time stable equilibria for fractional order systems (Shen and Lam, 2014), the commented paper (Wang et al., 2020) has shown finite time convergence of fractional order control systems via a couple of theorems. This comment demonstrates that the proofs of the given theorems in Wang et al. (2020) are incorrect.

Resonance response and chaotic analysis for an irrational pendulum system

We propose a novel rotating pendulum system with irrational nonlinearity, bi-stable phenomenon and quasi-zero stiffness characteristic, which bears a transition from standard double well dynamics to discontinuous double well dynamics by adjusting a geometric parameter. Nonlinear dynamical behaviors of the proposed irrational pendulum system subjected to viscus damping and periodic excitation are studied. To analytically demonstrate the primary resonance response for this perturbed irrational pendulum system, we introduce an approximate irrational system which is significant parallels with the original pendulum system from the perspective of qualitative analysis and quantitative calculation. Averaging method is applied to investigate the dynamic response of the perturbed irrational pendulum system with quasi-zero stiffness at origin. The effect of the internal and external parameters on the response curves are discussed. Numerical fitting technique and semi-analytical Melnikov method are used to detect the chaotic boundaries of the perturbed irrational pendulum system with two types of homoclinic orbits. Different types of periodic solutions and chaos are clarified in the perturbed irrational pendulum system by using numerical simulations. It is found that a class of chaos with oscillatory and rotational motions bears significant similarities to paroxysmal chaos due to sudden rotational motion and two chaotic motions formed by period-doubling bifurcation merge into one chaos.

A novel nonlinear oscillator consisting torsional springs and rigid rods

In this paper, we propose an archetypal double-winged novel nonlinear oscillator composed of torsional springs and rigid rods which behaves smooth and discontinuous dynamics and a kind of special collision pendulum depending on the varying of a geometrical parameter. The system behaves complicated equilibrium bifurcations with different geometrical states, leading to a series of buckling phenomenon and the loss of uniqueness near the limit points. A ‘collision’ parameter is introduced to determine the motions near the limit points which causes a discontinuous jump of velocity and exhibits the standard dynamics as an inverted pendulum, even though the mechanic model is rather different from a pendulum. We also distinguish the coexistence of periodic and chaotic motions and also a smooth period 1 located in a small region under the presence of damping and external force using amplitude–frequency analysis and numerical simulations. This research enriches the complicated dynamical behaviors of geometrical nonlinear oscillator and provide a novel structure which is of great potential in vibration isolation engineering.

TERM STRUCTURE MODELING OF SOFR: EVALUATING THE IMPORTANCE OF SCHEDULED JUMPS

As interest rate benchmarks move from LIBOR to overnight risk-free rates (RFR), it has become increasingly important for models to accurately capture the interest rate dynamics at the overnight tenor. Overnight rates closely track central bank policy rate decisions resulting, in highly discontinuous dynamics around scheduled meeting dates. In this paper, we construct a dynamic term structure model, which accounts for the discontinuous short-rate dynamics. We show that the model is able to jointly fit the overnight US policy rate, secured overnight financing rate (SOFR) and SOFR futures rates through the recent Fed hiking cycle. Comparing our model with a standard continuous time-homogeneous short-rate model, we find several indications that our model avoids the clear misspecification of the continuous model, in particular with regard to the short-rate dynamics around meeting dates of the Federal Open Market Committee (FOMC). Thiseffect begins to disappear as the term of the rates under consideration is increased, suggesting that diffusive dynamics are a reasonably accurate reflection of the evolution of market expectations embodied in longer-term interest rates.

ASYMPTOTIC EXPANSION OF THE SOLUTION FOR SINGULARPERTURBED LINEAR IMPULSIVE SYSTEMS

In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.

A pedagogical introduction to continuously monitored quantum systems and measurement-based feedback

In this manuscript we present a pedagogical introduction to continuously monitored quantum systems. We start by giving a simplified derivation of the Markovian master equation in Lindblad form, in the spirit of collision models and input-output theory, which describes the unconditional dynamics of a continuously monitored system. The same formalism is then exploited to derive stochastic master equations that describe the conditional dynamics. We focus on the two most paradigmatic examples of continuous monitoring: continuous photodetection, leading to a discontinuous dynamics with “quantum jumps”, and continuous homodyne measurements, leading to a diffusive dynamics. We then present a derivation of feedback master equations that describe the dynamics (either conditional or unconditional) when the continuous measurement photocurrents are fed back to the system as a linear driving Hamiltonian, a paradigm known as linear Markovian feedback. In the second part of the manuscript we focus on continuous-variable Gaussian systems: we first present the equations for first and second moments describing the dynamics under continuous general-dyne measurements, and we then discuss in more detail the conditional and unconditional dynamics under Markovian and state-based feedback.

Captive jump processes for bounded random systems with discontinuous dynamics

Stochastic captive jump processes are explicitly constructed in continuous time, whose non-linear dynamics are strictly confined by bounded domains that can be time-dependent. By introducing non-anticipative path-dependency, the framework offers the possibility of generating multiple inner tunnels within a master domain, such that a captive jump process is allowed to proceed either within a single inner tunnel or jump in between tunnels without ever penetrating the outermost shell. If a captive jump process is a continuous martingale or a pure-jump process, the uppermost confining boundary is non-decreasing, and the lowermost confining boundary is non-increasing. Under certain conditions, it can be shown that captive jump processes are invariant under monotonic transformations, enabling one to construct and study systems of increasing complexity using simpler building blocks. Amongst many applications, captive jump processes may be considered to model phenomena such as electrons transitioning from one orbit (valence shell) to another, quantum tunnelling where stochastic wave-functions can “penetrate” inner boundaries (i.e., walls) of potential energy, non-linear dynamical systems involving multiple attractors, and sticky concentration behaviour of pathogens in epidemics. We provide concrete, worked-out examples, and numerical simulations for the dynamics of captive jump processes within different geometries as demonstrations.

Friction modelling and the use of a physics-informed neural network for estimating frictional torque characteristics

This paper presents an exploration of friction modeling encompassing theoretical and practical aspects, utilizing a planar or 2D contact system. Various white-box friction models, including static and dynamic variants, are introduced, highlighting the superior capability of dynamic models in comprehensively capturing friction effects, substantiated through numerical simulation. Practical aspects of friction measurement and data-driven friction modeling are elucidated. The discourse extends to the development of grey-box and black-box friction models. A significant contribution lies in the proposition of a physics-informed neural network-based friction modeling approach, presenting it as an advanced and preferable alternative for friction estimation. To exemplify its efficacy, a case study of a torsion-based frictional contact scenario, employing Physics-Informed Neural Network (PINN) and the Nelder–Mead (N–M) algorithm for concurrent dynamics and friction model identification, was examined. Experimental data from a double torsion pendulum system, characterized by discontinuous dynamics, is employed for training. Results demonstrate the PINN’s superiority, providing more accurate representation of stick–slip phases at the contact zone and exhibiting faster performance compared to the N–M algorithm. The paper concludes by deliberating on challenges, prospects, and future directions in friction modeling.

Network Identification for Diffusively Coupled Networks With Minimal Time Complexity

The theory of network identification, namely identifying the (weighted) interaction topology among a known number of agents, has been widely developed for linear agents. However, the theory for nonlinear agents using probing inputs is far less developed, relying on dynamics linearization, and thus cannot be applied to networks with non-smooth or discontinuous dynamics. We use global convergence properties of the network, which can be assured using passivity theory, to present a network identification method for nonlinear agents. We do so by linearizing the steady-state equations rather than the dynamics, achieving a sub-cubic time algorithm for network identification. We also study the problem of network identification from a complexity theory standpoint, showing that the presented algorithms are optimal in terms of time complexity. We demonstrate the presented algorithm in two case studies with discontinuous dynamics.

Efficient Mode Transition Control for DM-PHEV With Mechanical Hysteresis Based on Piecewise Affine H ∞ Strategy

A dual motor plug-in hybrid electric vehicle (DM-PHEVs) can achieve higher power and better fuel economy through Mode Transition Process (MTP) from pure electric to hybrid driving modes. In a DM-PHEV, the MTP is more complex, with more components to be managed. As well as being a combination of a discrete stage transition and a system with continuous state evolution, the several actuators exhibit significant discontinuous dynamics and different characteristics from each other, particularly mechanical hysteresis. This makes the design of a coordinated controller challenging. In this paper, a two-layer coordinated control strategy is proposed. The upper layer is based on a stage-dependent piecewise-affine (PWA) model which is used to develop a PWA-static output feedback H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> strategy (PWA-SOF). The lower layer is based on a simplified actuator lag model, and a H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> design technique is used to develop a robust torque controller that reduces the effect of mechanical hysteresis. The resulting strategy is described as a piecewise-affine modified static output feedback (PW-MSOF) algorithm. (While the individual elements are not novel contributions to control theory, the combination and application to this problem is.) Performance indices are defined and hardware-in-the-loop (HiL) test shows that the new controller can effectively suppress the vehicle jerk without adversely affecting other aspects of system behaviour.

Effect of sequence pattern on conformation of DOPA-Peptide conjugate aggregates: a discontinuous molecular dynamics simulation study

ABSTRACT Underwater adhesives are critical for many applications, including marine coatings, sealants, and medical devices. Research on natural underwater adhesives has shown that L-3,4-dihydroxyphenylalanine (DOPA) and amyloid nanostructures are vital to their adhesive abilities. The fusion of DOPA-containing chains and amyloid-forming peptides creates a new space for designing underwater adhesives capable of multi-surface adhesion. One critical question for this design is the interplay between the DOPA and amyloid-forming peptide regions. Here we investigate the effect of the sequence pattern of DOPA-containing chains on the aggregation conformation of conjugates. Discontinuous molecular dynamics simulations were performed for fourteen DOPA-amyloid conjugates with different sequence patterns along the DOPA-containing portion. The amyloid-forming portion is represented by KLVFFAE from the Aβ42 peptide. The structural properties of the DOPA-amyloid conjugates are characterised by the percentages of ordered secondary structures and residue-residue contact maps. The results showed that certain patterns of DOPA and glycine in the DOPA-containing tail allowed the KLVFFAE portions of the conjugates to form distinct ordered β-sheets, and the DOPA-containing portion and the KLVFFAE portion of the conjugates to remain separated both within the same chain and amongst different chains. Among the designs, the most promising sequences are KLVFFAE-G-YYGYYGYY (where Y represents DOPA) and KLVFFAE-G-YYYYGGGG.

Observer-Based Fuzzy Adaptive Tracking Control of Nonlinear Strict-Feedback Systems via a Unique Event Triggering Approach

It is known that a controller designed via backstepping contains all virtual control signals constructed at the previous design steps. When this controller is used to generate an event-triggered control signal which is discontinuous, those virtual control signals are simultaneously triggered and the corresponding triggered virtual signals are utilized for the real control operation, they are also discontinuous. This fact results in the non-continuity of the virtual error dynamics. This problem is ignored in the existing backstepping-based event-triggered control schemes, in which control design and stability analysis are implemented according to the continuous virtual error dynamics, namely, all the virtual control signals are required to be continuous. Based on this observation, this paper focuses on the backstepping-based event-triggered fuzzy adaptive control for nonlinear strict-feedback systems with unavailable states. To do so, a concurrent event generator is first introduced, furthermore, the backstepping-based output feedback control design is given according to the discontinuous virtual error dynamics. Then, the Lyapunov approach for nonlinear impulsive systems is used to provide the closed-loop performance analysis. The results show that with the event-based fuzzy adaptive controller, all the closed-loop signals meet bounded requirements and the Zeno phenomenon is prevented. Ultimately, the proposed theoretical results are supported by two simulation examples.