Smooth Dynamical Systems Research Articles

Bifurcation analysis of 3D-PWS systems with two transversal switching boundaries: A case study in power electronics

This paper addresses the nonlinear analysis of PWS dynamical systems with two transverse switching boundaries through a real case study: a cascade of two buck converters connected to a common bus in a dc microgrid. Two-parameter bifurcation analysis is proposed to study local and global dynamics on a power electronic circuit feeding a constant power load and controlled by means of a sliding mode control. This electronic circuit is modeled as a 3D piecewise smooth dynamical system with two transversal switching boundaries, having sliding motion in just one of them. Complete characterization of phenomena associated with the bifurcations induced by two transversal switching boundaries is one of the main contributions of this work. Numerical continuation methods are employed to obtain bifurcation sets and bifurcation diagrams to understand the dynamic behavior of the study system and also to verify the analytical results.

Pseudo-Bautin Bifurcation in 3D Filippov Systems

Consider a piecewise linear dynamical system in [Formula: see text], divided by a switching plane, with the Teixeira Singularity (TS). Under the antiparallelism hypothesis, it is known that the system undergoes the so-called pseudo-Hopf bifurcation (pH). In this paper, we consider a particular position of the antiparallel vectors to yield a two-parametric unfolding of the TS. The bifurcation analysis gives us, besides the curve of pH bifurcation points, a curve of saddle-node bifurcation points for crossing limit cycles. We call this phenomenon the pseudo-Bautin bifurcation, since it exhibits the exact same behavior as the Bautin bifurcation for smooth dynamical systems.

Two regularizations of the grazing-sliding bifurcation giving non equivalent dynamics

We present two ways of regularizing a one parameter family of piece-wise smooth dynamical systems undergoing a codimension one grazing-sliding global bifurcation of periodic orbits. First we use the Sotomayor-Teixeira regularization and prove that the regularized family has a saddle-node bifurcation of periodic orbits. Then we perform a hysteretic regularization and show that the regularized family has chaotic dynamics. Our result shows that, in spite that the two regularizations will give the same dynamics in the sliding modes, when a tangency appears the hysteretic process generates chaotic dynamics.

Transient behaviors and equilibria-analysis-based boundary crisis analysis in a smooth 4D dynamical system

This paper addresses the transient behavior mechanism analysis of a new chaotic system. Interesting transient behaviors of the new system are analyzed under the attractor merging crisis. The transient chaos and transient quasi-periodic behaviors are observed, under the boundary crisis, when the system has two stable equilibria. Furthermore, the reason of the two transient behaviors is revealed, with the conditions and relevant threshold for the boundary crisis are found, by introducing a simplified method of Lorenz time series. The lifetime of transient quasi-periodicity and parameters are found to meet the power-law distribution and have long memory, which indicates that the lifetime of transient quasi-periodicity is predictable. Therefore, the transient quasi-periodicity has potential values in the field of weak signal amplitude prediction.

Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle

This paper introduces class G containing Cartesian products of orientation-changing rough transformations of the circle and studies their dynamics. As it is known from the paper of A. G. Maier non-wandering set of orientation-changing diffeomorphism of the circle consists of 2q periodic points, where q is some natural number. So Cartesian products of two such diffeomorphisms has 4q1q2 periodic points where q1 corresponds to the first transformation and q2 corresponds to the second one. The authors describe all possible types of the set of periodic points, which contains 2q1q2 saddle points, q1q2 sinks, and q1q2 sources; 4 points from mentioned 4q1q2 periodic ones are fixed, and the remaining 4q1q2−4 points have period 2. In the theory of smooth dynamical systems, a very useful result is that, given a diffeomorphism f of a manifold, one can construct a flow on a manifold with dimension one greater; this flow is called the suspension over f. The authors introduce the concept of suspension over diffeomorphisms of class G, describe all possible types of suspension orbits and the number of these orbits. Besides that, the authors prove a theorem on the topology of the manifold on which the suspension is given. Namely, the carrier manifold of the flows under consideration is homeomorphic to the closed 3-manifold T2×[0,1]/φ, where φ:T2→T2. The main result of the paper says that suspensions over diffeomorphisms of the class G are topologically equivalent if and only if corresponding diffeomorphisms are topologically conjugate. The idea of the proof is to show that the topological equivalence of the suspensions ϕt and ϕ′t implies the topological conjugacy of ϕ and ϕ′.

Homoclinic Boundary-Saddle Bifurcations in Planar Nonsmooth Vector Fields

In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interactions with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection to cross or slide along the discontinuity. Even the simplest case of connection to a regular saddle, which hits a discontinuity as a parameter is varied, is surprisingly complex. In this paper, we construct bifurcation diagrams for nonresonant saddles in the plane, unfolding the homoclinic connection to a boundary saddle in a nonsmooth dynamical system. As an application, we exhibit such diagrams for a model of a forced pendulum.

Bifurcation and chaos in a smooth 3D dynamical system extended from Nosé-Hoover oscillator

The chaotic system with complicated dynamical behaviors has high potential in practical applications. This paper reports a simple smooth 3D dynamical system that is derived from the Nosé-Hoover oscillator and has rich dynamical behaviors, such as fold-Hopf bifurcation, saddle-node bifurcation, transient chaos, and conservative chaos (hidden attractors), which respectively correspond to four cases of equilibrium: infinitely many equilibria, two equilibria, four equilibria and no equilibrium. We first concentrate on the theoretical investigation of degenerate fold-Hopf and saddle-node bifurcations. Then, the power-law distribution of average lifetime of transient chaos is calculated and the coexistence of steady state and periodic motion after transient chaos is partially exhibited by time series. For the case of no equilibrium, we further study the conservative and ergodic dynamics, and hidden attractors with extremely rare properties by time-domain waveforms, phase portraits, histograms and Poincaré sections. Finally, the FPGA-based experimental results are presented and they are consistent with the numerical simulation results.

Novel bursting patterns and the bifurcation mechanism in a piecewise smooth Chua’s circuit with two scales

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise smooth resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation.

Moment Lyapunov exponent and stochastic stability of a vibro-impact system driven by Gaussian white noise

In this paper, the stochastic stability and pth moment Lyapunov exponent of a vibro-impact system driven by Gaussian white noise are studied. Firstly, the smooth stochastic dynamic system is given by introducing a non-smooth transformation. Thereafter, the first-order approximate solution of the pth moment Lyapunov exponent is obtained through applying the L.Arnold perturbation method, which are well consistent with those results calculated by the Monte Carlo simulation. Finally, the effects of the noise, the coefficient of restitution, the damping coefficient, and the natural frequency on the stochastic stability of the vibro-impact are discussed. Due to the existence of impact factor, the natural frequency has a significant effect on the stochastic stability of the system.

Computation of periodic orbits for piecewise linear oscillator by Harmonic Balance Methods

In this paper, Harmonic Balance based methods, namely Incremental Harmonic Balance Method and the method of Harmonic Balance with Alternating Frequency and Time traditionally used to compute periodic orbits of smooth nonlinear dynamical systems, are employed to investigate the dynamics of a non-smooth system, specifically a piecewise linear oscillator with a play. The Incremental Harmonic Balance Method was used to compute the period one orbits, including those exhibiting grazing and large impacts. The method of Harmonic Balance with Alternating Frequency and Time was implemented to calculate more complex orbits and multi stability. A good agreement between obtained approximate solutions and numerically calculated responses indicates robustness of the implemented HBMs, which should allow to effectively study the global dynamics of non-smooth systems.

Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method

<p style='text-indent:20px;'>This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. <b>200</b> words.</p>

A Note on the Accurate Computation of Structural Properties for Dynamic Control Systems

In analysing the observability, identifiability, and controllability of smooth (possibly non-linear) dynamic control systems, the authors identified numerous ways to improve both the accuracy and efficiency of their computations. We summarize a few techniques that help to pinpoint exactly what kind of linear dependencies exist between parametric output sensitivities. These relations, in turn, can be traced back with the help of computer algebra software to an exact re-parametrizations of the original model. The suggested techniques to increase the accuracy of results will be presented on the basis of a few examples.

Controlling coexisting attractors of a class of non-autonomous dynamical systems

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system’s trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information of the desired attractor. The behaviour of the control is studied numerically for both non-smooth and smooth dynamical systems, using a soft-impact and a Duffing oscillators as examples. Special attention is given to identify the parametric regions where the proposed control strategy is applicable by using path-following methods implemented via the continuation platform COCO. It is shown that the proposed control concept can be implemented through either using an external control input or varying a system parameter. Finally, extensive numerical results are presented to validate the proposed control methods.

Chains in 3D Filippov systems: A chaotic phenomenon

This work is devoted to the study of global connections between typical generic singularities, named T-singularities, in piecewise smooth dynamical systems. Such a singularity presents the so-called nonsmooth diabolo, which consists on a pair of invariant cones emanating from it.We analyze global features arising from the communication between the branches of a nonsmooth diabolo of a T-singularity and we prove that, under generic conditions, such communication leads to a chaotic behavior of the system. More specifically, we relate crossing orbits of a Filippov system presenting certain crossing self-connections to a T-singularity, with a Smale horseshoe of a first return map associated to the system. The techniques used in this work rely on the detection of transverse intersections between invariant manifolds of a hyperbolic fixed point of saddle type of such a first return map and the analysis of the Smale horseshoe associated to it.From the specific case discussed in our approach, we present a robust chaotic phenomenon for which its counterpart in the smooth case seems to happen only for highly degenerate systems.

Carnot metrics, dynamics and local rigidity

AbstractThis paper develops new techniques for studying smooth dynamical systems in the presence of a Carnot–Carathéodory metric. Principally, we employ the theory of Margulis and Mostow, Métivier, Mitchell, and Pansu on tangent cones to establish resonances between Lyapunov exponents. We apply these results in three different settings. First, we explore rigidity properties of smooth dominated splittings for Anosov diffeomorphisms and flows via associated smooth Carnot–Carathéodory metrics. Second, we obtain local rigidity properties of higher hyperbolic rank metrics in a neighborhood of a locally symmetric one. For the latter application we also prove structural stability of the Brin–Pesin asymptotic holonomy group for frame flows. Finally, we obtain local rigidity properties for uniform lattice actions on the ideal boundary of quaternionic and octonionic symmetric spaces.

On the Correspondence between Replicator Dynamics and Assignment Flows

AbstractAssignment flows are smooth dynamical systems for data labeling on graphs. Although they exhibit structural similarities with the well‐studied class of replicator dynamics, it is nontrivial to apply existing tools to their analysis. We propose an embedding of the underlying assignment manifold into the interior of a single probability simplex. Under this embedding, a large class of assignment flows are pushed to much higher‐dimensional replicator dynamics. We demonstrate the applicability of this result by transferring a spectral decomposition of replicator dynamics to assignment flows.

Isochronous Attainable Manifolds for Piecewise Smooth Dynamical Systems

Considering the concept of attainable sets for differential inclusions, we introduce the isochronous manifolds relative to a piecewise smooth dynamical systems in \(\mathbb {R}^{2}\) and \(\mathbb {R}^{3}\), and study how analytical and topological properties of such manifolds are related to sliding motion and to partially nodal attractivity conditions on the discontinuity manifolds. We also investigate what happens to isochronous manifolds at tangential exit points, where attractivity conditions cease to hold. In particular, we find that isochronous curves in \(\mathbb {R}^{2}\), which are closed simple curves under attractivity regime, become open curves at such points.

The Curvature Property of a Linear Dynamical System

In this work a two-dimensional smooth autonomous dynamical system is regarded as a three-dimensional Riemannian manifold and it is shown that the scalar curvature of a linear dynamical system $\text{d}x/\text{d}t=ax+by$, $\text{d}y/\text{d}t=cx+dy$ is non-positive. The manifold is scalar-flat iff $b=-c$ and $a=d=0$.

Existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle

Existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle

A survey on the modeling of hybrid behaviors: How to account for impulsive jumps properly

We propose an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. These classes of systems are chosen due to their dominant prevalence in physics, mathematics, and control engineering research communities. A comparison of these frameworks is provided and their applicability depending on the character of the hybrid behavior is discussed. In particular, we show that special care should be taken when equations with Dirac delta function are interpreted as impulsive differential equations. We also provide insights on the stability and attractivity analysis of hybrid behaviors, highlight their essential differences to the respective stability concepts for smooth dynamical systems, and discuss specific phenomena which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques.