Area Of Dynamical Systems Research Articles

Dynamische Systeme

This workshop continues a series of workshops whose current format originated in 1981 under then-organizers Moser and Zehnder, and whose latest iteration took place in July 2023. The general goal of this series of workshops is to discuss the latest developments in the field of dynamical systems, broadly construed, and its connections with neighboring areas of mathematics such as differential geometry, partial differential equations, and more recently contact and symplectic geometry. We continued this tradition, bringing in new participants working in areas of dynamical systems and its connections with other areas of mathematics that are currently highly active and/or showing great promise for future development. Key focus areas for the 2023 workshop include spectral rigidity for planar domains, chaotic and oscillatory motions in celestial mechanics, conformal symplectic dynamics, and relations between dynamics.he workshop by the grant DMS-2230648, “US Junior Oberwolfach Fellows”.

Chimera states of phase oscillator populations with nonlocal higher-order couplings

The chimera states underlying many realistic dynamical processes have attracted ample attention in the area of dynamical systems. Here, we generalize the Kuramoto model with nonlocal coupling incorporating higher-order interactions encoded with simplicial complexes. Previous works have shown that higher-order interactions promote coherent states. However, we uncover the fact that the introduced higher-order couplings can significantly enhance the emergence of the incoherent state. Remarkably, we identify that the chimera states arise as a result of multi-attractors in dynamic states. Importantly, we review that the increasing higher-order interactions can significantly shape the emergent probability of chimera states. All the observed results can be well described in terms of the dimension reduction method. This study is a step forward in highlighting the importance of nonlocal higher-order couplings, which might provide control strategies for the occurrence of spatial–temporal patterns in networked systems.

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry. This year special emphasis where laid on different kinds of spectra (in contact geometry, in Riemannian geometry, in dynamical systems and in symplectic topology).

A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces

The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is Topological Dynamics. Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area has been recently called Topology. As an illustration, some topological properties of the space of all transitive interval maps are described. For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we survey some results on two new notions: Space and Compactness. A Slovak space, as a dynamical analogue of a rigid space, is a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in the theory of dynamical systems and means the collection of states which attract this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness is that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

Neurocomputing approach to matrix product state using quantum dynamics

During the last three decades, quantum neural computation has received a relatively high amount of attention among researchers and academic communities since the model of quantum neural network has been proposed. Matrix product state is the well-designed class of tensor network states, which plays an important role in processing of quantum information. The area of dynamical systems help us to study the temporal behavior of systems in time. In our previous work, we have shown the relationship between quantum finite state machine and matrix product state. In this paper, we have used the proposed unitary criteria to investigate the dynamics of matrix product state with quantum weightless neural networks, where the output qubit is extracted and fed back (iterated) to input. Further, we have used Von Neumann entropy to measure possible entanglement of output quantum state. Finally, we have plotted the dynamics for each matrix product state against iterations and analyzed their results.

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser–Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry. This year special emphasis where laid on symplectic methods with applications to dynamics. The workshop was dedicated to the memory of John Mather, Jean-Christophe Yoccoz and Krzysztof Wysocki.

Designing architectures of convolutional neural networks to solve practical problems

The Convolutional Neural Network (CNN) figures among the state-of-the-art Deep Learning (DL) algorithms due to its robustness to support data shift, scale variations, and its capability of extracting relevant information from large-scale input data. However, setting appropriate parameters to define CNN architectures is still a challenging issue, mainly to tackle real-world problems. A typical approach consists in empirically assessing different CNN settings in order to select the most appropriate one. This procedure has clear limitations, including the choice of suitable predefined configurations as well as the high computational cost involved in evaluating each of them. This work presents a novel methodology to tackle the previously mentioned issues, providing mechanisms to estimate effective CNN configurations, including the size of convolutional masks (convolutional kernels) and the number of convolutional units (CNN neurons) per layer. Based on the False Nearest Neighbors (FNN), a well-known tool from the area of Dynamical Systems, the proposed method helps estimating CNN architectures that are less complex and produce good results. Our experiments confirm that architectures estimated through the proposed approach are as effective as the complex ones defined by empirical and computationally intensive strategies.

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry related to Hamiltonian dynamics.

KAM for PDEs

In the last years much progress has been achieved in the theory of quasi-periodic solutions of PDEs, that we shall call in a broad sense “KAM theory for PDEs”. Many new tools and ideas have been developed in this field (and are in current progress) establishing new links with other areas of dynamical systems (like normal forms) and PDE analysis (like micro-local analysis). We provide an overview to the state of the art in KAM theory for PDEs.

Global attractors for a third order in time nonlinear dynamics

Long time behavior of a third order (in time) nonlinear PDE equation is considered. This type of equations arises in the context of nonlinear acoustics [12,20,22,24] where modeling accounts for a finite speed of propagation paradox, the latter results in hyperbolic nature of the dynamics. It will be proved that the underlying PDE generates a well-posed dynamical system which admits a global and finite dimensional attractor. The main difficulty associated with the problem studied is the lack of Lyapunov function along with the lack of compactness of trajectories, which fact prevents applicability of standard tools in the area of dynamical systems.

How well do students in secondary school understand temporal development of dynamical systems?

Despite difficulties understanding the dynamics of complex systems only simple dynamical systems without feedback connections have been taught in secondary school physics. Consequently, students do not have opportunities to develop intuition of temporal development of systems, whose dynamics are conditioned by the influence of feedback processes. We present a research study on students' understanding of temporal development of simple dynamical systems. Students participating in the study were attending the final two years of a technical secondary school (gimnazija) program. Based on written equations for the rate of change of some quantity students had to qualitatively determine dynamical development. The study confirmed the initial hypotheses with regard to poor knowledge in the area of dynamical systems, irrelevant of year in secondary school or final grade in physics or mathematics. The results showed that most students understood the development of linear systems without feedback and based on equations, students were able to forecast the dynamical development of changing quantities. Issues arose in understanding systems with feedback connections that influence the nonlinear dynamical development of changing quantity. Especially with negative feedback connections that provide stabilization of changing quantities. The reason could be that in most of the cases they turned towards linear dynamical development of changing quantities. Frequently, they incorrectly concluded that the temporal development of changing quantities is the same to temporal development of current, which determines the state of quantity. As a response in overcoming such issues, we recommend a geometrical consideration of one-dimensional dynamical systems.

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry related to Hamiltonian dynamics. Highlights were the solution of a fifty year old problem in Arnold diffusion and a KAM-result on quasi-linear perturbations of the KdV-equation.

Dynamische Systeme

This workshop continued the biannual series at Oberwolfach on Dynamical Systems that started as the ”Moser-Zehnder meeting” in 1981. The main themes of the workshop are the new results and developments in the area of dynamical systems, in particular in Hamiltonian systems and symplectic geometry related to Hamiltonian dynamics. Highlights were new results on Arnold diffusion and a new approach to the study of Hamiltonian systems based on pseudoholomorphic curve methods.

$\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems

The optimal $\mathcal{H}_2$ model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal $\mathcal{H}_2$ approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov- and interpolation-based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation-based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for $\mathcal{H}_2$ model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based first-order necessary conditions for $\cHtwo$ optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.

SIGEST

If you are not familiar with dynamical systems in general or tippe top inversion in particular, do not read the paper in the SIGEST section of this issue, “Dissipation-Induced Heteroclinic Orbits in Tippe Tops” by N. Bou-Rabee, J. Marsden, and L. Romero. Just kidding, of course, but to be more precise: don't read it until after you consider taking a few seconds to view the short movie that you can find by clicking on the link “Tippe Top Inversion” at Dr. Bou-Rabee's webpage, www.acm.caltech.edu/~nawaf or directly by typing in the longer address http://epubs.siam.org/mm/SJADAY/030601351/6013501.mpg. Then you will be fully motivated to read and enjoy this fascinating paper. Tippe top inversion is a classical problem in applied mathematics. (For an extensive scientific and historical background on tippe tops, including a picture of Nobel Prize winning physicists Wolfgang Pauli and Niels Bohr playing with a tippe top, see http://www.fysikbasen.dk/English.php?page=Vis&id=79.) A tippe top is a sphere, or a large section of one, with a small cylindrical stem attached (see cover photo for an example). Counterintuitively, the spinning tippe top can defy gravity and for a period of time invert from spinning on its large spherical portion (the “noninverted” state) to spinning on its stem (the “inverted” state). This inversion is possible because the center of mass of the tippe top does not coincide with the center of the spherical portion, and because of the tradeoff between gravitational potential energy and rotational kinetic energy. The inversion is possible only in the presence of friction, which means that for a mathematical model to simulate tippe top inversion, friction must be included. The behavior of the tippe top is perhaps the best known example of “dissipation-induced instability,” and this type of instability is at the heart of the behavior analyzed in this paper. The term “heteroclinic orbit” refers to a path in phase space that connects the two equilibrium points of a dynamical system; a dissipation-induced heteroclinic orbit is one that exists due to dissipation. The key contribution of this paper is to show that the stabilities of the noninverted state and the inverted state of the tippe top may be studied using modified Maxwell–Bloch equations, which also are important in a number of other fields of applied mathematics, including nonlinear optics. The authors show that the conditions for the existence of the heteroclinic orbit that connects these two states are described completely by the modified Maxwell–Bloch equations. This result quickly has become influential in the area of dynamical systems. This paper originally was published as “Tippe Top Inversion as a Dissipation-Induced Instability” in the SIAM Journal on Applied Dynamical Systems, 3 (2004), pp. 352–377. For the SIGEST publication, the authors have revised the paper significantly, augmenting the introductory section to make the paper more accessible to a general audience, streamlining some of the analysis, and adding an interesting final section on related work. This paper offers a fascinating glance into the world of dynamical systems via a classical problem, and a toy that has fascinated children and Nobel Prize winners alike.

Modal interactions in self-excited oscillators under external primary resonance

Abstract The dynamics of two-degree-of-freedom oscillators including Rayleigh and Duffing type non-linearities is investigated. The method of multiple scales is first applied and a set of averaged equations is derived for cases of primary external resonance. These equations admit two types of constant solutions. The first type involves the directly excited mode only, while for some parameter combinations the non-linear damping terms excite the second mode also, even when no internal resonance is present. Stability and bifurcation analyses are then presented. Emphasis is placed on deriving explicit conditions on the system parameters that will lead to forms of the evolution equations which can be studied in detail, by utilizing results from the area of dynamical systems. To illustrate the effectiveness and accuracy of the analysis, numerical results are presented for a dynamic model of a specific practical system: namely, a three-parameter study is first carried over in order to reveal basic response features of a metal cutting system. Codimension one, two and three bifurcations are determined and their effect on the interaction and transition between the response modes is investigated. A representative sample of response diagrams and results from direct integration are also presented, providing an overall picture of the dynamics.