The investigation of chaos synchronization spans three decades, resulting in the development of several methods such as identical, phase, and generalized synchronization. However, conventional methods often fail to detect synchronized patterns in systems lacking fully unison dynamics. To address this limitation, delta synchronization of Poincaré chaos is introduced, aiming to explain partially synchronized patterns. This novel synchronization type is built upon unpredictability, a concept that reveals chaotic dynamics in systems based on characteristic time sequences—specifically, sequences of convergence and separation. The presence of unpredictability guarantees Poisson stable motion and sensitivity by examining a single trajectory (or single initial condition set) of a system. Numerically, delta synchronization captures the common characteristic time sequences of unpredictability in both coupled and uncoupled systems. This method is applied to various models in this thesis, encompassing distinctive dynamics defined by ordinary, partial, and delay differential equations. In the case of unidirectionally coupled gas-discharge semiconductor systems, the absence of generalized synchronization is noted. For Mackey-Glass delay systems, generalized synchronization occurs only after surpassing a well-known threshold. Importantly, delta synchronization is demonstrated to occur in regions where generalized synchronization is absent for these models. Additionally, the same phenomenon is observed in the uncoupled Hindmarsh-Rose neural network for noise intensity domains where identical synchronization is absent, yet delta synchronization exists. This model is constructed with Markovian noise, and noise-induced synchronization is investigated. In the domains of generalized and identical synchronization, a stronger form of delta synchronization—complete synchronization of unpredictability—is detected.

Starting from an SU(N) matrix quantum mechanics model with massive deformation terms and by introducing an ansatz configuration involving fuzzy four- and two-spheres with collective time dependence, we obtain a family of effective Hamiltonians, Hn,(N=16(n+1)(n+2)(n+3)) and examine their emerging chaotic dynamics. Through numerical work, we model the variation of the largest Lyapunov exponents as a function of the energy and find that they vary either as ∝(E−(En)F)1/4 or ∝E1/4, where (En)F stand for the energies of the unstable fixed points of the phase space. We use our results to put upper bounds on the temperature above which the Lyapunov exponents comply with the Maldacena-Shenker-Stanford (MSS) bound, 2πT, and below which it will eventually be violated.

We study the effects of addition of the Chern-Simons (CS) term in the minimal Yang-Mills (YM) matrix model composed of two 2 ×2 matrices with 𝑆⁢𝑈⁡(2) gauge and 𝑆⁢𝑂⁡(2) global symmetry. We obtain the Hamiltonian of this system in appropriate coordinates and demonstrate that its dynamics is sensitive to the values of both the CS coupling, 𝜅, and the conserved conjugate momentum, 𝑝𝜙, associated to the 𝑆⁢𝑂⁡(2) symmetry. We examine the behavior of the emerging chaotic dynamics by computing the Lyapunov exponents and plotting the Poincaré sections as these two parameters are varied and, in particular, find that the largest Lyapunov exponents evaluated within a range of values of 𝜅 are above what is computed at 𝜅 =0, for 𝜅⁢𝑝𝜙 <0. We also give estimates of the critical exponents for the Lyapunov exponent as the system transits from the chaotic to nonchaotic phase with 𝑝𝜙 approaching to a critical value.

In this thesis, chaotic dynamics emerging from Yang-Mills matrix models are investigated. Firstly, we investigate the Yang-Mills two-matrix models with ChernSimons term using both analytical and numerical methods. In particular, we obtain the Poincaré sections and Lyapunov exponents at several different values of the parameters of the model, revealing the detailed structure of the chaotic dynamics. In the second part of the thesis, we focus on a massive deformation of the bosonic part of the Banks-Fischler-Shenker-Susskind (BFSS) matrix model. Using an ansatz involving fuzzy-2 and fuzzy-4 sphere configurations we determine reduced effective Hamiltonians through which we study the emerging chaotic dynamics.

We focus on an SU(N) Yang-Mills gauge theory in 0 + 1-dimensions with
the same matrix content as the bosonic part of the BFSS matrix model, but with mass deformation terms breaking the global SO(9) symmetry of the latter to SO(5)×SO(3)×Z2.
Introducing an ansatz configuration involving fuzzy four and two spheres with collective
time dependence, we examine the chaotic dynamics in a family of effective Lagrangians
obtained by tracing over the aforementioned ansatz configurations at the matrix levels
N = 1/6 (n + 1)(n + 2)(n + 3), for n = 1, 2, · · · , 7. Through numerical work, we determine
the Lyapunov spectrum and analyze how the largest Lyapunov exponents(LLE) change as
a function of the energy, and discuss how our results can be used to model the temperature
dependence of the LLEs and put upper bounds on the temperature above which LLE values
comply with the Maldacena-Shenker-Stanford (MSS) bound 2πT , and below which it will eventually be violated.

We introduce a new type of chaos synchronization, specifically the delta synchronization of Poincaré chaos. The method is demonstrated for the irregular dynamics in coupled gas discharge-semiconductor systems (GDSSs). It is remarkable that the processes are not generally synchronized. Our approach entirely relies on ingredients of the Poincaré chaos, which in its own turn is a consequence of the unpredictability in Poisson stable motions. The drive and response systems are in the connection, such that the latter is processed through the electric potential of the former. The absence of generalized synchronization between these systems is indicated by utilizing the conservative auxiliary system. However, the existence of common sequences of moments for finite convergence and separation confirms the delta synchronization. This can be useful for complex dynamics generation and control in electromagnetic devices. A bifurcation diagram is constructed to separate stable stationary solutions from non-trivial oscillatory ones. Phase portraits of the drive and response systems for a specific regime are provided. The results of the sequential test application to indicate the unpredictability and the delta synchronization of chaos are demonstrated in tables. The computations of the dynamical characteristics for GDSSs are carried out by using COMSOL Multiphysics version 5.6 and MATLAB version R2021b.
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We explore the chaotic dynamics of the mass-deformed Aharony-Bergman-Jafferis-Maldacena model. To do so, we first perform a dimensional reduction of this model from 2+1 to 0 +1 dimensions, considering that the fields are spatially uniform. Working in the ’t Hooft limit and tracing over ansatz configurations involving fuzzy 2-spheres, which are described in terms of the Gomis–Rodriguez-Gomez–Van Raamsdonk–Verlinde matrices with collective time dependence, we obtain a family of reduced effective Lagrangians and demonstrate that they have chaotic dynamics by computing the associated Lyapunov exponents. In particular, we focus on how the largest Lyapunov exponent, 𝜆𝐿, changes as a function of 𝐸/𝑁2. Depending on the structure of the effective potentials, we find either 𝜆𝐿 ∝(𝐸/𝑁2)1/3 or 𝜆𝐿 ∝(𝐸/𝑁2−𝛾𝑁)1/3, where 𝛾𝑁⁡(𝑘,𝜇) are constants determined in terms of the Chern-Simons coupling 𝑘, the mass 𝜇, and the matrix level 𝑁. Noting that the classical dynamics approximates the quantum theory only in the high-temperature regime, we investigate the temperature dependence of the largest Lyapunov exponents and give upper bounds on the temperature above which 𝜆𝐿 values comply with the Maldacena-Shenker-Stanford bound, 𝜆𝐿 ≤2⁢𝜋⁢𝑇, and below which it will eventually be not obeyed.
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The delta synchronization is a useful method to analyze appearance of chaotic synchronization in gas discharge systems. In recent studies, the generalized synchronization method has been implemented in various gas discharge systems. However, synchronization is not detected with this conventional method. In our previous study, we introduced the delta synchronization method and applied it to the gas discharge-semiconductor system (GDSS) via the one-dimensional ‘simple’ fluid model approach. In the present study, we implement this method in the more detailed (in terms of plasma chemical reactions and treatment of the electron transport) fluid model, namely the ‘extended’ fluid model or ‘local mean energy approximation’ model. The description of the GDSS model is given, and a bifurcation diagram demonstrates the system’s transition to the chaotic regime. The unpredictable motion, which proves the existence of Poincaré chaos, and the delta synchronized motion are confirmed by the numerical simulations, and corresponding algorithms are given. The time sequences corresponding to the unpredictability and delta synchronization are presented in tables. For consistency, the absence of generalized synchronization is also shown via the auxiliary system approach. The numerical characteristics indicating the degrees of chaos and synchronization are described and implemented in the analysis. These features are also used to compare our results to the simpler model.

The paper explores noise-induced synchronization in uncoupled Hindmarsh–Rose neurons, introducing two distinctive elements: the application of Markovian noise and an analysis of synchronization via unpredictability. The noise is defined as an unpredictable and continuous process with characteristics proper for stochasticity. While identical synchronization is also investigated, the primary focus is to reveal synchronization in noise intensity domains that elude conventional detection methods, through delta synchronization within the neural system. Furthermore, a stronger form of synchronization, namely complete synchronization of unpredictability, is found to emerge in the domain with identical synchronization. The research findings are substantiated by numerical outcomes assessing unpredictability and synchronization, alongside comprehensive tables displaying characteristic time sequences for synchronization.

This study presents a novel concept in chaos synchronization, delta synchronization of chaos, which reveals the presence of chaotic models evolving in unison even in the absence of generalized synchronization. Building upon an analysis of unpredictability in Poincaré chaos, we apply this approach to unilaterally coupled time-delay Mackey–Glass models. The main novelty of our investigation lies in unveiling the synchronization phenomenon for a coupling constant below the synchronization threshold, an unattainable domain for conservative methods. Furthermore, we rigorously examine the coexistence of generalized synchronization and complete synchronization of unpredictability, which is a special case of delta synchronization, above the threshold. Therefore, the threshold is no longer a requirement for the synchronization of chaos in view of the present results. Additionally, transitions to fully chaotic regimes are demonstrated via return maps, phase portraits, and a bifurcation diagram. The findings are substantiated by tables and novel numerical characteristics.