NETWORK AS A CHAOTIC DYNAMICAL SYSTEM

2007 ◽  
Vol 17 (10) ◽  
pp. 3529-3533 ◽  
Author(s):  
SYUJI MIYAZAKI ◽  
YASUSHI NAGASHIMA
Keyword(s):  
Dynamical System ◽  
Network Structure ◽  
Transition Matrix ◽  
Chaotic Dynamics ◽  
Large Deviation ◽  
Piecewise Linear ◽  
Directed Network ◽  
Chaotic Dynamical System ◽  
One Dimensional ◽  
Finite Time Lyapunov Exponents

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.

HORSESHOE CHAOS IN A HYBRID PLANAR DYNAMICAL SYSTEM

2012 ◽  
Vol 22 (08) ◽  
pp. 1250202 ◽  
Author(s):  
QING-JU FAN
Keyword(s):  
Dynamical System ◽  
Cross Section ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Buck Converter ◽  
Two Dimensional ◽  
Topological Horseshoe ◽  
Piecewise Linear System ◽  
Shift Map

In this paper, we study the chaotic dynamics of a voltage-mode controlled buck converter, which is typically a switched piecewise linear system. For the two-dimensional hybrid system, we consider a properly chosen cross-section and the corresponding Poincaré map, and show that the dynamics of the system is semi-conjugate to a 2-shift map, which implies the chaotic behavior of this system. The essential tool is a topological horseshoe theory and numerical method.

scholarly journals New Chaotic Oscillator Derived from Class C Single Transistor-Based Amplifier

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Jiri Petrzela
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
High Frequency ◽  
Piecewise Linear ◽  
Chaotic Oscillator ◽  
Chaotic Dynamical System ◽  
Transient Behaviour ◽  
Local Polynomial ◽  
Good Accordance ◽  
Class C

This paper describes a new autonomous deterministic chaotic dynamical system having a single unstable saddle-spiral fixed point. A mathematical model originates in the fundamental structure of the class C amplifier. Evolution of robust strange attractors is conditioned by a bilateral nature of bipolar transistor with local polynomial or piecewise linear feedforward transconductance and high frequency of operation. Numerical analysis is supported by experimental verification and both results prove that chaos is neither a numerical artifact nor a long transient behaviour. Also, good accordance between theory and measurement has been observed.

ANTIMONOTONICITY AND CHAOTIC DYNAMICS IN A FOURTH-ORDER AUTONOMOUS NONLINEAR ELECTRIC CIRCUIT

2000 ◽  
Vol 10 (08) ◽  
pp. 1903-1915 ◽  
Author(s):  
I. M. KYPRIANIDIS ◽  
I. N. STOUBOULOS ◽  
P. HARALABIDIS ◽  
T. BOUNTIS
Keyword(s):  
Fourth Order ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Electric Circuit ◽  
State Variables ◽  
One Dimensional ◽  
Nonlinear Resistor ◽  
Negative Conductance ◽  
Theoretical Considerations

In this paper we study the dynamics of a fourth-order autonomous nonlinear electric circuit with two active elements, one linear negative conductance and one nonlinear resistor with a symmetrical piecewise-linear v–i characteristic. Using the capacitances C1 and C2 as the control parameters, we observe the phenomenon of antimonotonicity and the formation of "bubbles" in the development of bifurcations, resulting typically in reverse period-doubling sequences. We also find a crisis-induced intermittency, when the spiral attractor suddenly widens to a double-scroll attractor. We have plotted several bifurcation diagrams of reverse period-doubling sequences and computed the scaling parameter δ versus the control parameter C2 for the different regimes, where bubbles evolve. Thus, besides the usual Feigenbaum constant δ → δF = 4.6692…, we also observe, in some cases, a convergence of δ to [Formula: see text], as expected from theoretical considerations. Finally, by plotting a return map associated with one of the state variables, we demonstrate the strongly one-dimensional character of the dynamics and discuss the dependence of this map on the parameters of the system.

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

2020 ◽  
Author(s):  
Heather M Russell ◽  
Julianna Tymoczko
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Transition Matrix ◽  
Group Representation ◽  
Jones Polynomial ◽  
Basis Matrix ◽  
Combinatorial Structures ◽  
Quantum Invariants ◽  
One Dimensional ◽  
Link Diagrams

Abstract Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

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