Complex and Fractional Dynamics

Machado, J. A. Tenreiro; Lopes, António M.

doi:10.3390/e19020062

Cited by 7 publications

(3 citation statements)

References 22 publications

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“…The test consists of creating a random dynamic process for the data and then studying how the scale of the stochastic process changes with time [67][68][69]. This test has been widely adopted as a suitable tool to confirm the chaotic behavior in fractional-order dynamical systems [26,33,40] because it is binary (minimizing issues of distinguishing small positive numbers from zero); the nature of the vector field, as well as its dimensionality, does not pose practical limitations; and it does not suffer from the difficulties associated with phase space reconstruction.…”

Section: Test 0-1 For Chaosmentioning

confidence: 99%

“…In recent years, fractional calculus has received much attention due to fractional derivatives providing more accurate models than their integer-order counterparts. Many examples have been found in different interdisciplinary fields [25], ranging from the description of viscoelastic anomalous diffusion in complex liquids, D-decomposition technique for control problems, chaotic systems; to macroeconomic models with dynamic memory, forecast of the trend of complex systems, and so on [26][27][28][29][30][31][32][33][34]. Those works have demonstrated that fractional derivatives provide an excellent approach to describing the memory and hereditary properties of real physical phenomena.…”

Section: Introductionmentioning

confidence: 99%

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A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Muñoz‐Pacheco

Zambrano-Serrano

Volos

et al. 2018

Entropy

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In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

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Section: Test 0-1 For Chaosmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Muñoz‐Pacheco

Zambrano-Serrano

Volos

et al. 2018

Entropy

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“…Dynamical networks, including brain networks 1 , quantum networks 2 , financial networks 3 , gene networks 4 , protein networks 5 , cyber-physical system networks 6 (e.g. power networks 2 , healthcare networks 7 ), social networks 8 , and physiological networks 9 , exhibit not only an intricate set of higher-order interactions but also exhibit distinct long-term memory dynamics where both the recent and more distant past states influence the state's evolution. Regulating these long-term memory dynamical networks in a timely fashion becomes critical to avoid a fullblown catastrophe.…”

mentioning

confidence: 99%

The role of long-term power-law memory in controlling large-scale dynamical networks

Reed,

Ramos,

Bogdan

et al. 2023

Sci Rep

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Controlling large-scale dynamical networks is crucial to understand and, ultimately, craft the evolution of complex behavior. While broadly speaking we understand how to control Markov dynamical networks, where the current state is only a function of its previous state, we lack a general understanding of how to control dynamical networks whose current state depends on states in the distant past (i.e. long-term memory). Therefore, we require a different way to analyze and control the more prevalent long-term memory dynamical networks. Herein, we propose a new approach to control dynamical networks exhibiting long-term power-law memory dependencies. Our newly proposed method enables us to find the minimum number of driven nodes (i.e. the state vertices in the network that are connected to one and only one input) and their placement to control a long-term power-law memory dynamical network given a specific time-horizon, which we define as the ‘time-to-control’. Remarkably, we provide evidence that long-term power-law memory dynamical networks require considerably fewer driven nodes to steer the network’s state to a desired goal for any given time-to-control as compared with Markov dynamical networks. Finally, our method can be used as a tool to determine the existence of long-term memory dynamics in networks.

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