2012
DOI: 10.1007/s11071-012-0424-0
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Competitive modes as reliable predictors of chaos versus hyperchaos and as geometric mappings accurately delimiting attractors

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Cited by 25 publications
(18 citation statements)
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“…Competitive modes have recently been used to study a number of chaotic or hyperchaotic dynamical systems [18][19][20][21][22][23][24][26][27][28][29][30][31][32]. In this approach, one treats each member of the dynamical system as a single oscillator equation.…”
Section: Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…Competitive modes have recently been used to study a number of chaotic or hyperchaotic dynamical systems [18][19][20][21][22][23][24][26][27][28][29][30][31][32]. In this approach, one treats each member of the dynamical system as a single oscillator equation.…”
Section: Resultsmentioning
confidence: 99%
“…For the general N-dimensional form of the system, we found that there may exist up to N À 1 modes which are competitive. It has previously been shown in the literature that the occurrence of two modes which are intermittently competitive is a type of necessary condition for chaos [18][19][20]. Similarly, it has been observed that for hyperchaotic models, the existence of three intermittently competitive modes is a type of necessary condition for the existence of hyperchaos.…”
mentioning
confidence: 84%
“…The requirements (A)-(D) essentially tell us that a condition for chaos is that two or more equations in (40) behave as oscillators (g i > 0), and that two of these oscillators lock frequencies at one or more times. In practice, we find that the frequencies agree at a countably infinite collection of time values [13,18]. The frequencies should be functions of time (i.e., we have nonlinear frequencies), and there should be at least one forcing function which depends on a state variable.…”
Section: Competitive Modes Analysis: a Check For Chaosmentioning
confidence: 97%
“…As stated before [13], the competitive modes analysis gives an interesting link between the geometry of phase space possibly yielding chaotic trajectories (recall that the competitive modes requirements appear to be a necessary, albeit not sufficient, condition for chaos [14,15,16,17,18,19]). Conversely, the differential elimination may cast light into the geometry of solutions in the space of derivations.…”
Section: Introductionmentioning
confidence: 99%
“…Many efforts for analyzing hyperchaotic dynamics have been reported in Refs. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”
Section: Introductionmentioning
confidence: 99%