Are Perpetual Points Sufficient for Locating Hidden Attractors?

Recently, a new category of attractors that are called hidden attractors are of special interest [Leonov & Kuznetsov, 2014; Leonov et al., 2014; Leonov et al., 2011a; Leonov et al., 2015b; Leonov et al., 2011b; Leonov et al., 2012; Leonov et al., 2015a; Leonov & Kuznetsov, 2013a, 2013b; Bragin et al., 2011; Kuznetsov et al., 2011, 2010; Kuznetsov, 2016]. Hidden attractors are not associated with saddle points or unstable equilibria, and thus they can be found only by a numerical search through the space of initial conditions to find those within their basin of attraction. Many new chaotic attractors have been discovered in this category, such as flows without any equilibrium points, with only stable equilibria, or with a line of equilibrium points [Jafari & Sprott, 2013, 2015; Jafari et al., 2013; Jafari et al., 2015b; Kingni et al., 2014; Lao et al., 2014; Pham et al., 2014a; Pham et al., 2014b; Pham et al., 2014c; Pham et al., 2014d; Pham et al., 2015; Shahzad et al., 2015; Tahir et al., 2015; Pham et al., 2016c; Goudarzi et al., 2016; Pham et al., 2017; Kingni et al., 2017; Pham et al., 2016b; Panahi et al., 2016; Pham et al., 2016a; Barati et al., 2016; Pham et al., 2016d]. Multistability and coexisting attractors are other topics that have received increasing attention in nonlinear dynamics [Angeli et al., 2004; Pisarchik & Feudel, 2014; BlazejczykOkolewska & Kapitaniak, 1996, 1998; Kapitaniak, 1985; Maistrenko et al., 1997; Silchenko et al., 1999]. The importance of these dynamical categories is their sensitivity to perturbations and initial conditions. On the other hand, knowledge of the basin and their classification are especially important for hidden attractors since the basin does not contain the small neighborhood of any equilibrium points [Sprott & Xiong, 2015]. One important structural feature in nonlinear dynamics is the fixed point [Ott, 2002; Strogatz, 2014]. Recently, Perpetual Points (PPs) were