Simple Chaotic Flows With One Stable Equilibrium

Abstract

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

New Chaotic Dynamical System with a Conic-Shaped Equilibrium Located on the Plane Structure

“…Strange attractors can also be observed in dynamical systems with a single unstable node equilibrium, as previously demonstrated [19]. The most surprisingly unstable fixed points, which are normally responsible for strange attractor excitation and extreme sensitivity of dynamical system behavior to the initial conditions, can be changed into a single stable fixed point [20,21] or a pair of stable equilibria [22]. Chaos can be observed in isolated dynamical systems with fixed points degenerating into plane geometrical structures such as a line [23], circle [24], rounded square and square [25].…”

Megastability: Coexistence of a countable infinity of nested attractors in a periodically-forced oscillator with spatially-periodic damping

“…These systems include dynamical systems with no equilibrium points [13][14][15][16][17][18][19][20][21], with only stable equilibria [22][23][24][25][26][27], with curves of equilibria [28][29][30], with surfaces of equilibria [8,9], and with non-hyperbolic equilibria [31,32]. Many of these examples belong to a new category of dynamical systems with hidden attractors [33][34][35][36][37][38][39][40][41][42].…”

Hyperchaotic Chameleon: Fractional Order FPGA Implementation

“…Control of such hidden oscillations is a big challenge because of the multistability nature of the systems [6,7]. Chaotic attractors are with no equilibrium points [8][9][10][11][12][13][14][15], with only stable equilibria [16][17][18][19], and with curves of equilibria [20]. Fractional order with no equilibrium systems with its FPGA implementation has also been reported recently [21,22].…”