Abstract
A hyperjerk system described by a single fourth-order ordinary differential equation of the form has been referred to as a snap system. A damping-tunable snap system, capable of an adjustable attractor dimension () ranging from dissipative hyperchaos () to conservative chaos (), is presented for the first time, in particular not only in a snap system, but also in a four-dimensional (4D) system. Such an attractor dimension is adjustable by nonlinear damping of a relatively simple quadratic function of the form , easily tunable by a single parameter A. The proposed snap system is practically implemented and verified by the reconfigurable circuits of field programmable analog arrays (FPAAs).
Highlights
Studies of chaotic systems have received great attention due to many practical applications in science and technology [1,2,3]
A three-dimensional (3D) chaotic system is expressed by a set of three coupled first-order ordinary differential equations (ODEs), whereas a fourdimensional (4D) chaotic system is expressed by a set of four coupled first-order ODEs
An field programmable analog arrays (FPAAs) is based on programmable analog building blocks, e.g., integrators, filters, and switched-capacitor circuits, whereas an field programmable gate array (FPGA) is based on configurable logic modules or look-up tables and does not involve analog building blocks
Summary
Studies of chaotic systems have received great attention due to many practical applications in science and technology [1,2,3]. Chaos is typically measured by Lyapunov exponents (LEs) and a Lyapunov dimension (DL). The latter is alternatively known as a Kaplan–Yorke dimension (DKY) or an attractor dimension [5]. The attractor dimension (DL) classifies whether chaos (or hyperchaos) in an n-dimensional, or n-th-order, system is dissipative (DL < n) or conservative (DL = n). Not all undamped systems can exhibit chaos, conservative chaos (or hyperchaos) is found in an undamped system where the average damping along the trajectories is zero (α = 0) [5]. Conservative chaos (or hyperchaos) is of special interest [13], but appears to be the minority of reported chaotic (or hyperchaotic) systems, whereas dissipative chaos (or hyperchaos) appears to be the majority.
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