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In this paper, we investigate a quadratic chaotic system modeling self-excited and hidden attractors which is described by a system of three nonlinear ordinary differential equations with three real parameters. The primary goal is to establish the existence of two limit cycles that bifurcate based on the system's nature as an electronic circuits model, specifically via Hopf bifurcation. Notably, the application of the first and second Lyapunov coefficients is utilized to demonstrate the bifurcation of two limit cycles from an equilibrium point near a Hopf critical point. Furthermore, employing the first-order averaging theory enables us to confirm the presence of unstable periodic orbits originating from the zero-Hopf equilibrium. [ABSTRACT FROM AUTHOR]

Numerous recently introduced chaotic systems exhibit straightforward algebraic representations. In this study, we explore the potential for identifying a global analytic first integral in a generalized 3-dimensional chaotic system (2). Our work involves detailing the model of a new 3-D chaotic system characterized by three Lyapunov exponents—positive, zero, and negative. We depict the phase trajectories, illustrate bifurcation patterns, and visualize Lyapunov exponent graphs. The investigation encompasses both local and global analytic first integrals for the system, providing results on the existence and non-existence of these integrals for different parameter values. Our findings reveal that the system lacks a global first integral, and the presence or absence of analytic first integrals is contingent upon specific parameter values. Additionally, we present a formal series for the system, demonstrating 3D and 2D projections of the system (2) for a given set of initial conditions achieved by selecting alternative values for parameters a, b, c, d, r and l. [ABSTRACT FROM AUTHOR]

In this paper, new solutions to nonlinear pseudo-hyperbolic equations with non-local conditions by residual power series (RPS) method is given. This method is based on the Taylor series formula and the residual error function. A new analytical solution to this equation is given. As an applications we have tested some examples to know the efficiency of current technique. The results obtained showed that the proposed method is effective, accurate and has fast convergent. [ABSTRACT FROM AUTHOR]

This article will give the residual power series method (RPSM) for solving pseudo hyperbolic partial differential equations with nonlocal conditions, RPSM is essentially based on general formula of Taylor series with residual error function. A new analytical solution is investigated. The analytical solution is designed to find the approximation solutions by RPSM and compare the obtained results from the current method with the exact solution that detects the precision, reliability, and rapid convergence of the proposed method. Finally at different times through the graphical representation of obtained results are given. [ABSTRACT FROM AUTHOR]