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In this chapter, low-dimensional nonlinear dynamical systems are discussed. The stability and bifurcations of the 1-dimensional systems are presented. The higher order singularity and stability for 1-dimensional nonlinear systems are developed.

In this chapter, basic concepts of nonlinear dynamical systems are introduced. A local theory of equilibrium stability for nonlinear dynamical systems is discussed. The spiral stability of equilibriums in nonlinear dynamical systems is presented through the Fourier series base. The higher order singularity and stability for nonlinear systems on the specific eigenvectors are developed. The Lyapunov function stability is briefly discussed, and the extended Lyapunov theory for equilibrium stability is also presented.

In this chapter, dynamical systems with infinite equilibriums are discussed through the local analysis. A method for equilibriums in nonlinear dynamical systems is developed. The generalized normal forms of nonlinear dynamical systems at equilibriums are presented for a better understanding of singularity in nonlinear dynamical systems. The dynamics of infinite-equilibrium dynamical systems is discussed for the complexity and singularity of nonlinear dynamical systems. A few examples are presented for complexity and singularity in infinite-equilibrium systems.

In this chapter, the hyperbolic bifurcations of equilibriums on the eigenvectors in nonlinear dynamical systems are discussed, and the Hopf bifurcation of an equilibrium on a specific eigenvector plane is presented. Based on the Fourier series base, the transformation for the spiral stability is introduced for the Hopf bifurcation of equilibriums. The Hopf bifurcation of equilibriums in the second-order nonlinear dynamical systems is discussed from the Fourier series transformation.

In this chapter, a global analysis of equilibrium stability in 1-dimensional nonlinear dynamical systems is presented. The classification of dynamical systems is given first, and infinite-equilibrium systems are defined. The 1-dimensional dynamical systems with single equilibrium are discussed first. The 1-dimensional dynamical systems with two and three equilibriums are discussed. Simple equilibriums and higher order equilibriums in 1-dimensional dynamical systems are analyzed, and herein a higher order equilibrium is an equilibrium with higher order singularity. The separatrix flow of equilibriums in 1-dimensional systems in phase space is illustrated for a better understanding of the global stability of equilibriums in 1-dimensional dynamical systems.

In this paper, periodic motions in a first-order, time-delayed, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motion. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation tree of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

In this chapter, symmetric periodic motions in a periodically forced van der Pol oscillator is presented. To obtain the periodic motions, the discrete maps of the van der Pol oscillator are developed through the discretization of the corresponding differential equations. Through mapping structures of periodic motions, stable and unstable periodic motions are obtained. A sequence of symmetric periodic motions to chaos with \(1(S) \triangleleft 3(S) \triangleleft \; \cdots \triangleleft (2l - 1)(S) \triangleleft \cdots\) is presented. The numerical simulations of the periodic motions are completed for illustration of motion complexity. This chapter is for a memory of Valentin Afraimovich for 20-year friendship.

In this paper, the experimental dynamics of a Duffing oscillatory system are studied for periodic motions. A Duffing oscillatory circuit is developed for the experimental study of periodic motions on the bifurcation trees. The coexisting asymmetric periodic motions are obtained experimentally. The analytical periodic motions in the Duffing oscillator are presented for comparison with experimental results. Because of hardware and data leakage of experimental instruments, the experimental result accuracy is much lower than the analytical results of periodic motions. To improve the experimental results accuracy, the high quality hardware and instruments should be adopted and the high resolution data acquisition systems should be adopted.

In this paper, period motions in a periodically forced, damped, double pendulum are analytically predicted through a discrete implicit mapping method. The implicit mapping is established via the discretized differential equation. The corresponding stability and bifurcation conditions of the period motions are predicted through eigenvalue analysis. Numerical simulation of the period motions in the double pendulum is completed from analytical predictions.Copyright © 2018 by ASME

In this paper, period-1 motions varying with excitation frequency in a periodically forced, nonlinear spring pendulum system are predicted by a semi-analytic method. The harmonic frequency-amplitude for periodical motions are analyzed from the finite discrete Fourier series. The stability of the periodical solutions are shown on the bifurcation trees as well. From the analytical prediction, numerical illustrations of periodic motions are given, the comparison of numerical solution and analytical solution are given.

In this paper, the analytical solutions of periodic evolutions of the periodically diffused Brusselator are obtained through the generalized harmonic balanced method. Stable and unstable solutions of period-1 and period-2 evolutions in the Brusselator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis, and the corresponding Hopf bifurcations are presented on the analytical bifurcation tree of the periodic motions. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectra show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

In this paper, frequency responses of periodic motions to chaos in a periodically forced, damped, quadratic nonlinear oscillator are investigated through the finite Fourier series analysis of discrete solutions of periodic motions. The discrete solutions of periodic motions are obtained from mapping structures of discrete nodes, and the corresponding stability and bifurcation analysis of periodic motions are completed by the eigenvalue analysis of fixed points in discrete nonlinear dynamical systems. The frequency–amplitude characteristics for bifurcation trees of periodic motions to chaos are discussed, and the quantity levels of harmonic amplitudes for different harmonic orders are illustrated clearly. Numerical results of periodic motions are illustrated to show harmonic frequency responses effects on periodic motions.

In this paper, analytically predicted are complex periodic motions in the periodically forced, damped, hardening Duffing oscillator through discrete implicit maps of the corresponding differential equations. Bifurcation trees of periodic motions to chaos in such a hardening Duffing oscillator are obtained. The stability and bifurcation analysis of periodic motion in the bifurcation trees is carried out by eigenvalue analysis. The solutions of all discrete nodes of periodic motions are computed by the mapping structures of discrete implicit mapping. The frequency-amplitude characteristics of periodic motions are computed that are based on the discrete Fourier series. Thus, the bifurcation trees of periodic motions are also presented through frequency-amplitude curves. Finally, based on the analytical predictions, the initial conditions of periodic motions are selected, and numerical simulations of periodic motions are carried out for comparison of numerical and analytical predictions. The harmonic amplitude spectrums are also given for the approximate analytical expressions of periodic motions, which can also be used for comparison with experimental measurement. This study will give a better understanding of complex periodic motions in the hardening Duffing oscillator.

In this paper, obtained are analytical solutions for period-1 motions to chaos in a two-degree-of-freedom (2-DOF) oscillator with a nonlinear hardening spring. From the finite Fourier series transformation, a dynamical system of coefficients of the finite Fourier series is developed to determine existence, stability and bifurcations of periodic motions in such a 2-DOF nonlinear oscillator. The equilibriums of such a dynamical system of coefficients give the analytical solutions of period-m motions, and the corresponding stability and bifurcations of period-m motions are determined through the eigenvalue analysis of equilibriums. Analytical bifurcation trees of period-1 motions to chaos are presented through frequency–amplitude curves. From the frequency–amplitude curves, the harmonic effects on the periodic motions can be discussed and nonlinear behaviors of periodic motions can be determined. Displacements, velocity, and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator. Through the analytical solutions, the complex dynamics of the 2-DOF nonlinear oscillator is studied. The analytical solutions presented herein for periodic motions in such 2-DOF systems can be used to further discuss the corresponding nonlinear behaviors, and can also be applied to engineering for design.

In this paper, periodic motions of a periodically forced, damped, Duffing oscillator with double-well potential are analytically predicted through discrete implicit mappings. The discrete implicit maps are obtained from the differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions of the Duffing oscillator are predicted analytically, and the corresponding stability and bifurcation analysis of periodic motions are carried out through the eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are performed by the numerical method of the differential equation to verify the semi-analytical prediction, and the corresponding harmonic amplitudes are computed through discrete Fourier series of the analytically predicted node points of periodic motions, and the complexity of periodic motions can be measured by the harmonic amplitude. The frame work presented in this paper can provide a semi-analytical method to find periodic motions and to determine routes of periodic motions to chaos rather than numerical simulation only in nonlinear dynamical systems, and the stable and unstable periodic motions and even chaos can be predicted analytically.

In this paper, the analytical solutions of period-1 motions of a buckled nonlinear Jeffcott rotor are developed, and the corresponding stability and bifurcation of period-1 motion are also analyzed by eigenvalue analysis. The Hopf bifurcations of period-1 motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the buckled nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the buckled nonlinear Jeffcott rotor.

In this chapter, time-delay effects on periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are reviewed and further discussed. Bifurcation trees of periodic motions to chaos varying with time-delay are presented for such a time-delayed, Duffing oscillator. From the analytical prediction, periodic motions in the time-delayed, hardening Duffing oscillator are simulated numerically. Through numerical illustrations, time-delay effects on period-1 motions to chaos in nonlinear dynamical systems are strongly related to the distributions and quantity levels of harmonic amplitudes.

In this paper, periodic motions of a periodically forced oscillator with a nonlinear isolator are studied through generalized harmonic balanced method. Both symmetric and asymmetric period-1 motions are obtained. Stability and bifurcation of the periodic motions are determined through eigenvalue analysis. Numerical illustrations of both symmetric and asymmetric are given. From the harmonic amplitude spectrums, the harmonic effects on periodic motions are determined, and the corresponding accuracy of approximate analytical solutions can be observed.

In this paper, bifurcation trees of period-1 motions to chaos are investigated in a parametrically excited pendulum. To construct discrete mapping structures of periodic motions, implicit discrete maps are developed for such a pendulum system. The bifurcation trees from period-1 motions to chaos are predicted semi-analytically through period-1 to period-4 motions. The corresponding stability and bifurcation analysis are carried out through eigenvalue analysis. Finally, numerical simulations of periodic motions can be completed through numerical methods. Such simulation results are illustrated for verification of the analytical predictions.

In this paper, analytical solutions of periodic motions in a 2-DOF self-excited Duffing oscillator are investigated through a semi-analytical method. The semi-analytical method discretizes the self-excited Duffing oscillator for the discrete implicit mappings. Through the implicit mapping, period-1 motion varying with excitation frequency are presented, and the corresponding stability and bifurcation are discussed via the eigenvalues analysis. The Neimark and saddle-node bifurcations of the periodic motion are obtained. Initial conditions for numerical simulations are from analytical solutions. Numerical and analytical solutions of periodic motions are illustrated for comparison.

In this paper, symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, softening Duffing oscillator is analytically predicted through a discrete implicit mapping method. Such a method is based on the discretization of the corresponding differential equation. The stability and bifurcations of the symmetric and asymmetric period-1 motions are determined through eigenvalue analysis. Numerical simulation of the period-1 motions in the time-delayed softening Duffing oscillator is presented for verification of the analytical prediction.

In this paper, the analytical solutions of periodic evolution of Brusselator are investigated through the general harmonic balanced method. Both stable and unstable, period-1 and period-2 solutions of the Brussellator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectrums show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

This paper studies bifurcation trees of periodic motions in a parametric, damped Duffing oscillator. From the semi-analytic method, the corresponding differential equation is discretized to obtain the implicit mapping. From implicit mapping structure, the periodic nodes of periodic motions are computed, and the bifurcation trees of period-1 to period-4 motions are presented and the corresponding stability and bifurcation are carried out by eigenvalue analysis. From the analytical predictions, numerical simulations are completed, and the trajectory, harmonic amplitudes and phases of period-1 to period-4 motions are illustrated.

In this paper, the bifurcation trees of periodic motions in a parametrically excited pendulum are studied using discrete implicit maps. From the discrete maps, mapping structures are developed for periodic motions in such a parametric pendulum. Analytical bifurcation trees of periodic motions to chaos are developed through the nonlinear algebraic equations of such implicit maps in the specific mapping structures. The corresponding stability and bifurcation analysis of periodic motions is carried out. Finally, numerical results of periodic motions are presented. Many new periodic motions in the parametrically excited pendulum are discovered.

In this paper, period-3 motions in a double-well Duffing oscillator with time-delay are predicted by a semi-analytical method. The implicit mapping structures of period-3 motions are constructed through the implicit mappings obtained by discretization of the corresponding differential equation. Complex period-3 motions are predicted through nonlinear algebraic equations of the implicit mappings in the mapping structures and the corresponding stability and bifurcation are carried out through eigenvalue analysis. Numerical and analytical results of complex period-3 motions are obtained and the corresponding frequency-amplitude characteristics are presented.Copyright © 2017 by ASME

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

In this paper, periodic motions of a periodically forced, coupled van der Pol-Duffing oscillator are predicted analytically. The coupled van der Pol-Duffing oscillator is discretized for the discrete mapping. The periodic motions in such a coupled van der Pol-Duffing oscillator are obtained from specified mapping structures, and the corresponding stability and bifurcation analysis are carried out by eigenvalue analysis. Based on the analytical prediction, the initial conditions of periodic motions are used for numerical simulations.

In this chapter, the analytical bifurcation trees of travelable period-1 motions to chaos in a periodically excited pendulum will be presented with varying excitation amplitude. The analytical prediction is based on the implicit discrete maps obtained from the midpoint scheme of the corresponding differential equation. Using the discrete maps, mapping structures will be developed for various periodic motions, and analytical bifurcation trees of periodic motions to chaos can be obtained.

In this paper, bifurcation trees of period-3 motions to chaos in a periodically excited, Duffing oscillator with a linear delay are investigated through the Fourier series. The analytical solutions of period-m motions are presented and the stability and bifurcation of such period-m motions in the bifurcation trees are discussed by eigenvalue analysis. Two independent symmetric period-3 motions were obtained, and the two independent symmetric period-3 motions are not relative to chaos. Two bifurcation trees of period-3 motions to chaos are presented through period-3 to period-6 motion. Numerical illustrations of stable and unstable period-3 and period-6 motions are given by numerical and analytical solutions. The complicated period-3 and period-6 motions exist in the range of low excitation frequency.

In this paper, analytical routes of periodic motions to chaos in a parametric, quadratic oscillator are investigated. The analytical solutions of period-m motions in such a parametric oscillator are presented from the finite term Fourier series, and the stability and bifurcation of the period-m motions are discussed through the eigenvalue analysis. The routes of periodic motions to chaos in such parametric oscillator are illustrated through harmonic amplitudes varying with excitation amplitude in the finite term Fourier series solution. From the routes of period-m motion to chaos, numerical illustrations of periodic motions are given through trajectories and analytical harmonic amplitude spectrum.

In this paper, complex period-1 motions in a periodically forced Duffing oscillator with a time-delay feedback are investigated, and the symmetric and asymmetric, complex period-1 motions exist in lower excitation frequency. The analytical solutions of complex period-1 motions in such a Duffing oscillator are obtained through the finite Fourier series, and the corresponding stability and bifurcations of complex period-1 motions are discussed by eigenvalue analysis. The frequency–amplitude characteristics of complex period-1 motions in the periodically forced Duffing oscillator with a time-delay feedback are discussed. Complex period-1 motions generated numerically and analytically are illustrated. As excitation frequency is close to zero, the complex period-1 motions need almost infinite harmonic terms in the Fourier series to express the analytical solutions. From this study, the initial time-delay in the time-delayed, nonlinear systems should be uniquely determined to achieve a specific periodic motion.

In this paper, analytical solutions of periodic motions in a periodically excited, time-delayed, quadratic nonlinear oscillator are obtained through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The analytical bifurcation tree of period-1 motion to chaos in such a time-delayed, quadratic oscillator is presented through period-1 to period-8 motion. Numerical illustrations of stable and unstable periodic motions are given by numerical and analytical solutions. Compared to dynamical systems without time-delay, the timedelayed dynamical systems possess different periodic motions and the bifurcation trees of periodic motions to chaos are also distinguishing.

In the original publication, Fig. 1a, b was published incorrectly. The corrected figure is given below.

In this paper, analytical solutions for period- $$m$$ motions in a nonlinear rotor system are discussed. This rotor system with two-degrees of freedom is one of the simplest rotor dynamical systems, and periodic excitations are from the rotor eccentricity. The analytical expressions of periodic solutions are developed, and the corresponding stability and bifurcation analyses of period-m motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. Displacement orbits of periodic motions in nonlinear rotor systems are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the nonlinear rotor. Coexisting periodic motions exist in the nonlinear rotor.

Analytical period-m motions and bifurcation trees in a periodically forced, van der Pol-Duffing oscillator are obtained through the Fourier series, and the corresponding stability and bifurcation of such period-m motions are discussed. To verify the approximate, analytical solutions of period-m motions on the bifurcation trees, numerical simulations are carried out, and the numerical results are compared with analytical solutions. The harmonic amplitude distributions are presented to show the significance of harmonic terms in the finite Fourier series of the analytical periodic solutions. The bifurcation trees of period-m motion to chaos via period-doubling are individually embedded in the quasi-periodic and chaotic motions without period-doubling.

In this paper, the analytical dynamics of peri- odic flows to chaos in nonlinear dynamical systems is pre- sented from the ideas of Luo (Continuous dynamical systems, Higher Education Press/L&H Scientific, Beijing/Glen Car- bon, 2012). The analytical solutions of periodic flows and chaos in autonomous systems are discussed through the gen- eralized harmonic balance method, and the analytical dynam- ics of periodically forced nonlinear dynamical systems is pre- sented as well. The analytical solutions of periodic motions in free and periodically forced vibration systems are presented. The similar ideas are extended to time-delayed nonlinear sys- tems. The analytical solutions of periodic flows to chaos for time-delayed, nonlinear systems with/without periodic excitations are presented, and time-delayed, nonlinear vibra- tion systems will be also discussed. The analytical solutions of periodic flows and chaos are independent of small para- meters, which are different from the traditional perturbation methods. The methodology presented herein will provide the accurate analytical solutions of periodic motions to chaos in dynamical systems with/without time-delay. This approach can handle nonlinear dynamical systems with either single time-delay or multiple time-delays.

The stability and bifurcation analyses of periodic motions in a rotating blade subject to a torsional excitation are investigated. For high speed rotations, cubic geometric nonlinearity and gyroscopic effects of the rotating blade are considered. From the Galerkin method, the partial differential equation of the nonlinear rotating blade is simplified to the ordinary differential equations, and periodic motions and stability of the rotating blade are studied by the generalized harmonic balance method. The analytical and numerical results of periodic solutions are compared. The rich dynamics and co-existing periodic solutions of the nonlinear rotating blades are investigated.

In this paper, the projective function synchronization of chaos between two gyroscope systems with distinct behaviors is investigated under sinusoidal constraints. The objective of the research is to adjust a current gyroscope system to the idealized behaviors in design through synchronization. From the theory of discontinuous dynamical systems, the mechanism of function synchronization of chaotic motions is studied. The analytical conditions for the function synchronization are achieved, and the invariant domain of the function synchronization is obtained. Numerical illustrations for partial and full, projective function synchronization of two gyroscopes with different dynamical behaviors are carried out. The scaling factors in such function synchronization are satisfied through numerical results.

In this paper, analytical solutions for complex period-1 motions in a periodically forced, quadratic nonlinear oscillator are presented through the Fourier series solutions with finite harmonic terms, and the corresponding stability and bifurcation analyses of the corresponding period-1 motions are carried out. Many branches of complex period-1 motions in such a quadratic nonlinear oscillator are discovered and the period-1 motion patterns changes with parameters are presented. The parameter map for excitation amplitude and frequency is developed for different complex period-1 motions. For small excitation frequency, the period-1 motion becomes more complicated. For a better understanding of complex period-1 motions in such a quadratic nonlinear oscillator, trajectories and amplitude spectrums are illustrated numerically. From stability and bifurcations analysis of the period-1 motion, the analytical bifurcation trees of period-1 motions to chaos need to be further investigated.

In this paper, parameter characteristics of the projective synchronization for two gyroscopes with different dynamical behaviors are investigated. The projective synchronization conditions are presented from the theory of discontinuous dynamical systems. From such synchronization conditions, the parameter characteristics for partial and full projective synchronizations for two gyroscope systems are studied. The full projective synchronization can be achieved exactly in finite time instead of asymptotic synchronization in the traditional projective synchronization. The scaling factors in such synchronization are observed through numerical simulations.

Approximate, analytical solutions of period-m motions in a periodically forced, van der Pol oscillator are obtained through the Fourier series expression, and the corresponding stability and bifurcation analysis of such period-m motions are carried out. To verify the approximate, analytical solutions of period-m motions, numerical simulations are performed, and the numerical results are compared with analytical solutions. The harmonic amplitude distributions are presented to show the significance of harmonic terms in the finite Fourier series expression of the analytical periodic solutions. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.