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In this paper, period-1 motions to twin spiral homoclinic orbits in the Rössler system are presented. The period-1 motions varying with a system parameter are predicted semi-analytically through an implicit mapping method, and the corresponding stability and bifurcations of the period-1 motions are determined through eigenvalue analysis. The approximate homoclinic orbits are obtained, which can be detected through the periodic motions with the positive and negative infinite large eigenvalues. The two limit ends of the bifurcation diagram of the period-1 motion are at twin spiral homoclinic orbits. For comparison, numerical and analytical results of stable period-1 motion are presented. The approximate spiral homoclinic orbits are demonstrated for a better understanding of complex dynamics of homoclinic orbits. Herein, only initial results on periodic motions to homoclinic orbits are presented for the Rössler system. In fact, the Rössler system has rich complex dynamics existing in other high-dimensional nonlinear systems. Thus, the further studies of bifurcation trees of periodic motions to infinite homoclinic orbits will be completed in sequel.

Constructed motions and dynamic topology are new trends in solving nonlinear systems or system interactions. In nonlinear engineering, it is significant to achieve specific complex motions to satisfy expected dynamical behaviors (e.g., nonlinear motions, singularities, bifurcations, chaos, etc.), and complex motion application and control. To achieve such expected motions and global dynamical behaviors, mapping dynamics, constructed networks, random/discontinuous dynamic theorems, etc., are applied to quantitatively determine the complex motions. These theories adopt the symbolic dynamic abstracts and topological structures with nonlinear dynamics to investigate constructed complex motions to satisfy expected dynamical behaviors.

In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2 n motions [Formula: see text]. On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.

In this paper, nonlinear piezoelectric energy harvesting induced by a Duffing oscillator is studied, and the bifurcation trees of period-1 motions to chaos for such a piezoelectric energy-harvesting system are obtained analytically. Distributed-parameter electromechanical modeling of a piezoelectric energy harvester is presented first, and the electromechanically coupled circuit equation excited by infinitely many vibration modes is developed. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates, and eventually an infinite set of algebraic equations is obtained for the complex modal vibration responses and the complex voltage responses of the energy harvester beam. One single mode case is considered in this paper, and periodic motions with bifurcation trees are obtained through an implicit discrete mapping method. The frequency–amplitude characteristics of periodic motions are obtained for the nonlinear piezoelectric energy-harvesting systems, which provide a better understanding of where and how to achieve the best energy harvesting. This study describes about how the nonlinear oscillator induces piezoelectric energy harvesting through a beam system. The nonlinear piezoelectric energy harvesting is presented through a nonlinear oscillator. Due to the nonlinear oscillator, chaotic piezoelectric energy-harvesting states can get more energy compared to the linear piezoelectric energy-harvesting system.

The book is about the global stability and bifurcation of equilibriums in polynomial functional systems. Appearing and switching bifurcations of simple and higher-order equilibriums in the...

In this paper, the existence and bifurcations of higher-order complex periodic motions in a nonlinear, electromagnetically tuned mass damper system are studied through period-3, period-9, and period-12 motions. For such a nonlinear system, there exist three branches of period-3 motions, one branch of period-9 motions, and one branch of period-12 motion. The corresponding stability and bifurcations of periodic motions are determined. The frequency-amplitude characteristics for bifurcation routes of such higher-order periodic motions are presented. Numerical illustrations are presented for complex periodic motions. Higher-order unstable and stable periodic motions exist in nonlinear systems, which are difficult to control for energy harvesting and vibration reduction. The unstable higher-order periodic motions make the system responses more complicated. The higher-order stable and unstable periodic motions have been studied comprehensively. This study on periodic motions can help people better understand the dynamical behaviors of an electromagnetically tuned mass damper system under different excitation frequencies. The study of higher-order periodic motion stability and bifurcations will also benefit the new development and design of vibration reduction and energy harvesting systems.

This paper presented the dynamics of an infinite-equilibrium system based on a quadratic oscillator. equilibria, stability and singularity of such an infinite equilibrium system are discussed through the local analysis, and numerical studies of the periodically perturbed infinite-equilibrium systems are completed. The infinite-equilibrium boundaries in infinite-equilibrium systems can be artificially designed to control motions in the corresponding non-infinite-equilibrium systems. Through this study, one can have a better understanding of dynamics of infinite-equilibrium nonlinear systems. The authors believe infinite-equilibrium systems will have extensive applications in science and engineering.

In this paper, the global stability and bifurcations of equilibriums in the (2m)th-degree polynomial system are discussed for a better understanding the complexity of bifurcations and stability of equilibriums. The appearing and switching bifurcations for simple equilibriums are presented, and the appearing and switching bifurcations for higherorder equilibriums are discussed as well. The teethcomb-appearing, spraying-appearing, and sprinkler-spraying-appearing bifurcations for simple and higher-order equilibriums are presented. The antennaswitching bifurcations for simple and higher-order equilibriums are discussed and the parallel straw-bundle-switching and flower-bundleswitching bifurcations for simple and higher-order equilibriums are presented as well.

In this paper, dynamical systems with infinite-equilibriums is discussed through the local analysis. A computational method for equilibriums in nonlinear dynamical systems is developed which extends the Newton–Raphson method for solving nonlinear algebraic equations. The generalized normal forms of nonlinear dynamical systems at equilibriums are presented for a better understanding of singularity in nonlinear dynamical systems. At bifurcation points of equilibriums, the corresponding dynamical systems may become infinite-equilibrium systems. The dynamics of infinite-equilibrium dynamical systems is discussed through a few examples for demonstration of complexity and singularity in infinite-equilibrium systems.

In this paper, the complete bifurcation dynamics of period-3 motions to chaos are obtained semi-analytically through the implicit mapping method. Such an implicit mapping method employs discrete implicit maps to construct mapping structures of periodic motions to determine complex periodic motions. Analytical bifurcation trees of period-3 motions to chaos are determined through nonlinear algebraic equations generated through the discrete implicit maps, and the corresponding stability and bifurcations of periodic motions are achieved through eigenvalue analysis. To study the periodic motion complexity, harmonic amplitudes varying with excitation amplitudes are presented. Once more, significant harmonic terms are involved in periodic motions, and such periodic motions will be more complex. To illustrate periodic motion complexity, numerical and analytical solutions of periodic motions are presented for comparison, and the corresponding harmonic amplitudes and phases are also presented for such periodic motions in the bifurcation trees of period-3 motions to chaos. Similarly, other higher-order periodic motions and bifurcation dynamics for the nonlinear spring pendulum can be determined. The methods and analysis presented herein can be applied for other nonlinear dynamical systems.

In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two Neimark bifurcations (NBs) of period-1 motions, one NB of period-2 motions, and four NBs of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling (PD) bifurcations exist for the period-1, period-2, and period-4 motions. The unstable PD bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable PD bifurcation is the third order source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

In this paper, periodic motions and homoclinic orbits in a discontinuous dynamical system on a single domain with two vector fields are discussed. Constructing periodic motions and homoclinic orbits in discontinuous dynamical systems is very significant in mathematics and engineering applications, and how to construct periodic motions and homoclinic orbits is a central issue in discontinuous dynamical systems. Herein, how to construct periodic motions and homoclinic orbits is presented through studying a simple discontinuous dynamical system on a domain confined by two prescribed energies. The simple discontinuous dynamical system has energy-increasing and energy-decreasing vector fields. Based on the two vector fields and the corresponding switching rules, periodic motions and homoclinic orbits in such a simple discontinuous dynamical system are studied. The analytical conditions of bouncing, grazing, and sliding motions at the two energy boundaries are presented first. Periodic motions and homoclinic orbits in such a discontinuous dynamical system are determined through the specific mapping structures, and the corresponding stability is also presented. Numerical illustrations of periodic motions and homoclinic orbits are given for constructed complex motions. Through this study, using discontinuous dynamical systems, one can construct specific complex motions for engineering applications, and the corresponding mathematical methods and computational strategies can be developed.

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

In this paper, analytical solutions of periodic motions in a 1-D nonlinear dynamical system are obtained through the generalized harmonic balance method with prescribed-computational accuracy. From this method, the 1-D dynamical system is transformed to a nonlinear dynamical system of coefficients in the Fourier series. The analytical solutions of periodic motions are obtained by equilibriums of the coefficient dynamical systems, and the corresponding stability and bifurcations of periodic motions are completed via the eigenvalue analysis. The frequency-amplitude characteristics of periodic motions are analyzed through the different order harmonic terms in the Fourier series, and the corresponding quantity levels of harmonic amplitudes are determined. From such frequency-amplitude characteristics, the nonlinearity, singularity and complexity of periodic motions in the 1-D nonlinear systems can be discussed. Displacements and trajectories of periodic motions are illustrated for a better understanding of periodic motions in the 1-D nonlinear dynamical systems. From this study, the periodic motions in the 1-dimensional dynamical systems possess similar behaviors of periodic motions in the van der Pol oscillator.

This chapter presents a Yin-Yang theory for nonlinear discrete systems with positive and negative iterations of discrete iterative maps. Based on the Yin-Yang theory, the complete dynamics of discrete dynamical systems are discussed. The critical dynamics of forward and backward discrete systems are discussed from the explicit expressions of discrete systems.

In this chapter, as in Luo (2019, 2020), discrete systems with infinite-fixed-points are discussed through the local analysis. A method for fixed-points in nonlinear discrete systems is developed. The generalized normal forms of nonlinear discrete systems at fixed-points are presented for a better understanding of singularity in nonlinear discrete systems. The dynamics of infinite-fixed-point discrete systems is discussed for the complexity and singularity of nonlinear discrete systems. A few examples are presented for complexity and singularity in infinite-fixed-point discrete systems.

In this Chapter, the stability and bifurcation of the quartic nonlinear discrete systems will be presented

In this Chapter, the global stability and bifurcations of period-1 fixed-points in the (2m)th-degree polynomial discrete system are presented. The parallel-appearing, spraying-appearing, sprinkler-spraying-appearing bifurcations for simple and higher-order period-1 fixed-points are presented, and the antenna-switching, straw-bundle-switching bifurcations and flower-bundle-switching bifurcations for simple and higher-order period-1 fixed-points are presented. From the period-doubling bifurcation, the period-2 fixed-point solutions and the corresponding period-doubling renormalization of such a forwarded (2m)th-degree polynomial discrete system are discussed. For multiple iterations, the appearing bifurcations of period-n fixed-points and the corresponding period-doublization of the forward (2m)th-degree polynomial discrete system are presented as well.

In this Chapter, as in Luo (2019, 2020), a global analysis of fixed-point stability in 1-dimensional nonlinear discrete systems is presented. The classification of discrete systems is discussed first, and infinite-fixed-point discrete systems are presented. The 1-dimensional discrete systems with a single fixed-point are discussed first. The 1-dimensional discrete systems with two and three fixed-points are discussed. Simple fixed-points and higher-order fixed-points in 1-dimensional discrete systems are analyzed, and herein a higher-order fixed-point is a fixed-point with higher-order singularity. The discrete flow of fixed-points in 1-dimensional systems in alike phase space are illustrated for a better understanding of the global stability of fixed-points in 1-dimensional discrete systems.

In this Chapter, the global stability and bifurcation of quadratic nonlinear discrete systems are discussed. Appearing and switching bifurcations of simple period-1 fixed-points are discussed. Period-1 and period-2 bifurcation trees with global stability are presented for forward and backward quadratic discrete systems. The period-2 appearing and period-doubling renormalizations of quadratic discrete systems are discussed, and the period-n appearing and period-doublization in quadratic discrete systems are presented. Similarly, backward period-2 appearing, backward period-doubling renormalization, and backward period-n appearing and period-doublization are also discussed. The forward and backward period-n bifurcations are presented, and the corresponding bifurcation dynamics can be determined as well.

In this Chapter, as in Luo (2019, 2020) one-dimensional and two-dimensional nonlinear discrete systems are discussed from the local theory of stability and bifurcations in Chap. 1. The linear discrete systems are discussed first, and the higher order singularity and stability for one-dimensional and two-dimensional nonlinear discrete systems are presented. A few special cases in 1-dimensional maps are discussed for a better understanding of the local theory.

In this paper, infinite homoclinic orbits existing in the Lorenz system are analytically presented. Such homoclinic orbits are induced by unstable periodic orbits on bifurcation trees through period-doubling cascades. Each unstable periodic orbit ends at its corresponding homoclinic orbit. Traditional computational methods cannot obtain homoclinic orbits from the corresponding unstable periodic orbits. This is because unstable periodic orbits in the Lorenz system cannot be achieved in numerical simulations. Herein, the stable and unstable periodic motions to chaos on the period-doubling cascaded bifurcation trees are determined through a discrete mapping method. The corresponding homoclinic orbits induced by the unstable periodic orbits are predicted analytically. A period-doubling bifurcation tree of period-1, period-2, and period-4 motions are generated as an example. The homoclinic orbits relative to unstable period-1, period-2, and period-4 motions are determined. Illustrations of homoclinic orbits and periodic orbits are given. This study presents how to determine infinite homoclinic orbits through unstable periodic orbits in three-dimensional or higher-dimensional nonlinear systems.

In this paper, the existence and bifurcations of periodic motions in a discontinuous dynamical system is studied through a discontinuous mechanical model. One can follow the study presented herein to investigate other discontinuous dynamical systems. Such a sampled discontinuous system consists of two subsystems on boundaries and three subsystems in subdomains. From the theory of discontinuous dynamical systems, switchability conditions of a flow at and on the boundaries are developed. From such switchability conditions, grazing motions of a flow at boundaries are discussed, and sliding motions of a flow on boundaries are presented. Based on the motions in each domain and on each boundary, generic mappings are introduced. Using the generic mappings, mapping structures for specific periodic motions are developed. Based on the grazing conditions and appearance and vanishing conditions of sliding motions, parametric dynamics of the existences of the specific periodic motions are presented. In addition, the traditional saddle-node bifurcation, Neimark bifurcations and period-doubling bifurcation are used for parametric dynamics of periodic motions. Bifurcation trees of periodic motions varying with a system parameter are presented first, and phase trajectories of periodic motions are illustrated. The [Formula: see text]-functions are presented for the illustration of the motion switchability at the boundaries and sliding motions on the boundaries. Codimension-2 parametric dynamics of periodic motions are studied and how to develop the 2D parametric maps for specific periodic motions are presented. In the end, periodic motions with grazing are illustrated.

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.

In this paper, complex symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, hardening Duffing oscillator are predicted through an implicit discrete map. The implicit discrete map is obtained by the discretization of a second-order differential equation of the time-delayed Duffing oscillator. Using the theory of nonlinear discrete systems, period-1 motions in the time-delayed Duffing oscillator are determined analytically from the fixed points of the mapping structures of discrete nodes under the computational accuracy of $$10^{-10}$$ , and the corresponding stability and bifurcation of period-1 motions are determined by eigenvalue analysis. From the discrete nodes of period-1 motions, the finite discrete Fourier series is employed to determine the frequency–amplitude characteristics of period-1 motions in the time-delayed, Duffing oscillator. The stable and unstable, symmetric and asymmetric period-1 motions are presented, and the corresponding stability and bifurcations are illustrated clearly. Presented are the quantity levels of harmonic amplitudes which indicate the accuracy of the semi-analytical solutions of period-1 motions in such a time-delayed oscillator. From the analytical prediction, numerical simulations of complex period-1 motions in the time-delayed, hardening Duffing oscillator are completed. The amplitude spectrums of period-1 motions in the time-delayed Duffing oscillator are given, and the approximate, analytical expressions of period-1 motions can be obtained. For small excitation frequency, discrete time step size should be reduced to keep the same computational accuracy of discrete nodes. This method can be applied to the time-varying time-delayed nonlinear dynamical systems and other nonlinear dynamical systems.

In this paper, the existence of periodic motions of a discontinuous delayed system with a hyperbolic switching boundary is investigated. From the delay-related [Formula: see text]-function, the crossing, sliding and grazing conditions of a flow to the switching boundary are first developed. For this time-delayed discontinuous dynamical system, there are 17 classes of generic mappings in phase plane and 66 types of local mappings in a delay duration. The generic mappings are determined by subsystems in three domains and two switching boundaries. Periodic motions in such a delay discontinuous system are constructed and predicted analytically from specific mapping structures. Three examples are given for the illustration of periodic motions with or without sliding motion on the switching boundary. This paper shows how to develop switchability conditions of motions at the switching boundary in the time-delayed discontinuous systems and how to construct the specific periodic solutions for the time-delayed discontinuous systems. This study can help us understand complex dynamics in time-delayed discontinuous dynamical systems, and one can use such analysis to control the time-delayed discontinuous dynamical systems.

In this paper, an independent bifurcation tree of period-2 motions to chaos coexisting with period-1 motions in a periodically driven van der Pol–Duffing oscillator is presented semi-analytically. Symmetric and asymmetric period-1 motions without period-doubling are obtained first, and a bifurcation tree of independent period-2 to period-8 motions is presented. The bifurcations and stability of the corresponding periodic motions on the bifurcation tree are determined through eigenvalue analysis. The symmetry breaks of symmetric period-1 motions is determined by the saddle-node bifurcations, and the appearance of the independent bifurcation tree of period-2 motions to chaos is also due to the saddle-node bifurcations. Period-doubling cascaded scenario of period-2 to period-8 motions are predicted analytically, and unstable periodic motions are also obtained. Numerical simulations are performed to illustrate motion complexity in such a van der Pol–Duffing oscillator. Such nonlinear systems can be applied in nonlinear circuit design and fluid-induced oscillations.

In this chapter, the global stability and bifurcations of equilibriums in the (2m)th-degree polynomial system are discussed for a better understanding of the complexity of bifurcations and stability of equilibriums. The appearing and switching bifurcations for simple equilibriums are presented, and the appearing and switching bifurcations for higher order equilibriums are discussed as well. The parallel appearing bifurcations, spraying-appearing bifurcations, and sprinkler-spraying-appearing bifurcations for simple and higher order equilibriums are presented. The antenna-switching bifurcations for simple and higher order equilibriums are discussed and the parallel straw-bundle-switching bifurcations and flower-bundle-switching bifurcations for simple and higher order equilibriums are presented as well.

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 paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

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, 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, 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.

In this paper, the global sequential scenario of bifurcation trees of periodic motions to chaos is studied for a first-order, time-delayed, nonlinear dynamical system with periodic excitation. The periodic motions of such a first-order time-delayed system is obtained semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential scenario of bifurcation trees is given by [Formula: see text] where [Formula: see text] is a global bifurcation tree of an asymmetric period-[Formula: see text] motion to chaos, and [Formula: see text] is a global bifurcation tree of a symmetric period-[Formula: see text] motion to chaos. Each bifurcation tree of a specific periodic motion to chaos is presented in detail. Numerical simulations of periodic motions are performed from analytical predictions. From finite Fourier series, harmonic amplitudes and phases for periodic motions are obtained for frequency analysis. Through this study, the rich dynamics of the first-order, time-delayed, nonlinear dynamical system is presented.

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.

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.

In this chapter, the concepts of nonlinear dynamical systems will be introduced. The local theory of nonlinear dynamical systems will be briefly discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system at equilibrium will be discussed. The higher order singularity and stability for nonlinear systems on the specific eigenvectors will be developed.

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.

In this chapter, the basic iterative solutions for memorized linear discrete systems will be presented. The iterative solutions of the memorized linear discrete systems with distinct and repeated eigenvalues will be discussed. The stability of memorized linear discrete systems will be discussed from the oscillatory, monotonic, and spiral convergence and divergence. The invariant, flip, and circular critical boundaries of the stability on the direction of the specific eigenvector will be classified. The 1-dimensional, memorized, linear discrete systems will be discussed.

In this chapter, the discretization of time-delay continuous systems will be presented through time-delay and nontime-delay discrete nodes. The explicit and implicit discrete maps will be discussed for numerical predictions of time-delayed continuous systems. Basic discrete schemes are presented which include forward and backward Euler methods, midpoint and trapezoidal rule method. The second-order Runge–Kutta method is discussed first, and then the explicit Runge–Kutta methods of the third and fourth orders are presented.

In this chapter, basic concepts of memorized nonlinear discrete systems will be presented. The local theory of stability and bifurcation for memorized nonlinear discrete systems will be discussed.

In this chapter, from Luo (2014), period-m flows to quasi-periodic flows in time-delayed nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic flows in autonomous time-delayed systems will be discussed, and the analytical solutions of quasi-periodic flows in periodically forced, time-delayed nonlinear dynamical systems will be presented.

In this chapter, the basic theory of nonlinear time-delay dynamical systems will be introduced. Local theory and bifurcation theory for nonlinear time-delay dynamical systems will be discussed. The stability switching and bifurcation on specific eigenvectors of the linearized, time-delayed system at equilibrium will be discussed. The higher-order singularity and stability for nonlinear time-delay systems on the specific eigenvectors will be developed. The lower dimensional, nonlinear, time-delay systems will be discussed for a better understanding of the stability and bifurcation theory.

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.

In this paper, the partial and full chaotic synchronizations of two nonlinear gyroscope systems with/without noise are investigated. From analytical conditions for synchronization and non-synchronization of two gyroscope systems, the parameter characteristic study is completed for a better understanding of the synchronization dynamics of two gyroscope dynamical systems. The boundaries of the parameter map for synchronization are determined by the onset and vanishing conditions of synchronization. The simple feedback control can make the noised gyroscope system synchronizing with chaotic behaviors of the expected gyroscope system. The methodology presented in this paper is different from other techniques for synchronization. The partial synchronization is an important phenomenon to be observed in engineering applications.

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).