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Chaos, Bifurcations, and Their Control 1999, 2015, 2018 CHAOS, BIFURCATIONS, AND THEIR In the discrete-time setting, a nonlinear dynamical sys- CONTROL tem is described by either a difference equation, ( ) ,, , +1 = ; =0 1 … (2) 1. NONLINEAR DYNAMICS or a map, ( ) Unlike linear systems, many nonlinear dynamical systems → ,, , do not show orderly, regular, and long-term predictable ∶ ; =0 1 … (3) responses to simple inputs. Instead, they display complex, where notation is similarly defined. Repeatedly iterat- random-like, seemingly irregular, yet well-defined output ing the discrete map backward yields behaviors. Such dynamical phenomenon is known as chaos ( ) ( ) ( ) today. ⋯ = −1 = ( −2) = = 0 The term chaos, originated from the Greek word , was designated “the primeval emptiness of the universe where the map can also be replaced by a function ,ifthe before things came into being of the abyss of Tartarus, the system is given via a difference equation, leading to underworld. … In the later cosmologies, Chaos generally ( ) ( ) ◦⋯◦ designated the original state of things, however conceived. = ⏟⏟⏟ 0 = 0 The modern meaning of the word is derived from Ovid, times who saw Chaos as the original disordered and formless ◦ mass, from which the maker of the Cosmos produced the where “ ” denotes composition operation of functions or ordered universe” (1). There also is an interpretation of maps. chaos in ancient Chinese literature, which refers to the Dynamical system (eq. 1)issaidtobenonautonomous spirit existing in the center of the universe (2).Inmod- when the time variable appears separately in the system ern scientific terminology, chaos has a fairly precise but function (e.g. a system with an external time-varying rather complicated definition by means of the dynamics of force input); otherwise, it is said to be autonomous and is a generally nonlinear system. For example, in theoretical expressed as ( ) physics, “chaos is a type of moderated randomness that, ̇ = ; ,∈( , ∞) (4) unlike true randomness, contains complex patterns that 0 are mostly unknown” (3). Bifurcation, as a twin of chaos, is another prominent 1.2. Classification of Equilibria phenomenon of nonlinear dynamical systems: Quantita- For illustration, consider a general two-dimensional au- tive change of system parameters leads to qualitative tonomous system: change of system properties such as the number and the stability of system equilibria. Typical bifurcations include ⎧ ̇ = (,) ⎪ transcritical, saddle-node, pitchfork, hysteresis, and Hopf ⎨ (5) bifurcations. In particular, period-doubling bifurcation is ⎪ ⎩ ̇ = (,) a route to chaos. To introduce the concepts of chaos and bifurcations as well as their control (4,5,6,7,8,9),somepre- , with given initial conditions ( 0 0),where and are liminaries on nonlinear dynamical systems are in order. two smooth nonlinear functions that together describe the vector field of the system. 1.1. Nonlinear Dynamical Systems The path traveled by a solution of system (eq. 5), start- ing from the initial state , ,isasolution trajectory,or A nonlinear system refers to a set of nonlinear equations, ( 0 0) , which can be algebraic, difference, differential, integral, orbit, of the system, and is sometimes denoted ( 0 0). functional, or abstract operator equations, even a certain For autonomous systems, two different orbits will never combination of these. A nonlinear system is used to de- cross each other (i.e., never intersect) on the x–y plane. scribe a physical device or process that otherwise cannot This – coordinate plane is called the (generalized) phase be well defined by a set of linear equations of any kind. plane (phase space, in the higher dimensional case). The Dynamical system is used as a synonym of a mathematical orbit family of a general autonomous system, correspond- or physical system, in which the output behavior evolves ing to all possible initial conditions, is called solution flow with time and/or with other varying system parameters in the phase space. (10). Equilibria,orfixed points,ofsystem(eq.5), if they exist, In general, a continuous-time dynamical system is de- are the solutions of two homogeneous equations: scribed by a differential equation, (,)=0 and (,)=0 ( ) ̇ , , ,̄ ̄ = ; ∈[0 ∞) (1) An equilibrium is denoted ( ).Itisstable if all the nearby orbits of the system, starting from any initial con- where = () is the state of the system, is a vector of ditions, approach it; it is unstable, if the nearby orbits are variable system parameters,and and are continuous moving away from it. Equilibria can be classified, accord- or smooth (differentiable) nonlinear functions of compara- ing to their stabilities, as stable or unstable node or focus, ble dimensions, which have an explicit formulation for a and saddle point or center, as summarized in Figure 1.The specified physical system in interest. type of equilibria is determined by the eigenvalues, 1,2, J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering.Copyright© 2018 John Wiley & Sons, Inc. DOI: 10.1002/047134608X.W1007.pub2 2 Chaos, Bifurcations, and Their Control y yy xx x Stable node Unstable node Stable focus yy y x x x Unstable focus Saddle Center Figure 1. Classification of two-dimensional equilibria: stabilities are determined by Jacobian eigenvalues. [ ] → solution, as ∞, is called the steady state of the solution, of the system Jacobian, ∶= ,with ∶= ∕, while the solution trajectory between its initial state and the steady state is the transient state. ∶= ∕, and so on, all evaluated at (,̄ ̄).Ifthetwo For a given dynamical system, a point in the state Jacobian eigenvalues have real parts ℜ{ } ≠ 0, the equi- 1,2 space is an -limit point of the system state orbit () if, for librium (,̄ ̄), at which the linearization was taken, is said every open neighborhood of , the trajectory of () will to be hyperbolic. enter at a (large enough) value of . Consequently, () will repeatedly enter infinitely many times, as → ∞. Theorem 1 (Grobman–Hartman) If (,̄ ̄) is a hyper- The set of all such -limit points of () is called the - bolic equilibrium of the nonlinear dynamical system (eq. limit set of (), and is denoted Ω.An-limit set of () 5), then the dynamical behavior of the nonlinear system is is attracting, if there exists an open neighborhood of Ω qualitatively the same as (topologically equivalent to) that [ ] [ ] such that whenever system orbit enters , it will approach ̇ → of its linearized system, = , in a neighborhood Ω as ∞.Thebasin of attraction of an attracting point ̇ is the union of all such open neighborhoods. An -limit set of the equilibrium (,̄ ̄). is repelling, if the system orbits always move away from it. This theorem guarantees that for the hyperbolic case, For a given map and a given{ initial} state 0,an -limit one can study the linearized system instead of the original → set is obtained from the orbit ( 0) as ∞.This - nonlinear system, with regard to the local dynamical be- limit set Ω is an invariant set of the map, in the sense havior of the system within a (small) neighborhood of the that (Ω ) ⊆ Ω . Thus, -limit sets include equilibria and equilibrium (,̄ ̄). In general, “dynamical behavior” refers periodic orbits. to such nonlinear phenomena as stabilities and bifurca- An attractor is an -limit set having the property that tions, chaos and attractors, equilibria and limit cycles, all orbits nearby have it as their -limit sets. Thus, a col- and so on. In other words, there exist some homeomorphic lection of isolated attracting points is not an attractor. maps that transform the orbits of the nonlinear system into orbits of its linearized system in a (small) neighbor- hood of the equilibrium. Here, a homeomorphic map (or a 1.4. Periodic Orbits and Limit Cycles homeomorphism) is a continuous map whose inverse ex- A solution orbit () of the nonlinear dynamical system ists and is also continuous. However, in the nonhyperbolic (eq. 1)isaperiodic solution if it satisfies ( + )=() case, the situation is much more complicated, where such p for some constant > 0. The minimum value of such is local topological equivalence does not hold in general. p p called the (fundamental) period of the periodic solution, while the solution is said to be p periodic. 1.3. Limit Sets and Attractors A limit cycle of a dynamical system is a periodic solution The most basic problem in studying the general nonlinear of the system that corresponds to a closed orbit on the dynamical system (eq. 1) is to understand and/or to analyze phase plane and possesses certain attracting (or repelling) the system solutions. The asymptotic behavior of a system properties. Figure 2 shows some typical limit cycles: (a) an Chaos, Bifurcations, and Their Control 3 (a) (b) (c) (d) (e) (f) Figure 2. Periodic orbits and limit cycles. inner limit cycle, (b) an outer limit cycle,(c)astable limit cycle,(d)anunstable limit cycle, and (e and f), saddle limit cycles. 1.5. Poincar´eMaps P (x) Assume that the general -dimensional nonlinear au- x tonomous system (eq. 4)hasa p periodic orbit, Γ,and Γ let ∗ be a point on the periodic orbit and Σ be an ( −1)- x* dimensional hyperplane transversal to Γ at ∗,asshown in Figure 3.SinceΓ is p periodic, the orbit starting from Σ ∗ ∗ will return to in time p. Any orbit starting from a ∗ point in a small neighborhood of on Σ will return Figure 3. Illustration of the Poincare´ map and cross section. and hit Γ at a point, denoted (), in the vicinity of ∗. Therefore, a map ∶ → can be uniquely defined by ∗ ∗ Σ, along with the solution flow of the autonomous system. Suppose that Σs( ) and Σu( ) are cross sections of the This map is called the Poincaremap´ associated with the stable and unstable manifolds of (), respectively, which system and the cross section Σ. For different choices of the intersect at ∗. This intersection always includes one con- ∗ cross section Σ,Poincare´ maps are similarly defined.
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