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... continuous time dynamical systems is of fundamental importance to get insights in the dynamical structure of a system. The distinction among regular and chaotic orbits can be easily made in a dissipative case whereas for conservative systems it is usually an hard task especially when we are dealing with many degrees of freedom or when the phase space is splitted into chaotic regions and regular islands. A large number of tools known as indicators of stability have been developed to accomplish this task: Lyapunov Characteristic Exponents (LCEs) [Wolf et al., 1985], [Rosenstein et al., 1993], [Skokos, 2010] and the indicators related to the Return Time Statistics [Kac, 1934], [Gao, 1999], [Hu et al., 2004] , [Buric et al., 2005] have been used from a long time as efficient indicators. Nevertheless, in 1 July 25, 2011 17:45 GEV˙indicator 2 the recent past, the need for computing stability properties with faster algorithms and for systems with many degrees of freedom resulted in a renewed interest in the technique and different dynamical indicators have been introduced. The Smaller Alignment Index (SALI) described in Skokos et al. [2002] and Skokos et al. [2004] ,the Generalized Alignment Index (...

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...o recent references for computation in continuous time systems and discrete time can be found in [23] and [24] respectively. A recent survey of the theory of Lyapunov exponents is also available from =-=[25]-=-. A drawback of LEs is that two dynamical systems having the same LEs can behave in a very different way [26]. In particular, if a chaotic system in Rn is characterised by LEs λ1, λ2, ..., λn, then a ...

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...e mean measuring LCEs, topological entropy and correlation dimension of chaotic orbit. 3.1 Lyapunov Characteristic Exponents The motion of the attractor exhibits sensitive dependence on initial conditions. This means that the two trajectories starting together with nearby locations will rapidly diverge from each other and therefore have totally different futures. The practical implication is that long-term prediction becomes impossible in a system Complexities in age structured predator-prey system 5943 where small uncertainties are amplified enormously fast. Lyapunov characteristic exponents [6, 7, 8, 9, 10, 11, 12, 13], is very effective tool for identification of regular and chaotic motions since this measures the degree of sensitivity to initial condition in a system. Plot of Lyapunov number and LCEs have been drawn for regular and chaotic motions of the considered map (1). Both are drawn at r = 1.5, b = 0.95, s = 0.2 and = 1.45 with initial point (1.1, 0.1, 0.1) given in Figure 3. Clearly this is the case of regularity of the system as at these points LCE is coming to be negative and Lyapunov number is less than 1. Figure 3(a): First row: plots of Lyapunov numbers with initial point (1.1, 0.1, 0.1) and...