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## Complexity Studies in Some Piecewise Continuous Dynamical Systems

### Citations

144 |
Lyupanov Characteristics Exponents for Smooth Dynamical Systems and For Hamiltonian Systems; A Method for Computing All of Them”, Meccanica V14-15
- Benettin, Galgani, et al.
- 1980
(Show Context)
Citation Context ...OF COMPLEXITIES For any complex system, some of the measurements which justify complexity are as follows: 2.1 Lyapunov Characteristic Exponents (LCEs) The motion be chaotic if the system exhibits sensitive dependence on initial conditions. That is two trajectories starting together with nearby positions, (initial conditions), will rapidly diverge from each other and have totally dissimilar features. The long-term prediction becomes impossible as the small qualms are amplified enormously fast. Lyapunov characteristic Complexity Studies in Some Piecewise Continuous Dynamical Systems 35 exponents[1,7, 8, 9, 10,22], is very effective tool for identification of regular and chaotic motions since it measures the degree of sensitivity to initial condition in a system. Actually, Lyapunov exponents (λ) provide measure the exponential divergence of orbits originating nearby. If λ > 0, then it implies the system is evolving chaotically, (or chaos is observed), and if λ < 0 then it implies the evolution is regular, (or regularity or ordered motion). For a smooth one dimensional map f and x0 an initial point the Lyapunov Exponent be defined by λ(x0) = lim log|f (x ) |log|f (x ) |log|f (x ) |.... log|f (xk 0 1 2 k... |

41 |
Computing the Lyapunov Spectrum of a Dynamical System from an Observed Time Series, Physical Review A 43(6)
- Brown, Bryant, et al.
- 1991
(Show Context)
Citation Context ...OF COMPLEXITIES For any complex system, some of the measurements which justify complexity are as follows: 2.1 Lyapunov Characteristic Exponents (LCEs) The motion be chaotic if the system exhibits sensitive dependence on initial conditions. That is two trajectories starting together with nearby positions, (initial conditions), will rapidly diverge from each other and have totally dissimilar features. The long-term prediction becomes impossible as the small qualms are amplified enormously fast. Lyapunov characteristic Complexity Studies in Some Piecewise Continuous Dynamical Systems 35 exponents[1,7, 8, 9, 10,22], is very effective tool for identification of regular and chaotic motions since it measures the degree of sensitivity to initial condition in a system. Actually, Lyapunov exponents (λ) provide measure the exponential divergence of orbits originating nearby. If λ > 0, then it implies the system is evolving chaotically, (or chaos is observed), and if λ < 0 then it implies the evolution is regular, (or regularity or ordered motion). For a smooth one dimensional map f and x0 an initial point the Lyapunov Exponent be defined by λ(x0) = lim log|f (x ) |log|f (x ) |log|f (x ) |.... log|f (xk 0 1 2 k... |

37 |
Dynamic complexity in predator–prey models framed in difference equations
- Beddington, Free, et al.
(Show Context)
Citation Context ...such systems vary in certain manner, we observe: Instability, bistability, bifurcations, chaos etc. Linearization of a nonlinear model representing a real system without proper justification often leads to complete wrong results. We can prove this fact for many nonlinear problems solved by linearizing the model e.g. damped and forced pendulum evolving chaotically, problems of viscous fluid flow etc. A complex system can be viewed as a system composed of many components which may interact with each other. Complexity of different type has been explained extendedly through some important articles[6,13,20,21] A complex system exhibits some (and possibly all) of the following characteristics: • Some degree of spontaneous order (spontaneous order is the spontaneous emergence of order out of seeming chaos). • Robustness of the order (robustness is the ability of a computer system to cope with errors during execution). • Sensitivity to initial condition (chaos). A System is chaotic if it possesses an strange attractor. • Numerosity (broken symmetry and the nature of the hierarchical structure of science). Actual motivation for this study is to see how complexities are arising for any piece wise contin... |

31 | Border collision bifurcations in two-dimensional piecewise smooth maps. Physical Review E,
- Banerjee, Grebogi
- 1999
(Show Context)
Citation Context ...cation diagram of one dimensional piecewise continuous map [11] The above bifurcation diagram clearly indicates period doubling & chaos adding phenomena. Chitkara, AK Kumra, N Saha,, LM 38 Plots of topological entropy & correlation dimension are shown in Figure 4: topological entropy for a=0.5 and –0.5≤ b ≤0.5 and correlation dimension curve for a=0.53 and b= –0.3. Using linear fit to the correlation curve data one obtains y x= −0 52914 0 256885. . (7) So, the correlation Dimension be obtained approximately as 0.53. 4.2 Dynamics of Lozi Map Lozi map[14], is the subject of many recent articles [3, 4, 5,18] focused on its various properties. Quadratic term of Hénon map is replaced with a piecewise linear contribution in the former which produces some interesting chaotic attractors. The equations governing Lozi system can be written from Henon system Figure 3(a): Bifurcation diagram for – 0.7 ≤ b ≤ 0.4 (left figure) and 3(b) LCE plot for – 0.5 ≤ b ≤ 0.5(right figure). Figure 4: Plots of topological entropy (left) and correlation curve (right). Topological entropy plot be obtained for a = 0.5 and – 0.5 ≤ b ≤ 0.5 and correlation curve be obtained for a = 0.53 and b = – 0.3. Complexity Studies in So... |

29 |
Un attracteur etrange? du type attracteur de Henon.,”
- Lozi
- 1978
(Show Context)
Citation Context ...es to cycle one, Figure 3(a). Figure 2: Bifurcation diagram of one dimensional piecewise continuous map [11] The above bifurcation diagram clearly indicates period doubling & chaos adding phenomena. Chitkara, AK Kumra, N Saha,, LM 38 Plots of topological entropy & correlation dimension are shown in Figure 4: topological entropy for a=0.5 and –0.5≤ b ≤0.5 and correlation dimension curve for a=0.53 and b= –0.3. Using linear fit to the correlation curve data one obtains y x= −0 52914 0 256885. . (7) So, the correlation Dimension be obtained approximately as 0.53. 4.2 Dynamics of Lozi Map Lozi map[14], is the subject of many recent articles [3, 4, 5,18] focused on its various properties. Quadratic term of Hénon map is replaced with a piecewise linear contribution in the former which produces some interesting chaotic attractors. The equations governing Lozi system can be written from Henon system Figure 3(a): Bifurcation diagram for – 0.7 ≤ b ≤ 0.4 (left figure) and 3(b) LCE plot for – 0.5 ≤ b ≤ 0.5(right figure). Figure 4: Plots of topological entropy (left) and correlation curve (right). Topological entropy plot be obtained for a = 0.5 and – 0.5 ≤ b ≤ 0.5 and correlation curve be obtained... |

29 | The Lyapunov characteristic exponents and their computation, - Skokos - 2009 |

28 |
Introduction to Discrete Dynamical Systems and Chaos, Discrete Mathematics and Optimization,
- Martelli
- 1999
(Show Context)
Citation Context ... 1 [21] defines the probability of a given reading as p A ii i= =µ( ), , ,1 2 …,N. Then the entropy of the partition be given by H(p) p logpi i i 0 N = − = ∑ (2) 2.3 Correlation Dimension Correlation dimension provides the dimensionality of the evolving system,[12]. It is a kind of fractal dimension and its numerical value is always non-integer. Being one of the characteristic invariants of nonlinear system dynamics, the correlation dimension actually gives a measure of complexity for the underlying attractor of the system. To determine correlation dimension one has to use statistical method, [16]. Chitkara, AK Kumra, N Saha,, LM 36 Consider an orbit O(x1) = {x1, x2, x3, x4, . . ….}, of a map f: U → U, where U is an open bounded set in Rn. To compute correlation dimension of O(x1), for a given positive real number r, we form the correlation integral, C(r) = lim 1 n(n 1) H(r x x ) ,i, j = 1, 2, 3, . . n i j i j n →∞ ≠− − −∑ . , n. (3) where H (x) = 0,x 0 1,x 0 < ≥ is the unit-step function, (Heaviside function). The summation indicates the number of pairs of vectors closer to r when 1 ≤ i, j ≤ n and i ≠ j. C(r) measures the density of pair of distinct vectors xi and xj that are cl... |

21 |
Local Lyapunov exponents computed from observed data.
- HD, Brown, et al.
- 1992
(Show Context)
Citation Context ...OF COMPLEXITIES For any complex system, some of the measurements which justify complexity are as follows: 2.1 Lyapunov Characteristic Exponents (LCEs) The motion be chaotic if the system exhibits sensitive dependence on initial conditions. That is two trajectories starting together with nearby positions, (initial conditions), will rapidly diverge from each other and have totally dissimilar features. The long-term prediction becomes impossible as the small qualms are amplified enormously fast. Lyapunov characteristic Complexity Studies in Some Piecewise Continuous Dynamical Systems 35 exponents[1,7, 8, 9, 10,22], is very effective tool for identification of regular and chaotic motions since it measures the degree of sensitivity to initial condition in a system. Actually, Lyapunov exponents (λ) provide measure the exponential divergence of orbits originating nearby. If λ > 0, then it implies the system is evolving chaotically, (or chaos is observed), and if λ < 0 then it implies the evolution is regular, (or regularity or ordered motion). For a smooth one dimensional map f and x0 an initial point the Lyapunov Exponent be defined by λ(x0) = lim log|f (x ) |log|f (x ) |log|f (x ) |.... log|f (xk 0 1 2 k... |

21 |
Lyapunov exponents from observed time series,
- Bryant, Brown, et al.
- 1990
(Show Context)
Citation Context |

4 |
Characterization of strange attractors. Physical review letters 50, 346,
- Grassberger, Procaccia
- 1983
(Show Context)
Citation Context ... is. A topological entropy measures the exponential growth rate of the number of distinguishable orbits as time advances in the system [2].However positivity of its value does not justify the system be chaotic. Consider a finite partition of a state space X denoted by P = { A1, A2, A3,. . . ., AN}. A measure μ on X with total measure μ = 1 [21] defines the probability of a given reading as p A ii i= =µ( ), , ,1 2 …,N. Then the entropy of the partition be given by H(p) p logpi i i 0 N = − = ∑ (2) 2.3 Correlation Dimension Correlation dimension provides the dimensionality of the evolving system,[12]. It is a kind of fractal dimension and its numerical value is always non-integer. Being one of the characteristic invariants of nonlinear system dynamics, the correlation dimension actually gives a measure of complexity for the underlying attractor of the system. To determine correlation dimension one has to use statistical method, [16]. Chitkara, AK Kumra, N Saha,, LM 36 Consider an orbit O(x1) = {x1, x2, x3, x4, . . ….}, of a map f: U → U, where U is an open bounded set in Rn. To compute correlation dimension of O(x1), for a given positive real number r, we form the correlation integral, C(... |

3 | A discrete predator–prey system with age-structure for predator and natural barriers for prey,
- Tang, Chen
- 2001
(Show Context)
Citation Context ...such systems vary in certain manner, we observe: Instability, bistability, bifurcations, chaos etc. Linearization of a nonlinear model representing a real system without proper justification often leads to complete wrong results. We can prove this fact for many nonlinear problems solved by linearizing the model e.g. damped and forced pendulum evolving chaotically, problems of viscous fluid flow etc. A complex system can be viewed as a system composed of many components which may interact with each other. Complexity of different type has been explained extendedly through some important articles[6,13,20,21] A complex system exhibits some (and possibly all) of the following characteristics: • Some degree of spontaneous order (spontaneous order is the spontaneous emergence of order out of seeming chaos). • Robustness of the order (robustness is the ability of a computer system to cope with errors during execution). • Sensitivity to initial condition (chaos). A System is chaotic if it possesses an strange attractor. • Numerosity (broken symmetry and the nature of the hierarchical structure of science). Actual motivation for this study is to see how complexities are arising for any piece wise contin... |

3 |
Comparison between covariant and orthogonal Lyapunov vectors,
- Yang, Radons
- 2010
(Show Context)
Citation Context |

2 |
Dynamics of a Hénon–Lozi-type map.
- Aziz-Alaoui, Robert, et al.
- 2001
(Show Context)
Citation Context ...cation diagram of one dimensional piecewise continuous map [11] The above bifurcation diagram clearly indicates period doubling & chaos adding phenomena. Chitkara, AK Kumra, N Saha,, LM 38 Plots of topological entropy & correlation dimension are shown in Figure 4: topological entropy for a=0.5 and –0.5≤ b ≤0.5 and correlation dimension curve for a=0.53 and b= –0.3. Using linear fit to the correlation curve data one obtains y x= −0 52914 0 256885. . (7) So, the correlation Dimension be obtained approximately as 0.53. 4.2 Dynamics of Lozi Map Lozi map[14], is the subject of many recent articles [3, 4, 5,18] focused on its various properties. Quadratic term of Hénon map is replaced with a piecewise linear contribution in the former which produces some interesting chaotic attractors. The equations governing Lozi system can be written from Henon system Figure 3(a): Bifurcation diagram for – 0.7 ≤ b ≤ 0.4 (left figure) and 3(b) LCE plot for – 0.5 ≤ b ≤ 0.5(right figure). Figure 4: Plots of topological entropy (left) and correlation curve (right). Topological entropy plot be obtained for a = 0.5 and – 0.5 ≤ b ≤ 0.5 and correlation curve be obtained for a = 0.53 and b = – 0.3. Complexity Studies in So... |

2 |
The basin of attraction of Lozi mappings.
- Baptista, Severino, et al.
- 2009
(Show Context)
Citation Context ...cation diagram of one dimensional piecewise continuous map [11] The above bifurcation diagram clearly indicates period doubling & chaos adding phenomena. Chitkara, AK Kumra, N Saha,, LM 38 Plots of topological entropy & correlation dimension are shown in Figure 4: topological entropy for a=0.5 and –0.5≤ b ≤0.5 and correlation dimension curve for a=0.53 and b= –0.3. Using linear fit to the correlation curve data one obtains y x= −0 52914 0 256885. . (7) So, the correlation Dimension be obtained approximately as 0.53. 4.2 Dynamics of Lozi Map Lozi map[14], is the subject of many recent articles [3, 4, 5,18] focused on its various properties. Quadratic term of Hénon map is replaced with a piecewise linear contribution in the former which produces some interesting chaotic attractors. The equations governing Lozi system can be written from Henon system Figure 3(a): Bifurcation diagram for – 0.7 ≤ b ≤ 0.4 (left figure) and 3(b) LCE plot for – 0.5 ≤ b ≤ 0.5(right figure). Figure 4: Plots of topological entropy (left) and correlation curve (right). Topological entropy plot be obtained for a = 0.5 and – 0.5 ≤ b ≤ 0.5 and correlation curve be obtained for a = 0.53 and b = – 0.3. Complexity Studies in So... |

2 |
Complex non-unique dynamics in simple ecological interactions.
- Kaitala, Heino
- 1996
(Show Context)
Citation Context ...such systems vary in certain manner, we observe: Instability, bistability, bifurcations, chaos etc. Linearization of a nonlinear model representing a real system without proper justification often leads to complete wrong results. We can prove this fact for many nonlinear problems solved by linearizing the model e.g. damped and forced pendulum evolving chaotically, problems of viscous fluid flow etc. A complex system can be viewed as a system composed of many components which may interact with each other. Complexity of different type has been explained extendedly through some important articles[6,13,20,21] A complex system exhibits some (and possibly all) of the following characteristics: • Some degree of spontaneous order (spontaneous order is the spontaneous emergence of order out of seeming chaos). • Robustness of the order (robustness is the ability of a computer system to cope with errors during execution). • Sensitivity to initial condition (chaos). A System is chaotic if it possesses an strange attractor. • Numerosity (broken symmetry and the nature of the hierarchical structure of science). Actual motivation for this study is to see how complexities are arising for any piece wise contin... |

2 |
Experimental characterization of nonlinear systems:a real-time evaluation of the analogous Chua’s circuit behavior.
- Rocha, GL, et al.
- 2010
(Show Context)
Citation Context ...oximately; for a = 2.8, b = –1, one gets 4 fixed points given by P1*= (–0.0676, –0.4054), P2*= (0.2083, –0.2083), P3*= (0.4054, 0.0676) and P4*= (–1.25, 1.25) approximately. Figure 8: LCE plots for above two cases. Figure 9: (Continued). Complexity Studies in Some Piecewise Continuous Dynamical Systems 43 For the case when a = 1.7, b = 0.5, taking initial points close to P1* and P2*as points (0.4, 0.2) and (–0.8, –0.4), time series and phase plots are obtained and shown in Figure 10 below. Both the orbits obtained show chaotic motion with strange attractors. 4.3 Chua’s Circuit Chua’s map,[15],[17] ,is related to Chua’s electric circuit theory that exhibits classic chaos theory behavior invented by Japanese Prof. Leon O. Chua in 1983. This means roughly that it is a “nonperiodic oscillator”; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never Figure 9: Some attractors of Lozi system when b = –1 and different values of a such that a < 1 – b, but closer to unity, and with different initial conditions. Figure 10: Plots of Lyapunov exponents, (left figure), and corresponding strange attractors, (right figure). Chitkara, AK Kumra, N Saha,, LM 44 “repeats... |

2 |
Dynamic complexities in predator–prey ecosystem models with age-structure for predator, Chaos, Solitons and Fractals,
- Xiao, Cheng, et al.
- 2002
(Show Context)
Citation Context |

1 |
Topological entropy.
- RL, AG, et al.
- 1965
(Show Context)
Citation Context ...gularity or ordered motion). For a smooth one dimensional map f and x0 an initial point the Lyapunov Exponent be defined by λ(x0) = lim log|f (x ) |log|f (x ) |log|f (x ) |.... log|f (xk 0 1 2 k 1→∞ −′ + ′ + ′ + + ′ )|( ) = lim log|f (x )| k k k 0 k 1 →∞ = − ′∑ (1) where, x1, x2, . . . . , xk–1 , . ., are iterates of x0 under f. 2.2 Topological Entropy Topological entropy provides the measure of complexity. More topological entropy means more complex the system is. A topological entropy measures the exponential growth rate of the number of distinguishable orbits as time advances in the system [2].However positivity of its value does not justify the system be chaotic. Consider a finite partition of a state space X denoted by P = { A1, A2, A3,. . . ., AN}. A measure μ on X with total measure μ = 1 [21] defines the probability of a given reading as p A ii i= =µ( ), , ,1 2 …,N. Then the entropy of the partition be given by H(p) p logpi i i 0 N = − = ∑ (2) 2.3 Correlation Dimension Correlation dimension provides the dimensionality of the evolving system,[12]. It is a kind of fractal dimension and its numerical value is always non-integer. Being one of the characteristic invariants of nonli... |

1 |
Dynamical systems with applications using MapleTM.
- Lynch
- 2009
(Show Context)
Citation Context ... approximately; for a = 2.8, b = –1, one gets 4 fixed points given by P1*= (–0.0676, –0.4054), P2*= (0.2083, –0.2083), P3*= (0.4054, 0.0676) and P4*= (–1.25, 1.25) approximately. Figure 8: LCE plots for above two cases. Figure 9: (Continued). Complexity Studies in Some Piecewise Continuous Dynamical Systems 43 For the case when a = 1.7, b = 0.5, taking initial points close to P1* and P2*as points (0.4, 0.2) and (–0.8, –0.4), time series and phase plots are obtained and shown in Figure 10 below. Both the orbits obtained show chaotic motion with strange attractors. 4.3 Chua’s Circuit Chua’s map,[15],[17] ,is related to Chua’s electric circuit theory that exhibits classic chaos theory behavior invented by Japanese Prof. Leon O. Chua in 1983. This means roughly that it is a “nonperiodic oscillator”; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never Figure 9: Some attractors of Lozi system when b = –1 and different values of a such that a < 1 – b, but closer to unity, and with different initial conditions. Figure 10: Plots of Lyapunov exponents, (left figure), and corresponding strange attractors, (right figure). Chitkara, AK Kumra, N Saha,, LM 44 “re... |

1 |
Bifurcations in Lozi map.
- Ros, Botella, et al.
- 2011
(Show Context)
Citation Context |